New York State College of Agriculture At Cornell University Ithaca, N. Y. Library MA 29.K69""*" ""'"«»»/ Library 1^ K XI Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924013703438 Appendix A. VOL. I. Census of the Commonwealth of Australia. The Mathematical Theory of Population, of its Character and Fluctuations, and of the Factors which influence them, BEING AN Examination of the general scheme of Statistical Representation, with deductions of necessary formulae; the whole being applied to the data of the Australian Census of 191 1, and to the elucidation of Australian Population Statistics generally. BY G. H. KNIBBS, C.M.G.. F.S.S., F.R.A.S., etc., ^ Member of the International Institute of Statistics, Honorary Member of the Soci^t^ de Statistique of Paris, and of the American Statistical Association, etc., etc. COMMONWEALTH STATISTICIAN. Published under Instructions from the MINISTER OF STATE FOR HOME AND TERRITORIES, Melbourne. ■« By Authority: McCARRON, BIRD & CO., Printers, 479 Collins Street, Melbourne [C.S.— No. 312.] FOREWORD. The following monograpH on the Mathematical Theory of Population, in form an appendix to the Report on the AustraHan Census of 1911, is intended to serve a double purpose. It aims on the one hand at supply- ing the elements of a mathematical technique, such as are needed for the analysis of the various aspects of vital phenomena that come under statistical review, and, on the other, at interpreting material made available by the first Census of AustraHa which has been carried out upon uniform lines and' by a central authority. The earUer portion of the appendix has consequently been almost wholly devoted to the creation of the requisite technique. Later technical solutions are introduced only when required by way of application to any statistical analysis under immediate review. In the realm of official statistics there is an enormous amount of accumulated material, which, decade after decade, remains unanalysed and uninterpreted.- This is due to several things', viz., to the fact that routine tabulations largely occupy the energies of the staffs of statistical bureaux; to the fact that much of the mass of material itself is defective and its correction involves more time than is available ; and perhaps still more to the fact that appropriate schemes of mathematical analysis have as yet either not been developed, or are regarded as inapphcable. The present analyses and interpretations have yielded many results ' which, it is believed, mil be seen to be of value. They have brought into clearer rehef the necessity for recognising that the variation of any one statistical element affects all other statistical elements, so that the satis- factory reduction of " crude data" to a common system is by no means an easy undertaking, and the comparability of the statistic of two com- munities can never be rigorously exact in all particulars. It is fortunate, however, that practically exactitude means merely " a precision sufficient for any particular purpose in view." In substance this monograph consists of two elements, viz., (i.) a technical one, and (ii.) an interpretative one. Formulse essential for the purposes of interpretation have been deduced, and their use has been illustrated by appHcation to the data of the AustraHan Censuses, or to intercensal statistical data which, subject otherwise to considerable uncertainty, could be adjusted only by means of information derived from the Census. Thus results of immediate value are obtained simul- taneously with an exposition of the theory and technique of the subject. FOBEWORD. The various formulae developed have been carefully checked through- out, but it is too much to hope that among so many results error has been completely avoided. The author will, therefore, be grateful if any discoverer of errors or misprints will communicate with him. As a rule corrections to data have been pushed as far as seemed to be desirable ; theoretically it is often possible to push them even still farther. It is doubted, however, whether the precision of the data'would justify this. An example will illustrate the point. In determining the ratios which reveal the age of maximum fecundity, if the number of women at risk be taken as the total of the same age-group, the denominator wiU be too large and the derived ratio too small. Hence allowances must be made for the diminution of risk for prior cases of child-birth. But there is no well-defined time-limit at which these allowances should stop. In general, however, their apphcabihty becomes more questionable as they become smaller. A synopsis shews the general treatment of the subject, and an index, at the end of this appendix, makes reference thereto easy. Where it has been deemed necessary to coin technical expressions their derivation has been indicated. Finally it may be mentioned that many of the formulae developed will be found serviceable in other investigations in which statistical methods are called into requisition. G. H. KNIBBS. Commonwealth Bureau of Census and Statistics. Melbourne, March 1917. CORRIGENDA Page 3. — Under figures in footnote : after " small figures" read " in brackets." Page 4. — ^§ 4, line 8 : for " an" read " on." Page 7. — Line 3 ; for " acurately" read " accurately." Line 3, footnote, for " Gesellsohaftsehre" read " Gesellschaftalehre." Page 8. — Sub-heading (iv.). For " interpolation" read " interpolations." Line 7, last paragraph, insert " the" after " given." Page 40. — ^Line H from bottom, after log x, insert " and k being log k." Line 9 from bottom, for " fc," " 21c," " 3k," read " k," " 2k," " 3(c." Page 55.— ^Line 1, for " of a curve" read " of the curve." Page 68.— Formula (197(i), for (" 1 — " read " (i — ." Line 13, after " above" add " the numerical coefficients remaining, of course, the same." Page 72. — In formula (211), the y should follow the sign of integration. Page 81. — Line 4 from bottom, for (" n" read (" h." Page 104. — ^Line 5, for" difference" read" the differences." Line 27, for " the comparison of" read " comparisons among." Page 144. — Lines 10 and 11, for" section" and "sections," read"Part" and"Parts," and for XII., read XI. Page 163.— Line 4, for " M " read " M." Line 7, for " 2Mr + " read " 2Mr,." Page 213. — Line 3 from bottom, for " occupying " read " occurring." Page 233. — Line 4 of paragraph, for " in part of the" read " in part the." Page 240. — ^Line 4, § 8, add after " maternity," " each birth being regarded a case of maternity." Page, 242. — ^Throughout table read " births" for " maternity " Page 277.— Table LXXXVIII., in " Duration," for 251-160, read "251-260," and for" 251-170" read" 261-270." Page 306. — Add to end of paragraph : — " Twins produced from one ovum have been called ' univiteUins ' and those from two ova ' bivitelUns *." Page 307. — ^Line 3 from bottom, for " uniovulate" read " uniovular." Note. FormulsB 374, and 396 are omitted. SYNOPSIS. THE MATHEMATICAL THEORY OF POPULATION, OF ITS CHARACTERS AND FLUCTUATIONS, AND THE FACTORS WHICH INFLUENCE THEM. I. Introductory. 1. General 2. SignificEuioe of analysis 3. The nature of the problem 4. Necessity for the mathematical ex pression of the conditions of the problem 5. Conception applies equally to a popula- tion de/octo or a population de/iwe 6. Nature of population fluctuations 7. Changes in the constitutions of popula- tions 8. Organic adjustments of populations 9. Continuous and finite fluctuations 10. Curves required to represent various fluctuations and the solution of the same II. Various Types of Population Fluctuations. 1. Mathematical conception of rate of in- crease 2. Determination of a population for any instant when the rate is constant 3. Kelation of instantaneous rate to the ratio of increase for various periods 4. Determination of the mean population for any period ; rate constant 5. Error of the arithmetical mean ; rate constant . . 6. Empirical expression for any population fluctuation 7. Mean population for any period ; rate not constant . . 8. Change, with change of epoch, of the coefficients expressing rate 9. Error of the arithmetical mean ; rate not constant . . 10. Expression of the coefficients in the em- pirical formula for rate in terms of the constant rate 11. Investigation of rate is complete only when its variations are ascertained 12. Rate is a function of elements that varies with time 13. Factors which secularly influence the rate of increase . . 14. Variations which depend on natm-al re- sources, irrespective of human interven- tion 15. Variations of rate of long periods 16. Representation of periodic elements in non-periodic form 17. Influence of natural resoittces disclosed by advancing knowledge . . 18. Influences of resources dependent upon huinan intervention 19. Effects of migration 20. Simple variation of rate, returning asymptotically to original value Formulse. (l)..(la) (2).. (4) (5) (6).. (7) (8).. (86) (9)..(9o) (10).. (10a) (11). .(12) (13).. (13a) (14) (15) (16).. (17) (18). .(19) (20) Tables. Fig. Page. 1 2 3 4 5 10 10 11 11 12 12 12 13 13 13 14 14 14 16 16 17 17 17 18 18 APPENDIX A. n. Various Types etc. — continued. 21. Examination of exponential curves ex- pressing variation of rate 22. Determination of constants of such expon- ential curves 23. Case of total non-periodic migration re- presented by an exponential curve 24. Simple variation of rate, returning asymptotically to a particular value . . 25. Examination of the preceding curve ' 26. Determination of the constants of the curve 27. Total non-periodic migration resulting in permanent increase but returning to original rate 28. The utility of the exponential curve of migration 29. Fluctuation of annual periodicity 30. Discontinuous periodic variations of rate 31. Empirical expression for secular fluctua- tion of rate 32. Growth of- various populations . . 33. Rate of increase of variotis populations . . 34. The population of the world and the rate of its. increase m. Oeteiminatiou of Cuive-constants and of in- termediate Values when the Data are Instantaneous Values. 1. General 2. Determination of constants where a fluctuation is represented by an integral function of one variable 3. Evaluation of the differences from the coefficients 4. Subdivision of intervals 5. Evaluation of constants of periodic fluctuations . . 6. Constants of exponential curves 7. Evaluation of the constants of various curves representing types of fluctuations 8. Polymorpluc and other fluctuations 9. Projective anamorphosis IV. Special Types of Curves and their Character- istics. 1. General 2. Curves of generalised probability 3. The method of evaluating the constants of the curves of generalised probability . . 4. Flexible Curves 5. Determination of the constants of a flex- ible curve . . 6. Generalised probability-curves derived from projections of normal curves 7. Development of type-curves 8. Evaluation of the constants of the pre- ceding type-curves 9. To determine the surface on which the pro- jeotion of a normal probability-curve wiU result in a given skew-curve 10.- Reciprocals of curves of the probability- type 11. Dissection of multimodal fluctuations into a series of unimodal elements Formulse. (20a).. (24) (25).. (30) (31). .(316) (32) (32a ..(36) (37).. (38) (39).. (39a) (40).. (42) (43).. (436) (44) . . (45a) (46).. (69) (70) (71) (72).. (101) (102). .(104) (105).. (122) (123).. (133) 134 (135.. 145) 146 (147).. (166) (167).. (176) (177).. (181) (182).. (183) (184) (185) Tables. II. III. IV., V. Fig. Page. — 19 — 21 — 22 2 22 23 6, 7 8 9-20 21-27 28-33 24 24 25 25 25 26 26 28 30 34 34 37 37 38 40 40 42 45 47 49 52 52 53 57 61 62 62 63 63 SYNOPSIS. V. Group Values, their Adjustment and Analysis. 1. Group-values and their limitations 2. Adjustment of group-values 3. Representation of group-values by equa- tions with integral indices 4. Formulae depending on successive differ- ences of group-heights . . 5. Formulae depending on the group-heights themselves 6. Formulae depending upon the leading differences in the groups or in group- heights 7. Determination of differences for the con- struction of curves 8. Cases where position of curve on axis of ordinates has a fixed value 9. Determination of group-values when con- stants are known 10. Curves of group-totals for equal intervals of the variable expressed as an integral function of the central value of the interval 11. Average values of groups VI. Summation and Integration for Statistical Aggregates. 1. General 2. Areal and volumetric summation formulae 3 . The value of groups in terms of ordinates 4. The value of group-subdivisions in terms of groups 5. Approximate computation of various moments 6. Statistical integrations 7 . The Eulerian integrals or Beta and Gamma functions 8. Table of indefinite and definite integrals and limits . . . , Vn. The Place o£ Graphics and Smoothing in the Analysis of Population-Statistics. 1. General 2. The theory of smoothing statistical data 3. Object of smoothing 4. Justification for smoothing process 5. Mode of application of smoothing processes 6. On smootlung by differencing • 7. Effect of changing the magnitude of the differences 8. Smoothing, by operations on factors . . 9. Logarithmic smoothing . . On the diHerenoe between instantaneous and grouped results Determination of the exact position and height of the mode 12. The testing of smoothed or graphic results 10 11. Formulse. (186) (187).. (189) (190)..(194d!) (195)..(197d)] (198)..(198d)] (199).. (200c) (201)..(209d) (210)..(210e) (211).. (216) (217).. (224) (225).. (252) (253).. (268) (269).. (274) (275).. (281) (282).. (288) Tables. Fig. 34 VI. VII. VIII. (289) (290) (291) (292).. (298) IX. Page. 64 64 65 66 67 69 69 72 72 73 75 75 80 81 82 84 84 35, 36 85 86 87 87 88 89 90 91 91 91 92 94 APPENDIX A. Vni. Conspectus of Fopulation-chaiacters. 1. General 2. Characters directly given or derivative . . 3. Characters in their instantaneoTis and progressive relations 4. Conspectus of population-characters 5. The range of the wider theory of population 6. The creation of norms 7. Homogeneity as regards populations 8. Population norms . . 9. Variation of norms 10. Norms representing constitution of popula- tion according to age . . 11. Mean age of a population 12. Population norm as a fvmction of age IX. Population in the Aggregate, and its Distribu- tion according to Sex and Age. 1. A census and its results 2. Causes of misstatement of age 3. Theory of error of statement of age . . 4. Characteristics of accidental misstatements and their fluctuations 5. Characteristics of systematic misstate- ment 6. Distribution of misstatement according to amount and age of persons . . 7. The smoothing of enumerated populations in age-groups 8. The error of linear grouping 9. Graphic process of eliminating systematic error ■^ 10. Summation methods ' 11. Advantages of graphic smoothing over summation and other methods 12. Graphs of Australian population distri- buted according to age and sex for various censuses 1 3. Growth of population when rate is identical for all ages 14. Growth of population where migration element is known 15. Growth of population rate of increase varying from age to age 16. The prediction of future population and its distrib\ition . . X. The Masculinity of Population. 1. General 2. Norms of masculinity and femininity . . 3. Various defifiitions of masculinity and femininity . . . . .... 4. Use of norms for persons and masculinity only 5. Relation between masculinity at birth and general masculinity of population 6. Masculinity of still and live nuptial and ex-nuptial births 7. Coefficients of ex-nuptial and still-birth masculinity 8. Masculinity of first-bom 9. Masculinity of populations according to age, and its secular fluctuations 10. Theories of masculinity . . Formulse. (299) to (306) (307) (308).. (309) (310) (311).. (323) (324).. (325) (326).. (330) (331) (332) (333).. (335) (336) (337).. (339) Tables. X. XI. XII. XIII. XIV. XV., XVI. XVII., XVIII XIX., XX. XXI. XXII. XXIII., XXIV, XXV., XXVI. XXVII. XXVIII, XXIX. XXX. XXXI. Fig. Page. 96 96 97 98 102 103 103 104 104 37 & 38 39 40 &41 42 43 &44 125 — 127 — 128 — 128 — 129 — 130 131 — 131 — 132 45,46 133 — 136 — 137 138 47 139 140 105 106 107 108 109 109 111 112 114 116 117 119 120 124 SYNOPSIS. XI. Natality. 1. General 2. Crude birth-rates . . 3. Influence of the births upon the birth- rate itself 4. Influence of infantile mortality on birth- rate 5. World-relation between infantile mortal- ity and birth-rate 6. Eesidual birth-rates 7. Determination of proportion of infantile deaths arising from births in the year of record, number of births constant 8. Equivalent year of birth in cases of infan- tile mortality 9. Proportion of infantile deaths arising from births in year of record, number of births increasing 10. Secular fluctuation in birth-rates 11. The Malthusian law , 12. Malthusian equivalent interval . . 13. The Malthusian coefficient and Malthusian gradient 14. Reaction of the marriage-rate upon the birth-rate 15. Annual periodic fluctuation of births 16. The subdivision of results for equaUsed quarters into values corresponding to equalised months 17. Equalisation of periods of irregular length 18. Determination of a purely physiological- annual fluctuation of birth-rate 19. Periodicities due to Easter Xn. Nuptiality. 1. General 2. The nuptial-ratio . . 3. The crude marriage-rate . . 4. Secular fluctuation of marriage-rates 5. Fluctuation of annual period in the fre- quency of marriage 6. General. — Conjugal constitution of the population . . • . . 7. Relative conjugal numbers at each age 8. The curves of the conjugal ratios 9. The norms of the conjugal ratios 10. Divorce and its secular increase . . 11. The abnormality of the divorce curve 12. Desirable form of divorce statistics 13. Frequency of marriages according to pairs of ages . . . . .... 14. Numbers corresponding to given differ- ences of age 15. Errors in the ages at marriage 16. Adjustment nvmibers for ages 18 to 21 in elusive 17. Probability of marriage of bride or bride groom of a given age to a bridegroom or bride of any unspecified age 18. Tabvdation in 5-year groups Formula. (340).. (341) (342).. (342a) (3426) (343) to (348) (349) (350), (351) (352),(353),(354) (355) to 362) (363).. (364) (365).. (366) (367).. (368) (369) (370), (371), (371a) (372).. (373) Tables. XXXII. XXXIII. XXXIV. XXXV. (375) to (395) (397), (398) (399) (400) XLV. XL VI. XL VII. — XL VIII. — XLIX. L. LI. (401).. (402) LII. LIII. (403) (404), (405) (406), (407) XXXVI, XXXVII. XXXVIII, XXXIX. XL. Fig. 48 49 to 52 XLI., XLII., XLIII. XLIV. LIV. LV. LVI., LVII. LVIIL, LIX. LX. 53 54 55 66 57, 58 59 60, 60o Page. 142 143 144 145 147 150 152 155 158 160 162 163 164 166 166 169 171 172 173 175 175 176 179 180 180 182 185 186 186 188 189 189 192 193 195 198 198 APPENDIX A. Xn. Nuptiality — coniimied. 19. Frequency'of marriage according to age representable by a system of cvirved lines . . . . . . '■ ■ 20. The error of adopting a middle value of a range 21. General theory of protogamio andgamio sm'f aces . . 22. Orthogonal trajectories . . 23. Critical characters on the protogamio sur- face 24. Apparent peculiarities of the protogamio frequency 25. The contours of the protogamic surface 27. Relative marriage frequency in various age-groups 28. The numbers of the unmarried and their masculinity . . '. . 29. The theory of the probability of marriages in age-groups 30. Masculmity of the unmarried in various age-groups 31. The probabihty of marriage according to pairs of ages 32. The relative numbers of married persons in age-groups 33. Conjugal age-relationships 34. Non-homogeneous groupings of data . . 35. Average differences in age of husbands and wives according to census 36. Average diSerences ofage at marriage 37. The gamic surface 38. Smoothing of surfaces 39. Solution for the constants of a surface re- presenting nine contiguous groups . . 40. Nuptiality and conjugality norms 41. The marriage-ratios of the unmarried Xni. Fertility and Fecundity and Reproductive Efficiency. 1. 2. 3. 4. 5. 6. 7. 8. 9. General Definitions . . . . ■ • The measurement of reproductive efficiency Natality tables Norm of population for estimating repro- ductive efficiency and the genetic index The natality-index Age of beginning and of end of fertility . . The maternity frequency, nuptial and ex- nuptial, according to age, and the female and male nuptial-ratios Nuptialand ex-nuptial maternity and their frequency -relations Maximum probabilities of marriage and maternity, etc. 11. Probabihty of a first-birth occurring with- in a series of years after marriage . . 12. Maximum probabiUties of a first-birth . . 13. Determination of the co-ordinates of the vertices . . 14. Average age of a group . . 15. Curves of probability for different inter- vals derived by projection . . Number of first-births according to age and duration of marriage 10, 16, FormulsB, (408), (409), (410), (411) (412) to (416) (417), (418), (419) (420) to (424) (425) to (435) (436) (437) (438), (439) (440), (441) (442) to (452) Tables. LXI., LXII. LXIII. . LXIV. LXV., LXVa LXVI., LXVII LXVIII. LXIX. LXX. (453) (454) (455), (*6) (457 to (461) (462) to (465) (466) (467) (468), (469) (470), (471) (472), (473) Fig. Page, 199 200 201 203 203 61, 62 63 64 65 LXXI., LXXII. LXXIII. LXXV. LXXVI. 66 to 70 71 I LXXVII. 208 208 211 212 214 218 223 223 224 224 225 226 228 229 230 232 232 233 233 235 236 237 237 238 240 243 245 245 248 249 250 250 251 SYNOPSIS. Xm. Fertility and Fecundity — continiied. 17. The nuptial protogenesic boundary and agenesic surface 18. Curve of nuptial protogenesic maxima . . 19. Bx-nuptial protogenesis . , 20. Average age for quinquennial age-groups of primiparae . . 21. Average interval between marriage and a first-birth, a function of age 22. The protogenesic indices 23. Exact evaluation of the average interval from a limited series of age-groups Evaluation of group intervals for an ex- tended number of groups Average interval for curves of the expon- ential type Positions of average intervals for groups of all first-births 27. The unprejudiced protogenesic interval . . 24, 25, 26 28. 29. 30. Protogenesic index based on age at and duration of marriage . . Protogenesic quadratic indices and quad- ratic intervals . . Correction of the protogenesic interval for H. population whose characters are not constant . . 31. Proportion of births occurring up to any point of time after marriage 32. Range of gestation period 33. Proportion of births attributable to pre- nuptial insemination . . 34. Issue according to age and duration of marriage 35. Initial and terminal non-linear character of the average issue according to dura- tion of marriage . . 36. The polygenesio, fecundity, and gamo- genesic distributions 37. Diminution of average issue by recent maternity 38. Crude fertility, according to age, corrected for preceding cases of maternity 39. Age of greatest fertility 40. Feoundity-oorrection for infantile mortality 41. Secular trend of reproductivity 42. Crude and corrected reproductivity 43. Progressive changes in the survival co efficients XIV. Complex Elements of Fertility and Fecundity. 1. General 2. Correspondence and correlation 3. Corrections necessary in statistics involv- ing the element of duration . . 4. Distribution of partially and wholly speci- fied quantities in tables of double entry 5. Unspecified cases follow a regular law . . 6. Number of children at a confinement — a function of age 7. Relative frequency of multiple births . . 8. Uniovular and diovular multiple births Formulse. (474) (475) to (4786) (479) to (490) (491) to (495o) (496) to (510) (511) to (517) (518) to (521) (522), (523.) (524) (525) (526), (527) (528) to (533) (534) (535), (536) (537) to (540) (541) (542) (543) to (547) (548),(549),(550) (o51),(552),(553) Tables. Fig. Page. 72,73 LXXVIII. LXXIX., LXXX. LXXXI. LXXXII. LXXXIII., LXXXIV. LXXXV.- LXXXVI. LXXXVII. LXXX VIII., LXXXIX. XC, XCI., XCII. XCIII. XCIV. xcv. XCVI. 74,75 76 to 79 80 81 XCVII. XCVIII. XCIX. C, CI. cii., cm., CIV. 82 255 256 257 ■257 257 260 261 262 264 267 268 271 272 274 276 276 278 279 282 285 286 289 290 291 292 293 295 297 297 298 300 302 303 305 306 APPENDIX A. StV. Complex Elements of Fertility — continued. 9. Small frequency of triovulation . . 10. Nuptial and ex-nuptial probability of twins, according to age 11. Probability of triplets according to age .. 12. Probability of twins, according to dura- tion of marriage 1 3. Probability of triplets according to duration of marriage 14. Remarkable initial fluctuation in the fre- quency of twins according to interval after marriage . . 15. Frequency of twins according to order of confinement 16. Secular fluctuations in multiple births . . 17. Comparison of nuptial and ex-nuptial fertility 18. Theory of fertility, sterility and fecundity 1 9. Past fecundity of an existing population . . 20. Fecundity during a given year 21. Nvimber of married women without child- ren, all durations of marriage 22. SteriEty-ratios according to age and duration of marriage . . 23. Curves of sterility according to duration of marriage 24. Fecundity according to age and duration of marriage 25. The age-genesic distribution 26. The durational genesic distribution 27. The age-fecundity distribution . . 28. The durational fecundity distribution . . 29. The age-polyphorous distribution 30. The durational polyphorous distribution 31. Fecimdity distribution according to age, duration of marriage and number of children borne 32. The duration and age-fecundity distri- butions . . 33. The duration and age-polyphorous dis- tributions 34. The age and durational fecundity distri- butions . . 35. The age and durational polyphorous distributions 36. Fecundity-distributions according to age at marriage 37. Complete tables of fecimdity 38. Digenesic surfaces and diisogenic contours 39. Diisogenic graphs and their significance . . 40. Diisogens, their trajectories and tangents 41. Digenesic age-equivalence in two popula- tions 42. Birth-rate equivalences for given age- differences 43. Diisogeny in Australia 44. Diisogeny generally 45. Multiple diisogeny 46. Twin and triplet frequency according to ages 47. Apparent increase of frequency of twins with age of husbands 48. Triplet diisogeny . . 49. Frequency according to age and according to order of confinerhent 50. Unexplored elements of fecundity Foimulie. (554) (555) (556) (557) (558),(559),(560) (561) (562) (563) (564) to (569) (570) (571) to (575) (576) (577),(578),(579) (580) to (586) (587) (588) to (591) (592) (593) (594) Tables. CV. CVI. CVII. CVIII. cix., ex. CXI., CXII. CXIII., CXIV. CXV., CXVI. CXVII, CXVIII. CXIX., CXX. CXXI. CXXII. CXXIII. CXXIV., CXXV. CXXVI., CXXVII., CXXVIII. Fig. {Page. CXXIX. cxxx. CXXXI. CXXXII., CXXXIII. 83 84 85 86 87 88 89, 90 91 92 93 94 95 96, 97 309 309 310 311 311 312 314 316 317 319 321 324 326 327 331 331 333 333 334 335 335 336 337 340 340 340 340 345 349 349 350 352 353 354 366 361 363 364 367 367 368 368 SYNOPSIS. XV. Mortality. 1. General 2. Secular changes in crude death-rates . 3. Secular changes in mortality according to age 4. The changes in the ratio of female to male mortality according to time and age . . 5. Secular changes in mortality vary with age 6. Fluent life-tables 7. Determination of the general trend of the secular changes in mortality 8. Modification of the general trend by age 9. Significance of the variations in the mor- tality improvement ratio 10. The plasticity curve 1 1. Rate of mortality at the beginning of life 12. Composite character of aggregate mortal- ity according to age . . 1 3. The curve of organic increase or decrease 14. Exact value of abscissa corresponding to the quotient of two groups . . 15. Absence of climacterics in mortality 16. Fluctuations of the ratio of female to male death-rates according to age 17. Rates of mortality as related to conjugal condition 18. Exact ages of least mortality 19. General theory of the variation of mor- taUty with age . . 20. The Gompertz-Makeham-Lazarus theory of mortality 21. Theory of an actuarial population 22. The relation between the mortaUty curve and the probabiUty of death 23. Limitations of the Gompertz theory and its developments 24. Senile element in the force of mortaUty . . 25. The force of mortality in earlier childhood 26. Genesic and gestate elements in mortaUty 27. Norm of mortality-rates . . 28. Number of deaths from particular causes 29. Relative frequency of deaths from par- ticular diseases according to age & sex 30. Death-rates from particular diseases ac- , cording to age and sex . 31. Rates of mortality during the first twelve months of life . . 32. Annual fiuctuation of death-rates 33. Studies of particular causes of death, voluntary death XVI. Migration. 1. Migration 2. Proportion bom in a country . . 3. Correlation, owing to migration between age and length of residence . . 4. The theory of migration 5. Migration-ratios for Australia 6. Periodic fluctuations in migration 7. Migration and age 8. Defects in migration records and the closure of results Formulae. (595) to (600) (601) (602) to (604) (605),(606),(607) (608),(609),(610) (611),(612),(613) (614), (615) ■ (616) to (627) (628) (629) to (629/) (630) to (638) (639) to (644) (645), (646) (647) to (649) (650), (651) (652) to (654) (655) to (660) (661), (662) Tables. CXXXIV. CXXXIVa CXXXV. CXXXVI., CXXXVII. CXXXVIII. CXXXIX.. CXL. CXLI. CXLII. CXLIII. CXLIV. CXLV. CXL VI. Fig. 98 99 100 101 CXL VII. CXL VIII. 102 103 CXLIX CLIIL, CLIV. CLV., CLVE. CLVII. CLVIII. to CLX. CLXI. to CLXIII. 104 105 106 Page. 370 373 374 375 378 380 382 382 387 389 389 392 394 395 399 399 400 401 402 405 407 408 410 411 412 413 413 414 414 415 415 424 426 429 429 431 431 433 435 439 439 APPPENDIX A. XVn. Miscellaneous. Formulae. Tables. Fig. Page- 1. General _ _ , 440 2. Subdivision of population and other groups (663) to (667) — — 440 3. The measure of precision in statistical results (668) — — 441 4. Indirect relations — — 107 442 5. Limits of uncertainty — CLXIV. — 443 6. The theory of happenings or " occurrence frequencies" (669) to (686) — — 444 7. Actual statistical curves do not coincide with elementary type-forms . . . — — . — 448 8. International norm-graphs and type- curves — — — 449 9. Tables for facilitating statistical com- putations — — — 450 10. Statistical integrations and general for- mulae — — — 450 Table of Integrals' and Limits . . — — — 451 XVm. Conclusion. 1 . The larger aim of population statistic 453 2. The impossibUity of any long-continued increase of population at the present rate — — — 454 3. Need for analysis of existing statistical material . . — _ — 455 4. The trend of destiny — — — 456 APPENDIX A. THE MATHEMATICAL THEORY OF POPULATION, OF ITS CHARACTER AND FLUCTUATIONS, AND OF THE FACTORS WHICH INFLUENCE THEM. L— INTRODUCTORY. 1. General.^The fundamental elements of social statistics are the fluctuations of the numbers and constitution of the population and of its various characteristics. These fluctuations are profoundly affected by many factors, only some of which are susceptible of physical ex- pression. For example, the extraordinary development, characteristic in the last few decades, of every branch of science and technology, and the skill with which acquired knowledge has been applied to the exploitation of Nature's resources, have probably created the possibility of develop- ing a considerably larger population than the world has yet carried, at least in historic times. On the other hand, the social standards have been so profoundly altered as to strongly counteract the effect indicated. Thus the raising of the standard of living, and an increased complexity in social organisation have held in check, more or less, that increase of population which might otherwise have been possible. The opposition of tendency involved by the coexistence of these two factors necessarily reinforces the interest, while it increeises the difficulty of the problems which depend for solution on an evaluation of the degree of influence exerted by particular factors. The interest of any theory is evident when we ask : " What, on the whole, is indicated by past statistical history as to the future populations of the various races of the world ? " This is a question, the correct answer to which is a necessary guide for national policy, and one which involves not only the accumulation of statistical facts that have now become available, but also a theory by means of which a forecast can be made as to what the immediate future has in store for each community. An interesting illustration of this may be drawn from the history of the United States. In the year 1815, Elkanah Watson predicted with extraordinary accuracy the population of the United States up to the year 1860, by some method which, though not absolutely doing so, was sensibly equivalent to simply assuming a constant rate of increase. As a matter of fact, had Watson actually assumed that the rate of in- crease from 1790 to 1800 would remain constant till 1860, he would have predicted the population with still greater accuracy than he actually did. This will be made apparent hereinafter ; see also Figs. 3 and 4, APPENDIX A. The more complex conditions of the world to-day and the rapidity of the development of the arts and sciences, make the accuracy of pre- diction for so lengthy a period extremely doubtful ; nevertheless an attempt to forecast the affairs of any country, to be well founded, must be based upon the results of a review, among other things, of aU the facts of its population development, and upon a study of this develop- ment in aU other parts of the world. Of no less interest is the constitution of a population in respect of age, sex and race, and the influence of birth-rates and death-rates there- upon. The effect of age at marriage, the reproductivity as measured by frequency of childbirth, and the age at which it occurs, the pro- bability of living at every age, and the variation of this probability with increasing scientific, hygienic and economic knowledge, are problems of the first order of importance. The attempt is here made to give a rough outhne of the theory of the subject, elucidating that theory where it seemed desirable by quanti- tative examples. 2. Significaiice of analysis. — ^The fluctuations in the number and constitution and other characters of populations present, ia general, complex and dissimilar changes, and depend upon elements which will not readily lend themselves to prediction. They would thus appear at first sight not to be amenable to mathematical analysis. Never- theless, when the fluctuations are analysed and expressed in mathe- matical form, their trend often becomes much more definite, and their true significance is more clearly revealed. ^ ^ An example will illustrate what is meant. The populations in the United States in 1790 and 1820 were respectively 3.93 and 9.64 milUons of people. If the number were supposed to iaorease at each instance at a uniform rate so as to give these numbers in the years mentioned, the deduced populations would be very nearly the actual ones, not only for the iutermediate decades, but even up to the year 1860, as is evident from the following table, viz. : — Year .. 1790 1800 1810 1820 1830 1840 1850 1860 Population supposed to increase at uni- form rate (millions) Actual population (millions) Difference (millions) 3.93 3.93 .00 5.30 5.31 .01 7.15 7.24 .09 9.64 9.64 .00 13.00 12.87 .13 17.53 17.07 .46 23.65 23.19 .46 31.89 31.44 .45 A remarkable prediction by Elkanah Watson is referred to later : see Figs. 3 and 4. This fact, viz., that the supposition made is approximately true, throws light on the other facts. Thus, that to accord with this supposition the figures for 1800 and 1810 are very slightly too small, while those for 1830 to 1860 are somewhat in excess ; and the excess is constant for 1840, 1850, and 1860 ; illustrate the value of the scheme of analysis by means of which the fundamental idea is ascertained. The deviations of the actual values from those computed on the assumption of uniform rate of increase may thus, indeed, become in turn the starting point of a further analysis undertaken with a view to the interpretation of the departure from the law of imiform increase, arbitrarily adopted as the norm of the phenonxena. INTRODUCTORY. For this reason it is proposed to develop the mathematical con- ceptions which may serve as the foundation of definite analyses of the fluctuation of any population ; to express these conceptions by formulae ; to so develop and resolve the formulae that they may be readily applied ; and, where necessary, to illustrate their application. 3. The nature of the problem. — ^An ideal theory of population is one which would enable the statistician not only to determine definitely the influences thereupon of the various elements of human development, and of the phenomena of Nature, but also to examine all facts of interest to mankind, as they stand in relation to population. And however hopeless may be the expectations of establishing such a theory with meticulous precision and in all detail, it nevertheless remains true that fluctuations of population can often be adequately understood only when they are analysed by means of definite mathematical conceptions. Moreover, since all important facts concerning population are susceptible of numerical expression, analjrtical conceptions formulated for the pur- pose of giving exactitude to a knowledge of its variations, should be ultimately cast, if possible, in a mathematical mould. * The total population-aggregates of some countries have been found to increase almost exactly at a uniform rate ; in general, however, the rate fluctuates. " Can the characteristics of such fluctuations be subsumed under any conception ? " is a question which naturally presents itself. * To revert to a previous illustration, for example, if we ask : " What uniform rate of increase would cause a population of 3.93 millions to become 9.64 millions in 30 years ? " the answer is that it would be necessary that each million persons should receive at each instant an addition at the rate of 29,910 persons per annum, that is to say, the rate of continuous increase would have to be 0.02991 per annum. More exactly, this would give the following figures, viz. : — 3,930,000; (+ 1,370,173) = 5,300,173; (+ 1,847,877) = 7,148,050; (-1- 2,492,128) = 9,640,178^^^^^^^ The differences, shewn by the small figures^o not in themselves disclose the fact that the increase is at a uniform rate, but on dividing each by the preceding popiUa- tion figures it is seen to be equivalent to adding 348,644 persons per million per decennium. Hence, obviously, the rate of increase was constant. This rate will be found to be equal to an increase of 30,361.8 annually per million of the population at the beginning of each year. The facts just indicated, viz., that starting with a population of 3,930,000, and uniform increases at the rate of 0.02991 per anniim, gives a population of 5,300,173 in ten years, etc. ; that an equivalent figure is given for the population if, at the end of each year, there is added to it an absolute increment of the amount of 0.0303618 of the population at its beginning ; that the figvu^es at the end of a decennium are given by adding an increment of 0.348644 of the population at the beginning of the decennium— can be elucidated only by formulating a definite conception of rate, and studying the consequences that flow therefrom. It is, for example, by no means immediately obvious that, used with the limitations above indicated, the three sets of figures will give identical results. The last will accurately give only decennial results ; the middle value only annual ; the rate of continuous increase is the only one which is appropriate to furnish correct results for any moment during the whole period under review : see Fig. 4. APPENDIX A. Such answer as may be given must, if it is to be explicit, obviously be in the form of a mathematical theory of the subject. Such a theory will be found to involve two elements, viz. : — {a) The appropriation of suitable conceptions of a mathematical character, and (6) The development of a scheme of using them. The propriety of the apphcation of such conceptions is to be measured by the extent to which they are capable of illuminating the actual facts, and of reducing them to system. What has been said regarding total population, appUes equally to each constituent part, viz., to the totals for each sex, to the number of both sexes or of either sex at birth or at a particular age, to the ratio of the sexes, to the fluctuations in the rates of birth or death, and to all the circumstances of migration. In other words, any fact, either of the condition or constitution of population at any moment, or of the relation of these at different moments can be readily subsumed under appropriate mathematical conceptions with suiScient precision for practical purposes. Again, in deaUng with the co-ordination of population with other related facts susceptible of statistical statement, the question often arises : " How can the nature of the relation be best defined or best disclosed ? " The selection of appropriate mathematical conceptions, and the means of bringing the facts under them, also constitute phases of the theory to be considered. 4. Necessity foi the mathematical expression of the conditions of the problem. — ^Although, in the nature of the case, the population of any territory necessarily changes through births and deaths by whole units, and in instances of immigration and emigration sometimes by relatively large groups of units, no appreciable error will ordinarily be committed, at least where the aggregate population is large, if all its fluctuations be supposed to take place continuously and by iuiinitesimal increments. This supposition, which might appear . ai^^nsuffioient consideration to be physically invahd, very fairly represents, after all, the, actual facts, in their totality.* 1 For, when all tjie oircufnstanoes are tak;en into account, it is obviojis that the extent or degree to whicli the individuals of a community participate in its economic and general life, pr in territorial occupation, passes through a wide range of values. These considerations have application even to, the circumstances of birth, and death, and even moreover to those of immigration and emigration. The ordinary involve- ment of a community by each individual through the circun;stances preceding birth and following upon death, s,hew clearly that in ijnany important respects the introduction and disappearance of a imit of the population is, virtually, not quite instantaneous. It is obvious, too, that this consideration would apply even if registration, or rather the statistical recognition of that fact, were contemporaneous with birth and death, which, however, it is not, since ordinarily it follows these events by a period of varying length. In cases of birth it also stretches over a longer period. It INTRODUCTORY. Thus the fluctuations of population therein may at least in ordinary cases, be represented with precision by an imaginary or fictitious popula- tion, the ideal fluctuations of which, varying with time, conform to all the laws of infinitesimal increment or decrement, in this way rendering those fluctuations amenable to a rigorous analysis by the methods of the infinitesimal calculus. Such an imaginary population, changing con- tinually by infinitesimal amounts, not only accurately represents the totaUty of facts, but is amenable to mathematical treatment. It is nevertheless important to bear in mind that actual pojJulation- changes may be oscillatory, as will later be shewn. 5. Conception applies equally to a population " de facto " or a population "de jure." — ^Population may be related to territory in two ways, viz., by actual presence, and by legal relationship therewith ; that is to say, the relationship may be " de facto " or " de jure " ; and official statements regarding population are of each kind. In some countries, as where the floating population is large, or where citizens are under special obHgations {e.g., military service, etc.), the main concern may be to ascertain the population which may be said to belong to, or to be domiciled in the place, the foreign migratory element, whatever its magnitude, being regarded as of relatively little moment. Again, where communal rights are exphcit and of an important character, the general reasons for deciding to adopt the " de jure " relationship for the official enumeration of population may be very cogent. ^ The association of a human being, however, with any particular territory, defimited by frontiers of any tj^e whatever, is, after aU, only one of degree, so that any criterion {e.g., nationality, domicile, etc.), other than that of mere presence in the territory, however necessary for certain purposes, is more or less indeterminate for others, particularly in countries where the freedom of movement of the individual is practic- ally unrestricted. The actual presence of an individual in any territory involves, in varying degree, ^ the whole scheme of general relationship which every unit has to the general community in which he finds himself, and which that community has to the territory it is occupying. He is is considerably influenced by legal prescriptions in regard thereto, as well as by the traditions and cireiuustaiioes of the community. Thus the registration of death must perforce quickly follow on its occurrence ; not so the registration of birth. In a sparsely -populated district, the registration of birth may be very late as com- pared with registration in a densely -populated area. We may remark in passing, that official estimates of population, at least when based upon accurate vital and migration records, as ordinarily kept and reported, are usually slightly in error as regards actual populations, viz., to an extent cor- responding to the want of balance between inclusions at the beginning of a period of record, really belonging to a previous record, and exclusions at the end of the period owing to complete information not being to hand. In an increasing population the error tends on the whole to be one of defect. ^ As, for example, in some of the Cantons in Switzerland. ^ The economics and general relationship of individual with a community passes through a wide range of values, and in each individual the value varies with his age. APPENDIX A. subject to the laws and to the same extent also the general civic and other responsibilities of the place, while the community, on the other hand, is concerned with his protection and well-being. Hence the " de facto " population may often be statistical desideratum. For other purposes obviously the " de jure " population is a necessity. For the general purposes of economics there are features character- istic of population which may be considered either in the " de jure " or the " de facto " relationship, which may call for specialisation in any mathematical treatment. For mere enumeration, however, the mathe- matical conception as above defined will apply with equal rigour to either. 6. Nature of population fluctuations.— The fluctuations of the en- tire population of the earth, if available for long periods, would probably disclose in their most general aspect the secular characteristics of its increase, which must have greatly varied. Merely local effects would to a large extent disappear in the total ; opposite periodicities, dependent on seasons, would be balanced by the inclusion of results from both hemispheres ; by taking quinquennial, decennial, or longer means or averages, the effect of minor fluctuations would be correspondingly eliminated ; and the broad outhnes of the facts of the growth of the world's population would be brought into reUef. Were the curve of secular increase of population for the entire earth available, it would obviously constitute the most suitable norm for general comparative purposes. Statistic unfortunately, has, however, not yet attained to this. All we can assert with certainity is that the present rate of in- crease can have existed for a relatively short time only. Limiting the consideration to particular countries, changes will be found exhibiting the following features, viz. : — (i.) The rate of appearance of individuals by birth, and disappear- ance by death is not, in general, uniform throughout the year, but shews more or less definitely an annual period. (ii.) The movement of floating population is also non-uniform, disclosing, in many instances, definite annual periodicity. (iii.) Improvements of natural conditions are in general followed by changed rate of increment to the population, which may have a period of a considerable number of years, or may be brief. (iv.) Variations of social and economic traditions profoundly affect the rate of increase of population. For the larger purposes of statistic, elements of the type (i.) and (ii.) are ordinarily negUgible ; while those of the type (iii.) and (iv.) are of the first order of importance. For minor purposes the converse may be true. Hence, the scheme of any investigation must be adapted to the element under consideration. INTRODUCTORY. In general, secular and long-period changes must be eliminated in order to accurately study minor and short-period changes ; and con- versely, minor periodic changes must be eliminated in order to acurately ascertain the characteristics of the secular changes. 7. Changes in the constitution of populations. — ^The ratio of the total numbers of each sex, the proportion of the sexes at each age, the relative birth, marriage, and death rates, the circumstances affecting fecundity, the consequences upon all of these of migration, of disease, of war, and of economic and social traditions and developments, as well as their fluctua- tions with the lapse of time, are necessarily matters of statistical concern. Such changes may be called " constitutive changes," or perhaps " organic changes," and their analysis and subsumption under mathematical ex- pressions are often of importance and are essential in various statistical analyses. 8. Organic adjustments of populations. — ^In reviewing the constitu- tion of population as a whole, it is obvious that organic adjustments occur.^ The nature and drift of such adjustment as has been indicated, or of the deviations of the actual constitution of a population at any moment from some norm adopted for comparison, and the changes in such devia- tions, can be effectively studied only by the estabhshment of a system of suitable mathematical relations. For such deviations to be made the subject of prediction, the law of their fluctuation with time, must, of course, be ascertained. The principles guiding the constitution of a norm will be illustrated hereinafter. 9. Continuous and finite fluctuations. — ^The scope of the mathe- matical theory of the fluctuation of population reveals its fundamental importance. Every form of fluctuation, whether of total population, or of its constitutive elements, of its characters, or of the influences to which these are subject, may ordinarily be regarded as changing continuously by infinitesimal increments or decrements within the period during which it is assumed to vary. In special cases the fluctua- tions may even be discontinuous. 1 In Europe, for example, of those bom living, there are about 105 male births to every 100 female births : of those still-bom the proportion is about 133 (see " Pie Geborenen nach dem Geschleoht," in " Statistik und Gesellsohaftsehre," by Prof. Dr. Georg von Mayr. Bd. II., § 56, p. 189), and the deviation from these figures for different countries is, in general, small. Nevertheless, in the total popula- tion of Europe there is a ratio of only about 97.6 males to 100 females. To war and unhealthy occupation, and accident, the death of a considerable number of males is directly attributed. Thus there are no less than about 108 deaths of males to 100 deaths of females, for a number of countries. Nevertheless, because of the larger number of male births, the percentage does not materially change. APPENDIX A. The aim of any definitive consideration of the subject is to express the fluctuations of population or of its constituent elements, and of its characters, in forms which will serve — (i.) To render intelligible the characteristics of such fluctuations. (ii.) To assist attempts at tracing the cause and effect of fluctuations. (iii.) To determine means and averages, etc. (iv.) To make aU required interpolation of values. (v,) To make prediction by extrapolation possible, or to make it possible by the result of a general analysis. (vi.) To bring into clear reUef the various characters of a population. 10. Curves required to represent vaxious fluctuations and the solution of the same. — ^When a curve or " graph " representing a series of statistical results can be defined with sufficient accuracy by some form which is susceptible of geometrical or algebraical representation, such definition constitutes an advance as regards the understanding of the essential nature of the facts : a clearer conception of the statistical results is attained. For example, if the rate at which a population is growing be constant, then the curve passing through the terminals of the ordinates (whose length represents the successive values of the population) plotted against distances along an axis representing time, is a curve which is concave upward. This curve is of character such that, it, instead of plotting the ordinates on the natural scale, their logarithms be plotted, the terminals will be found to lie upon a straight hne. Thus, if when the logarithms of the numbers of any population at different dates are plotted as ordinates, and the times as abscissse, the points are found to lie on a straight Une, we know that the rate of increase is constant. To thoroughly represent and to analyse the nature of the changes in the size of any population or the changes in its constituent elements or characters, a considerable command of schemes of curve-representation is a desideratum. For the mathematical representation of fluctuation, therefore, it is, in general, necessary to know the geometrical form or graph of various algebraic or other mathematical expressions ; in order that, given geometrical form or graph of a series of results, the mathe- matical expression appropriate to represent it wiU be "recognised. For this reason a considerable number of type-curves and a knowledge of their graphs must be at the disposal of the statistical analyst, so that the appropriate expression may be selected. As soon as it is decided upon, the mode of solving for the constants of the representative ex- pression becomes of importance. With this ia view, it has been found desirable to give a considerable number of formulae, and to indicate the methods by means of which the constants that make the expression definitive can be found. INTRODUCTORY. This has been the more necessary, because, after all, the scheme of statistical representation, or the "fitting of curves," is an art of much difficulty, and one which is only in its infancy. The fluctuations of the numbers representing population and its various characters make considerable demands in regard to knowledge of this kind, and consequently not only are formulae given herein from time to time, but their " graphs " are also drawn. These exhibit the character of the curves represented. It will be seen that the interpretation of statistical results therefore make considerable demands of what is called curve-tracing.^ 1 The "Spezielle algebraische iind transzendente ebene Kurven, Theorie und Geschiohte," of Dr. Gino Loria, 2 vols., Teubuer, Leipzig, 1910-1911 ; the "Samm- luug von Formehi der reinen und angewandten Mathematik," by Dr. W. Laska, Ft. Vieweg und Sohn, Braunschweig, 1888-1894; and Frost's well-known "Curve Tracing," give much valuable information in regard to the possibility of representing certain important forms. These works, however, are neither adequate nor exhaus- tive. The work of Felix Auerbach on " Physik in graphischen Darstellungen," Teubner, Leipzig, 1912, has also a large number of forms of importance to statis- ticians. n— VARIOUS TYPES OF POPULATION FLUCTUATIONS. 1 . Mathematical conception of rate of increase. — Whether diminish- ing or gaining, any actual population may be replaced by a " representa- tive population," assumed to change at every moment by infinitesimal amounts at some rate (p say) per unit of time. That is to say, p will denote the fraction of a unit which, at the instant under consideration, measures the rate of change of the population for a unit of time. Hence, if Pj be the population at the time t, and Pt + dt that at the time t -\- 8t, then where 8tis small we shall have (1) P.^st = Pt{l + pSt)=P^e'''' as the fundamental expression for its fluctuation. In other words- Pt pSt is the absolute change in the time St. If p be positive, the change is an increase ; if negative, it is a decrease. The rate p may be either constant, in which case we shall denote it by r, or it may on the other hand vary in some determinate way with time, in which case we shall retaia the Greek letter. If the fate be regarded as a function of time, then we should have (la) P^^^^ = Pt\p{l+t)dt] We shall consider initially the case where it is constant. 2. Determination of a population for any instant when the rate is constant.^ If increments of population be supposed to be added at N uniform intervals of time, extending over the period t, at the uniform rate r per unit of population per unit of time, then, putting P^ for the initial population and Pt for that at the end of the time t, we shall have, (2) Pt = Po ( 1 + ^)'"; = Poe' when N becomes finite. As usual e denotes the base of Napierian logarithms, viz. : — 2.7182818284590, etc. It is sometimes convenient to put this expression in the form of a series : thus, by the exponential theorem, we have (2a) Pt = Po(l +>t+~ + ~ + etc.) Taking logarithms of both sides of (2), we notice that 1 AVhen p is constant the investigation is analogous to that for determining the increase in a sum of money when interest is supposed to accrue at every instant of time. For a development of the theory of continuous interest and a kindred investi- gation of population, see a paper by J. M. Allen, Joiim. Inst. Actuaries, Vol XLI p. 305. TYPES OF POPULATION FLUCTUATIONS. 1 1 (3) log Pt =logPo + (»■ log e) t hence, if r be constant, the graph obtained by passing a line through the points formed by plotting as ordinates the logarithms of the population for successive years, quinquenniums or decenniums, opposite the cor- responding values of t as abscissae, wUI be a straight line, the tangent of whose angle with the axis of abscissae is r log e. We shall call this graph the partial^ logarithmic homologue of the graph of equation (2). The value of log^„ e is 0.4342944819032, etc., and of logj„ (log^^e) is 9.6377843113005, etc.^ Both are required in practical calculation, to, however, only few places of decimals. To find the constant rate of increase, we have (4) r= (log Pt -^ log Po )/{t\oge) 3. Relation of instantaneous rate to the ratio of increase for various periods. — We may call the constant r the constant rate of continuous increase, and similarly the variable p the instantaneous rate of con- tinuous increase. It is often necessary, however, to substitute for r the equivalent rate for a year, or for five or ten years, that is to say, to measure the ratio at which the population at the beginning of the period must be increased in order to give it its proper value at the end thereof. Calling this rt, we have (5).- r, ={Pt -P„ )/P„ = /'-I; ore''' =1 -f r^ 4. Determination of the mean population for any period : rate constant. — ^Let Pq denote the population at the beginning of any period and Pt the population after the time t : then, since /e""' dt = e'V^, the mean population P,„ is obviously I/-'. .. Po rt.,... PqK'-D ^ I , rtrH 2 (6)....-J^Ptdt=^'lertdt^ r^ =Po{ l + 2r+^+etc. ) a formula which is suitable for determining the mean from the initial population. This expression may be put also in the form, see (5) (7) (Pt — Po)/rt; or P^rt / rt by means of which, when the rate is constant and known, the mean population can be calculated, either from the absolute increase for a given period, or from the ratio of the increase for a given period to the initial population for that period. ^ Partial, because the values of t and not of the logarithms of t are not used as the abscissee. ' 9 is used instead of I. 12 APPENDIX A. 5. Error of the arithmetical mean : rate constant. — ^The arith- metical mean of the population at the beginning and end of any finite period differs, of course, from the true mean. The magnitude of this difference is sometimes required. From (2a) and (6) we obtain — „ ffH^ 2rH^ 3rH^ , n (8) -Pm = 4 (Po + -P. ) -n (2X! + 2ir + 2:5r + ^^- ) which may also be written — (8a) Pm=i{Po+ Pt)-Pt{2M-'2A-\+ ^X! " ^**'- >* When expressed in terms of the arithmetical mean itseH, the odd powers of r and t disappear, thus (8b) -Pm = i (^0 + -Pt) ( 1 -2:3^ + ^X! - -JVV.-^^ ) This last is the most convenient formula. The values of the coefficients are "I'j, Y^jj, 55rTC' ^^''• Remembering that the maximum value of r is about 0.03, all these series converge with sufficient rapidity. 6. Empirical expression for any population-fluctuation.— If the population of a country be determined at « + 1 different dates, then a curve of the w** degree can be arbitrarily drawn, passing through the graph of the coordinates. In the absence of any information as to the magnitude of the population between the given dates, the ordinate to the curve drawn from the terminal of the abscissa corresponding to the date may be assumed to be a probable value for the population at that date. The curve in question may be written ^ — (9) Pt = Pq {I +at+bt^ + ct^ + etc.) which, for purposes of practical calculations or computational check, may be found convenient in the form : — (9a). ...Pt=Po {1 +t [a + t{b+tc+ etc.)] [ 7. Mean population for any period : rate not constant. — ^Using the same notation as in II., 4, equation (6), we have — (io)....P. =,-^/;=p,* = Po{^ + I (h+h)+ I («,H<2«i+t, the several quantities in the brackets in (12) are dy/dt ; {d^y/dt^)/2 ! ; (d^y/dt^)/3 ! ; etc., and the coefficients can be written out by a reference to Pascal's triangle. They are, of course, simple " figurate numbers " of the second, third, fourth, etc., orders. That the coefficients must be altered when a new origin for t is selected, exposes one of the inherent limitations of the empirical equation. 9. Error of the arithmetical mean : rate not constant. — ^The arith- metical mean will always be in excess with either a uniform or a growing rate of increase. From (9) and (10a) we obtain — bt^ 2ct^ 3dt* (13) Pm =UPo + Pt)-Poi 273+ 2A + 2:5 + ^^- ) which may also be readUy expressed in terms of the mean itself, as in (8b), thus— ,,^ ^>,, 6 , 2a6-6r-, 5a^b-10b^-l5ac-36d,^ , , (13a)..P,„=i(Po+P,)-|l-3-,<^+-4T-«=' gi t*-eto.} This, however, is more tedious to use than (13). 10. Expression of the coefficients in the empirical formula for rate in terms of the constant rate. — ^If in equation (9), viz. : — P( = p^ (1 4- a< + 6<2 _^ c(3 ^ etc.) a=r; b=r^/2\; c = r^/3 I ; etc. the equation would express a constant rate, that is to say, it would be simply another form of equation (2a) ; and if a, b, c, etc., have not these values, the rate of increase is variable. 14 APPENDIX A. By substituting the corresponding values of r in (13a), it may eadily be seen to be identical with (8b) ; and similarly as regards (13) and (8). 11. Investigation of rate is complete only when its variations are ascertained. — Heverting to II., 1, equation (1) may be written — (14) 8P = Pplt = P.^(<)S« which may be regarded as the fundamental differential form for increase of population, the final form being required, since the rate p is rarely if ever, constant, even for short periods of time. Hence in its theoretical form, an investigation of the fluctuations of population cannot be com- plete tiU all variations of its rate of growth are definitively ascertained, in other words, <^(<) must be ascertained. 12. Bate is a function of elements that vary with time. — ^The rate at which population increases is dependent upon elements external to and beyond the control of man, as well as upon elements within him, more or less under control. Both change with the lapse of time. In Fig. 3, § 32, hereinafter, examples are given shewing the curve of popula- tion of different countries, and in Fig. 4, of the same section, the cor- responding logarithmic homologues of the populations. As already pointed out, the latter would be straight lines, if the rates of increase were constant. Hence, in the sense that it is dependent upon elements that vary with time, and may thus be directly related to the latter, the rate p =(<) may be investigated as a function of the elapsed time. 13. Factors which secularly influence the rate of increase. — ^Where not otherwise expressed, the rate of increase will be assumed to refer to total population. Let us consider primarily a community which grows by natural increase alone. This increase will be profoundly affected by four types of things, viz. : — (i.) The material natural resources of the occupied territory, (ii.) The various cosmic energies which facilitate man's development. (iii.) Knowledge which increases the power of utihsing natural resources, (iv.) Sociological and other analogous standards, which react upon human activities, particularly upon man's productiveness, and the magnitude and character of his consumption of what he has produced. Regarding (i.), it may be said that the natural resources of the ter- ritory occupied may be either actual or potential. Even without human intervention, a territory' may be prodigal of those forms of animal and vegetable life, for example, which provide immediately for human wants. Its climate and meteorology may be propitious. It may possess large stores of readily available wealth, or of energy convertible into wealth. TYPES OF POPULATION FLUCTUATIONS. 15 Or yet, again, though in the state of Nature infertile, it may respond to weJl-directed efforts to make it so. It may have large hidden resources which can be recognised, and can become available only through a con- siderable development of scientific and technical knowledge, and through practical abiUty in applying the same. Lastly, it may contain types of wealth^l as for example mineral wealth generally, which, though valueless per se to sustain life, may be made contributory to the growth of popula- tion through the part they play in the world-economy. All these may be summed up under two headings, viz. : — (i.) Natural fertility and resources of the territory independent of human action, (ii.) Wealth or resources dependent on human action. Both, however, are potentialities rather than actuaUties in regard to population : how they eventuate in respect thereto depends upon other and very subtle factors inhering in that order of things which concerns the general sociological and economic beliefs and in the traditions and activities of the people. For example, the general attitude of a people in respect to the question of fecundity and the prevaUing view as to what should constitute a reasonable standard of living, profoundly affect the rapidity of the increase of the population, and the reaching of the' time when natural limitations of fecundity operate severely. There is still another factor of an analogous nature that plays a part, the significance of which is each year becoming more manifest, viz. : — The attitude of a people toward the development of the intellectual powers of man, and toward the application of such powers to the avail- ment of the resources of Nature. Indeed, in general, the great advantages of the human being over the larger mammals is due to the efficiency in this direction of his intellectual endowment, and his power by systematis- ing to store and apply acquired knowledge. If we denote natural fertihty or wealth of resources of the territory, say, by w ; what may be called its geographical and climatic advantages by g ; its other available resources when better scientific knowledge is applied, or even when new wants are created by advancing civiUsation, by u ; the factors expressing themselves in the matter of fecundity by / ; through standard of Uving, including hygiene, by I ; through intellectual knowledge and its range, energy, and wisdom of appUcation by i ; then we must regard the increased population as really a function of all these, that is to say — (15) P = P„ ^ iw,g, u,f, I, i,. . . A) The influences of these elements are, in general, secular in character, i.e., they produce slow changes, some being manifest in the years of a decade, others only in many decades. They are all determining factors of the possibilities of population, but do not necessarily express its actuality. 16 APPENDIX A. Their specific character is such that ordinarily they produce gradual and more or less remote effects, rather than effects which are instantaneous and immediately of great magnitude. Such effect may tend towards a constant value, may increase, or diminish, but in all cases the consequent changes will be gradual. It is to be noted, however, that some of the factors may acquire for a short time an importance which, locally at any rate, may lead to rapid changes. Factors of the kind considered are probably either non-periodic, or if periodic their period is secular. A general solution, if it were possible, would presuppose that the way in which w, g, u, f, I and i, varied with elapsed time was determinable. This variation, however, is not susceptible of exact definition : never- theless, the form of the functions expressing their effect on the rate of increase p is not always wholly indeterminable. 14. Variations which depend on natnial resources, irrespective of human intervention. — ^This may include both periodic and non-periodic elements. The natural wealth of a territory, as unaffected by the inter- vention of man, is, in general though not invariably, a maximum initially, 1 though its values may oscillate between very wide Umits, owing to variations of meteorological or climatological factors. Where natural wealth is of a type that is subject to steady decUne, its effect on the rate of increase may be represented for all practical cases probably by a very simple function of the elasped time. 15. Variations of rate of long periods. — ^Any periodicity in meteoro- logical and other factors, affecting the natural wealth of a territory, however much their influence may be masked by other factors, wiU in most cases cause a collateral periodicity in rate of increase. This can be represented by such a formula as the following, viz. : — (16)..pt/po = 1+ [tto+ai sin (ai+ jr ) +a^sia{a^+Y )+etc.J+ Q in which Ti, T^, etc., will represent the lengths of the various periods to which the elapsed time t is related ; ai, a^, etc., are intervals deter- mining the epochs of Ti, T^, etc. ; and finally o^, a^, etc., are the amplitudes of the variation from the mean value. Thus necessarily — (17) Ug = — {tti sin ai -f a^ sin a^ + etc.) and Q wiU of course represent the effect of the other elements influencing the rate of increase to which reference will be made later. Equation (16) is specially suitable for representing fluctuations of long period, which are expressible in terms of a sine series. ' Examples could be drawn in recent times from America or Australia. It may, however, even in regions which nevertheless can be made habitable, be actually zero, as for example, in the Sahara, in Arizona, and in some parts of Australia. TYPES OF POPULATION FLUCTUATIONS. 17 (18) 16. Representation of periodic elements in non-periodic form. — Where T is exceedingly long as compared with t, the numerator of the expression (16) may take a much more simple form, available probably for all practical cases. For putting — ^A,=2 [(ai cos ai )/Ti ]; A, ^ —1 2 [(«< sin a,- ) / T/ ] ; 1 ^3= -^^ 2 [(ai cos ai )/T/]; ^, = + 1 i;[( («; sin a^ )/r/]. etc., etc., ; etc., etc. the limits of the summation being from i = 1 to i = w, and n being the number of periodic terms. Then remembering that Ug + 2 (ai sin tti ) = with the same limits, we can express (16) in the form (19)../3j/p=l+ao+aisin(ai+ ^ )+ etc.=l+.4i< + A,t^+. . + etc. which, with (18), connects the coefficients with the amplitude and epoch of the periodic fluctuations. The values of Ai, A^, etc., may be either positive, negative, or zero. 17. Influence of natural resources disclosed by advancing know- ledge. — ^Turning now to the question of the various terms in Q, viz., those representing in equation (15) the effect of m, /, I, and i on the rate of increase, we remark first of all that increased scientific knowledge, especially in physics and chemistry, suggests that possibly the available resources of Nature are practically without Umit, (that is m = oo ). This being so, the rate of increase may be regarded as dependent, not so much upon Nature's Umitations as upon the extent and character of our know- ledge, and of our energy and wisdom in applying it; that is, in the formula, it depends upon i, not upon %. We shaU find, however, that Nature's limitations are very real, for rates of increase of population which characterise many countries at the present time cannot be maintained for several thousand years. 18. Influences of resources dependent upon human intervention. — There is a narrower sense, however, in which % may represent specific and finite quantities, which can be sufficiently indicated by two or three illustrations. Territories hke portions of the Sahara in Africa, and of Arizona in America, apparently hopeless waste, may in response to the appUcation of artesian water, become fertile and habitable. In ordinary agriculture, land, practically valueless in the state of Nature, may become valuable by the appHcation of suitable fertilisers. The infertility of land which is due to the absence of the necessary micro-organisms, may, when once such organisms are introduced, quite disappear, and the potential wealth in the territory existing may have been quite undreamt of. Or yet again, the value to man of a natural product, utilisable in 18 APPENDIX A. the natural state, or after being treated technically, may be wholly unknown ; the discovery of its real value may so change the economic conditions of a territory as to greatly facilitate increase of population. In these and many other similar ways, natural resources reacting to man's operation may be found to be very great, though at first apparently non-existent. It would obviously therefore be very difficult to assign a form to the function which is in any way to represent the effect of natural resources. 19. Effects of migration. — ^Migration operates in several ways on the rate of increase of population, viz. : (i.) By the actual addition or withdrawal of the migrants ; (ii.) by the change of the constitution of the population, thus affecting its rate of fecundity ; (iii.) by consequential economic changes which favour or impair the rate of increase. A com- plete expression for its effects would therefore be elaborate in form. Since, however, the community changed by migration tends to adjust itself to the economic condition of the country, the real elaboration into each component element is unnecessary, and the resultant of all the elements operating may take a relatively simple form. Migration itself is of two forms — ^periodic and non-periodic. The population of countries, for example, which at certain seasons are visited by large numbers of tourists, or from which large numbers depart, may be taken as affording illustrations of periodic migration. The rate of influx or ef&ax is usually slow initially ; it then increases, becoming a maximum ; when it dechnes much in the same way. In form, the curve of absolute increase or decrease is somewhat similar to the probability curve, but the curve is probably rarely symmetrical with respect to the maximum ordinate. Non-periodic migration may, in addition to the effect of its absolute amount, change the final rate of increase or leave it as it was originally. Although both periodic and non-periodic migration may be actually discontinuous, no material error wiU ordinarily be committed if it be assumed to be continuous, provided that in amount it be negligibly small for the part of the year when it has actually ceased. So that there is no serious objection to the use of an essentially continuous function. 20. Simple variation of rate, returning asymptotically to ordinal value. — ^Non-periodic migration of population, frequent in new countries, may produce a simple variation of rate which ultimately disappears. Owing to the reputation the territory acquires in respect of some real or supposed advantage, immigration sets in, increasing in rate till a maximum is reached, and declining again till the original rate is restored. For the territory or territories from which the emigration takes place, the converse effect may be true. If the rate can be ascertained at several periods, the total effect on the population can then be deduced with fair accuracy. TYPES OF POPULATION FLUCTUATIONS. 19 The simplest variation of this type, and one which will probably represent most instances with sufficient precision, may be expressed in the form — (20). ■Pi/P^ = I +T7<« ■q being positive for cases of immigration, and negative for those of emigration. This form would be suitable for deductions as to population based on the determination of rate of increase at various times. By suitably selecting the unit of {, the parameter rj and the index- numbers m and n, equation (22) may be made to represent the very different circumstances which may obtain at the commencement, and during the development and passing away of the effect of migration on the original rate of increase. For example, it wiU express that type of migration in which the increments per unit of time to the rate of increase, though initially slow, grow and decrease with continually changing velocity, tiU the original rate is restored ; or, on the other hand, it will express that type where the migration effect on the rate is sudden. This is illustrated by the curves in Fig. 1., viz. : — Curve Tj^™-"' Curves graphed ,,=1 a wi = A w = h 1 c 1 6, 2 e 4 4 ^ / \ ^ / \ \ 2 \ / / ^< k 1 / ^ // s A / /' '~^. ■*^.c \ ■•V 16^ y K :^ "■■- v -~. ^_ «„ = 1/e: 1— m/nt 4 5 Fig. 1. 7 Values of t Curves y = tj*™ + "* in which the parameter ij is unity throughout. The possible varieties of change of rate of increase are obvious from the figure^ when it is re- membered also that the horizontal proportions can be maintained, and the vertical changed at pleasure by simply altering the value of ij. 21. Examination of exponential curves expressing variation of rate. — ^The curve of equation (20) demands special consideration. For brevity put E for {pt — p^ / p^, then we can re-express (20) in the form^ 1 An expression of still greater fitting power is 1/ = At'"'' e"* • See a paper on the curve by G. H. Knibbs. Journ. Roy. Soc. N.S. Wales, Vol. XLIV., pp. 341-367. 20 APPENDIX A. therefore (20a) R =rit' „4m-nt {2l)....dR/dt= J-^- ;w— w<(l +logc<)} (21a) ^ = 7,«-+««(« \ogt + ^ +«) (21b). dt dhj , , , , »* >o wi — m , dt and hence the value of t, which gives the maximum value for R, is found by solving the equation — (22) m/n = (' h a 1 K^ '' _j_ — -. — — -- -- — -- — r '?/ n ^ '^J Cuives r^t m + f\i Curves graphed a m = 4 ; w = 4 b 1 c i i . d i i *max — ^ 1 + 8 9 Fig. 2. 25. Examination of the preceding curve.— As in section (21), put 1 then (32a) iJ'=Tj<±'» + ' (33).... dW /dt = "nL n loge t m + nt ^ t ±m+nt\ and consequently the value of t which gives the maximum value for B' is found by solving the equation ^ = £^^t '^^''^ l^ads to— (34). ± m = t (log« t - 1) 24 APPENDIX A. For a maximum to correspond to a value of t greater than e, the 1 equation will be of the form «»> + «« (m and n being positive) ; or less than e, equation will be of the form (-»»+"« This may be solved for the series of values already given in Table I. for t (loge t — 1) : see section 21 hereinbefore. Similarly to the preceding case we take the logarithm of both sides of (32a), we have — (35) log R'= log 7] + log < / (± m + nt) Hence as before, finding ^' from observed values of R' we have — (36) '§, = log t / (log B' —log 7]) = ± m + nt which enables us to examine the vaHdity of the assumption, since it is the equation of a straight line of which the values of "g{' and t are respectively ordinates and abscissse. For the point of inflexion the second differential will be required : the sign of m being positive, it is — 26. Deteimination of the constants of the curve. — ^In this case the rate for < = 1 is known, and r/ = Ri = i', thus formula (29) holds when ^ is changed for ^', and similarly in regard to (30) changing the sign, that is — (37) m = (il',*3 - W,t,) I (<3 - t,) (38) n = {W^-3\)/{h-h) The test of (36) is necessary if there be more than three values of ^'. For the case of immigration tj is positive, for emigration negative. 27. Total non-periodic migration resultii^ in permanent increase but returning to original rate. — ^Where the migration effect on total population adds or subtracts its quota, but leaves the original rate practically undisturbed, the result may be expressed similarly to (31), i.e., (39) Pt = Po ieP« + y) {qt)^^^^^\ and if as supposed in section 23 the migration be itself influenced at every moment by the magnitude of the population, (39) will become — (39a) Pt =Po ep< {1 + •ij(g<)± »»+»»; TYPES OF POPULATION FLUCTUATIONS. 25 28. The utility of the exponential curve of migration. — ^Formulae (20) to (31b) are serviceable, when the population has to be determined by taking into account the rate of migration determined only at several suitable occasions, the intermediate migration being supposed to conform to the exponential curve assumed to represent all values intermediate to those determining it, and all future values so long as it is apphed. 29. Fluctuation of annual periodicity. — ^The instantaneous rate of increase of the population of any country, at least where the population is at aU numerous, must, during the course of the year, indicate a yearly period, since both the migration rate and the birth and death rates have, in general, a characteristic annual fluctuation. There is sometimes a difference, however, between the migration fluctuation, and that due to births and deaths, for the former, owing to local circumstances, is some- times conflned to a part of the year only, while the two latter extend over the entire year. The scheme of expressing long periodic fluctuations has already been indicated, viz., in equations (16) to (19). Continuous fluctuations of short periods may with advantage be put in the form — (40). . . .Pf/p = 1 + Oq + ai sin (ai + fiit) + a.^ (sin a^ + fx^t) + etc. where jx\ and /Aj are whole numbers or proper or improper fractions, deflning definitely ascertained periods, and where, as before, we must necessarily have — (41) ttj = —Z asma; see section (17) ; or yet again, if the true period is not known and a curve known by experience is to be empirically reproduced, then we may put 2tt 2tt (42). .pt/p^ = 1 + tto + «! sm (ai + — + a^ sm 2{a^ + -t) + ttg ain 3(a3-| t) -\- etc., the unit of t being the period (e.g., one year) embracing aU the fluctuations to be reproduced in the period following. 30. Discontinuous periodic variations of rate. — We may assume that the continuous rate is any function of t, i.e., pt = (f>{t) say. Suppose that superimposed on this curve, there is a migration effect existing for parts of the year only, reappearing at the corresponding times in each following year. Let us suppose further that in the intervals, there is no variation of rate through migration, the fluctuation being fully expressed by {t) above. Then, provided that suitable values are given to the constant oSq to the amphtudes ai, a^, etc., and to the epochal angles ai, a.. 2, etc., the fluctuation of rate may be represented by such an expression as — 277 (43) |0j//>„ = ^(«) ± V [Wo + ai sin (ai + — - «) + etc.J. 26 APPENDIX A. the + sign denoting immigration effects, and the — sign emigration effects. For the final term will have no real values when the quantity under the radical sign becomes negative : a^ must of course satisfy the conditions expressed by equation (17) hereinbefore. Similarly, fluctuations of other character may be represented by — (43a). . . . p^/p^ = cf>(t) ±V{ao + «! sin (ai + t/Ti) + etc. } or again by — (43b). . . .p^/p^ = (t) ± V{at +bt^ +ct' + etc.) Since only real values can have any meaning the expressions under the radical sign in (43), and (43a) and (43b) are discontinuous, the discon- tinuity extending from each value of t where the value of the expression changes from + to — , to where it changes from — to + again. 31. Empiiical expiessiou for secular fluctuations of rate. — ^For the purpose of prediction it is usual to deal either with mean population or the population at a particular date, say the end of the year. The fluctuations of rate may be empirically determined from past records and put in the exponential form, viz., (44) Pf/p = 1 + Tjt* + "»« + «t'+ etc. 7], k, m, n, etc., being integral- or fractional, positive or negative. Or again, it may be expressed in the form — (45) p^/p^ = 1 + ai + j8<2 + yt^ + etc. or yet again in the form — (45a) p^/p^ = \ -\- atv + ^ta + ytr + etc. in which p, q, r, etc., are in ascending order of magnitude, but not re- stricted to integral values. The fitting efficiency of this latter form is much greater than where the indices are restricted to integral values, ^ but the determination of the constants a, j8, y, etc., andp, q, r, etc., are not so convenient. 32. Growth of various populations. — ^Populations increase when the additions by birth and immigration together exceed the deductions through death and emigration together. The rate of increase differs greatly as between country and country, and differs from decade to decade, so that it cannot be regarded as in any sense uniform even for short periods of time. This is evident from Fig. 3, in which the growth of the popula- tions of a larger number of countries is shewn by their progression every decade, and is still more obvious in Fig. 4 (shewing their logarithmic homologues) by the changes in the slope of the Unes. In the following table, the populations, given in millions and decimals of a miUion, are those shewn on Fig. 3. 1 Obviously, since both the coefficients and indices are at our disposal, it is easy to see that attempts to apply (45) to the curve y=atP, where p is a proper or im- proper fraction, are invalid. It is also invalid for the curve y=atP + btP+a + etc. TYPES OF POPULATION FLUCTUATIONS. 27 The Populations of Various Countries from 1790 to 1910. The scale for the lower part of the figure denotes ten times the numbers of the scale for the higher part. The predicted population for the United States was based on the assumption that the rate for 1790 to 1800 would be maintained constant. On the scale of the figure this curve substantially agrees with the prediction by Elkanah Watson in 1815. 28 APPENDIX A. Table n. — Populations in Millions, of Various Countries. Years. COl IHTRT. 1790-9. 1800-9. 1810-9. 1820-9. 1830-9. 1840-9. 1850-9. Commonwealth .002 .005 .01 .03 .07 .19 .41 United Kingdom 15.90 1 17.91 1 20.89 1 24.03 1 26.71 1 27.37 Scotland 1.61 1 1.81 1 2.09 1 2.36 1 2.62 1 2.89 Ireland . . • • .. 5.40 1 5.94 1 6.80 1 7.77 1 8.18 1 6.55 Austria 15.59 16.58 17.53 Belgium 6 4.34 6 4.53 Denmark .93 4 1.22 1.28 1.41 France 26.93 1 29.87 1 31.89 1 33.40 1 34.71 Germany 23.18 6 24.83 2 27.04 1 29.77 32.79 2 35.96 Hungary 7 13.77 Italy 6 18.38 5 19.73 H 21.98 « 23.62 8 24.86 Norway .88 5 1.05 5 1.19 5 1.33 5 1.49 Portugal 8 3.92 Spain , , , 7 16.46 Sweden 2.19 2.35 2.40 1) 2.58 2.88 3.14 3.48 Finland U .71 .83 .86 1.18 1.37 1.45 1.64 Servia .40 U.S. America U 3.93 5.31 7.24 U 9.64 12.87 17.07 23.19 Years. CODNTRT. 18 60-9. 1870-9. 18 80-9. 1890-9. 1900-9. 1910-9. Commonwealth 1.15 1.65 2.23 3.65 3.77 4.43 United Kingdom 1 28.93 1 31.49 1 34.88 1 37.73 1 41.46 1 45.22 Scotland 1 3.06 1 3.36 1 3.74 1 4.03 1 4.47 1 4.76 Ireland i 5.80 1 5.41 1 5.17 1 4.70 1 4.46 1 4.39 Austria 9 20.39 22.14 23.90 26.15 28.57 Belgium 6 4.83 5.52 6.07 6.69 7.42 Denmark . 1.60 1.78 1.97 2.17 1 2.45 1 2.78 France 1 35.84 2 36.10 1 37.41 1 38.13 1 38.45 1 39.60 Germany . 1 38.14 1 41.06 45.23 49.43 56.37 64.93 Hungary . V 1.22 15.51 15.74 17.46 19.25 20.89 Italy 2 25.00 1 25.96 1 28.46 1 30.46 1 32.48 34.67 Japan 2 36.70 40.45 44.83 50.50 Norway . 1.70 b 1.82 1 1.99 (1 2.22 2.39 Portugal . 8 4.00 V 4.16 1 4.31 4.66 5.02 1 5.55 Spain V 16.43 7 17.55 7 18.32 18.61 19.59 Sweden 3.86 4.17 4.57 4.78 5.14 5.52 Finland 1.75 1.77 2.06 u 2.38 2.71 3.12 Servia u 1.00 4 1.35 4 1.90 2.16 2.49 2.91 U.S. America 31.41 38.56 50.16 62.62 76.21 93.35 33. Bate of increase of various populations. — ^Fig. 3 and the accom- panying table reveal directly only the relative magnitude of the popula- tions, but not their exact rate of growth. The latter is displayed on Kg. 4, in which (the scale being constant) the steepneas of slope of the line repre- sents the rapidity of the rate of increase. As before mentioned, this rate is very irregular from decade to decade, as would be revealed by dividing the population at the end of each decade by that at the beginning thereof and comparing the numbers ; i.e., by finding and comparing, for example, the values of P„/Po giving those of 1+ r. The rates tabulated here- under are the anrnud rates which, if maintained constant, would produce the populations at the end of the decades ; that is, they are the values of r found from log (1+ r) = (log P„— log Po)/n, where n is the inter- vening number of years. TYPES OF POPULATION FLUCTUATIONS. 29 Bates of Increase of Various Populations, 1790 to 1910. 90 1800 10 30 30 40 1850 60 70 80 90 1900 10 F denotes Finland ; N, Norway ; S, Servia. * The logarithms for Australia, Denmark, Finland, Ireland, Norway, Scotland, and Servia are shewn on the right of the figure ; for the others, on the left. Fig. 4. 30 APPENDIX A. Table in. — Annual Rate of Increase per 10,000 of Population of Various Countries. Approximate Decade. C'OLXTRY. 1790 to 1799 1800 to 1809 1810 1820 to to 1819 , 1829 1830 to 1839 1840 to 1849 1850 to 1859 1860 to 1869 1870 to 1879 1880 to 1889 1890 to 1899 1900 to 1909 C'wealth . . V. K'dom* Scotland* Ireland* . . Austria Belgiumt •• Denmark* France* . . Germany . . Hungary . . Italy Norway* .. Portugal . . Spain Sweden Finland . . Servia Japan* U. States . . 976 71 157 306 829 120 118 96 52t 43(a) 74(d) 21 36 315 1124 1 764 155 141 145 1 122 136 ' 134 52t ' 66 143 114 — 79(W 74(d)| 126 73 1 111 321 150 291 293 1052 106 105 52 62 80 § 46 108 83(c) 112 87 57 2i«\ 785 24 99 -225 56 43 97 39 77 114 103 124 311 1095 56 57 -122 152 64 127 32 66 41** 133 104 65 309 370 85 94 —70 41t 96** 107 7 74 86** 42 69 20 sot 77 12 190** 206 308 103 108 —45 41t 96** 102 36 108 17 92 56(a) 58(c) 30 1 92 153 348 267 351 79 75 —96 77 96 97 19 89 104 68 56(a) 66 45 145 216 116 224 180 95 104 —53 90 98 122 8 132 64 64 122 75 33((!) 73 131 143 110 198 163 87 63 —16 89 104 127 30 142 51 58 74 92 67 71 142 157 118 205 * Add 1 year to date for proper decade, t Add 6 years up to 1860 inclusive, t Kate for 20 years. ** Bate for 14 years. § Rate for 6 years, (a) Rate lor 16 years. (6) Bate for 9 years, (c) Rate for 13 years, (rf) Bate lor 24 years. 34. The population oi the world and the rate of its increase. — ^In (iealing with the magnitude of the population of any country and the rate of its growth, the most general comparison is that made with the entire population of the world and its rate of growth. This, however, is not well siscertaiaed. Recently, for example, the estimate for China's population has been reduced over 100 millions. The following table gives results of different estimates : — Table IV.— Estimates of World's Population.* Year. Authority. Estimate (Millions). Year. Authority. Estimate (MiUions). 1660 Riccioli 1,000 1813 Graberg v. Hemso 686 1685 Isaak Vossius 500 1816 A. Balbi 704 1740 Nio. Struyok 500 1822 Reichard 732 1672 Riccioli 1,000' 1824 G. Hassel 938 1742 J. P. Sussmilch 9S0 to 1,000 1828 G. Hassel 850 1753 Voltaire 1,600 1828 I. Bergius 893 1761 J. P. Sussmilch 1,080 1828 A. Balbi 737 1789 W. Black 800 to 1,000 1828 Balbi* 847 1804 Malte-Brvm* 640 1833 Stein 872 1804 Volney 437 1838 Franzl 9S0 1805 Pinkerton 700 1838 V. Rougemont 850 1805 Fabri 700 1840 OmaUus d'Halloy 750 1809 G. Hassel 682 1840 Bernoulli 764 1810 Abnanach de Gotha* 682 1840 V. Roon 864 1812 Morse 766 1843 Balbi 739 TYPES OF POPULATION FLUCTUATIONS. 31 Table IV. — Estimates ot World's Population*— oonSinited. Year. Authority. H. BerghauB Estimate (Millions). Year: Authority. Estimate (Millions). 1843 1,272 1880 Behm & Wagner 1,456 1845 Miohelot* 1,009 1882 Behm & Wagner 1,434 1854 V. Reden 1,135 1883 Behm & Wagner* 1,433 1889 Dieterioi 1.288 1886 Levasseur* 1,483 1866 E. Behm 1,360 1891 Ravens tein* 1,467 1868 Kolb 1,270 1896 Statesman's* Year 1.493 1868 E. Behm 1,375 1903 Jurasohek* [Book 1,512 1870 E. Behm 1,359 1906 Jurasohek* 1,538 1872 Behm & Wagner 1,377 1910 Annuaire Statistique 1873 Behm & Wagner 1,391 d. I. Rep. Fran9ai8e* 1874 Behm & Wagner* 1,391 Jurasohek* 1,610 1878 Levasseur* 1,439 1913 Knibbs* 1,632 1878 Levasseur 1,439 1914 Knibbs 1,649 * These will be found on the graph, Fig. 5. This table shews, for the period 1804 to 1914, rates of annual increase ranging between 0.0015 and 0.0121^ and averaging about 0.00864. We may obtain some idea of the present rate of growth by taking the weighted mean of the rate for the known countries ; that is, each rate of increase is weighted according to the population. In this way, it is found for the quinquennium 1906 to 1911, and for the group of countries in the Table V. hereinafter, that the general result is a rate of increase of 0.01159 per annum, or 1.159 per cent, of the population. Table V.— Annual Increase per 10,000 Population for the quinquennium 1906-1911. Country. Rate Years t Country. Rate. Yearst Ireland — 6 Switzerland + 121 57.6 France + 16 436 Netherlands 122 57.2 Jamaica 28 248 Denmark 126 66.4 Scotland 65 126 Grerman Empire 136 61.3 Norway 66 105 Finland 143 48.8 Belgium 69 101 Rilmania 148 47.2 Italy 80 87 Servia . . 155 45.1 Sweden 84 82.9 Chile 156 44.8 Hungary 84 82.9 United States . . 182 38 4 Austria 86 80.9 Commonwealth 203 34.6 Spain 87 80.0 New Zealand . . 256 27.4 England and Wales 104 67.0 Canada . . 298 23.6 Japan 108 64.6 Ceylon 120 58.1 Weighted Average* . + 115.9 60.1 * Weighted average according to population. f Years necessary for the population to be doubled in value at the rate indicated. The number of years n in which a population, increasing at the rate r, is doubled, may be very readily computed thus : — (1+r)" = 2 ; therefore n log, (1 + r) = log, 2 = 0.693147 consequently n == 0-69316 0.69315 log, (1+r) r(l- + 3-.) but when r is very small we may neglect powers higher than the second (that is ^ in the brackets) ; hence 0.69315 ,, , , , ... 0.693 , „ , ., n = ; — ( 1 + i»-), sensibly, = + 0.347- ' On taking the mean of Levasseur and Behm & Wagner, and again of Levasseur and Ravenstein. 32 APPENDIX A. Either this rate of increase must be enormously greater than has existed in the past history of the world or enormous numbers of human beings must have been blotted out by catastrophes of various kinds from time to time. For, putting the present population at 1,649,000,000, at the average rate of increase this number would be produced from a single pair of human beings in about 1782 years,* that is to say, since A.D. 132, or since Salvius JuKanus revised under Hadrian the Edicts of the Prstors. Even the rate given by the world -populations 1804 and 1914, viz. (0.0086) gives only 2397 years, carrying us back only to B.C. 483, or since the days of Darius I. of Persia. The profound significance of this fact, accentuated also by the extraordinary increase in the length of life (expectation of life at age 0), which has revealed itself of recent years, is obvious when the correlative food requirements are taken into account. The resources of Nature will have to be exploited in the future more successfully than in the past to maintain this rate of increase (0.01159), which doubles the population every 60.15 years, and would give for 10,000 years the colossal number 22,184. with 46 noughts (lO*^) after it. This number is so colossal that it is difficult to appreciate its magnitude. Assuming the earth to be a globe of 3960 miles radius, of a density 5.527 compared with water, that water weighs about 62J lbs. per cubic foot, and that a human being weighs on the average, say, 100 lbs. (7 st. 2 lbs.), the actual mass of the earth would be equivalent only to, say, 132,265 x 10^* persons; that is, it would require 16,771 X 10*' times as much " matter " as there is in the earth. Or, to consider it as a question of surface, allowing 1 J square feet per person, the earth's entire surface area would provide standing room for only 36,625 x 10" persons. That is, the population would be 60,570 x 10** times as great as there would be standing room if the whole earth's surface were available. It is evident from this that the rate of increase of human beings must have been more approximate to the rate for France at the present time, if the earth has been peopled for 10,000 years : the French rate, 0.0016, would require 12,842 years to give the present population from a single pair. This rate, however, would give a population of only 17.55 millions in 10,000 years. The foregoing analysis of the effect of the rate of increase, with which we are familiar, establishes the fact that the rate must have passed through great changes, and could not have been maintained for any long period, either at its present average, or that characteristic of the last century. (See II. § 12, 13, 14 and 15.) It is not improbable that the rate of the last quinquennium will not be long maintained ; and it is * Thus dividing by 2, we have 824,500,000 = (1.01159)" where n is the immber of years, that is, n = 1.782. TYPES OF POPULATION FLUCTUATIONS. 33 certain that however great human genius or effort may be, in enlarging the world's food supplies, that rate cannot possibly be maintained for many centuries. The contention of Malthus is thus placed beyond question, from a different point of view. The analysis also suggests that there are probably great oscilla- tioMs of the rate of increase, but since accurate records date back for so comparatively short a time, no general indication of their character can be given. THE WORLD'S POPULATION, 1806-1914. 17 16 15 14 o 13 S 12 ,(!'>- y '\ [o/ 7 / ■ / / '/ ,^ / / '/ >' / »' V / ./' f^ J V / / ^ " .S 11 a 10 H 9 8 7 10 20 30 40 50 60 70 SO 90 1900 10 Fig. 5. In Fig. 5 some of the estimates are shewn by black dots. The firm line drawn among these dots is intended to represent the probable development of the world's population. The thin broken line among the dots, though adhering more closely to the various estimates, is, however, of doubtful probability. The lower broken line represents a population increasing at a uniform rate from 640 millions in 1804 to 1649 milhons in 1914 ; i.e., 110 years. From the figure it is evident that the rate of increase in the early part of last century has fallen off, and the world's population increase will continue at a less rapid rate. Thus it is beyond question that there have been oscillations of rate, but their period has not yet been determined, and is perhaps not determinable, owing to lack of data. One thing is assured, viz., that the present rate of increase cannot be maintained for any lengthy period. in.— DETERlffllNATION OF CURVE-CONSTANTS AND OF INTER- MEDIATE VALUES WHEN THE DATA ARE INSTANTANEOUS VALUES. 1. Creneial. — The data of statistics are usually to hand in two essentially different forms, viz., [a) instantaneous values or numbers which are true at a given moment ; as, for example, the population of a country at a given instant ; and (b) group values or numbers belonging to some particular interval of time, as the number of births per month, or per annum, for a population of given magnitude. Some indications have already been given of suitable formulEe for instantaneous values, and in one or two instances the mode of deducing their constants was also furnished. We proceed to consider the solution for the .constants of equations which are appropriate for representing instantaneous values. In mathematical language, if «/ = / {x), then having chosen the form of the function, it remains to determine its constants from the data. In the case of group values, the equations must denote the value of the integral of the function between given limits, and the problem has special features, the study of which will be undertaken later (IV.) There are a considerable number of cases of importance, some of which are aperiodic, and others periodic. 2. Determination of constants wheie a fluctuation is represented by an integral function of one variable. — ^When, as is ordinarily the case , the data consist of values corresponding to equal intervals of time, as, for example, the population at the end of each quarter, at the end of each year, or at the end of each ten years, etc., the fluctuation may be empiric- ally represented by the equation. (46) y (or — )=a +hx + cx^ + dx^ + etc., in which, in the above illustration, x represents time. In this case the number of constants to be determined wiU depend upon the number of instants for which we have data. Two classes of cases arise, viz., (i.) oases in which the data furnish the initial value ; (?/„), that is, a in the equation above, and (ii.) cases in which the initial value is not furnished, but is for a unit interval of time before the first result available. In other words, in the equation above, we require a series of solutions for the cases where a has a fixed value, including zero, and when it is undeter- mined ; or what is the same thing, when we have either y, or y^ as the « CURVE- CONSTANTS AND INTERMEDIATE VALUES. 35 initial datum. If we have the value of y„, then n subsequent points will require an integral equation of the rath degree. If not, n points, including 2/1, necessitate an integral equation of the (w-l)th degree.^ If h denote the common interval of time (represented by distance between the ordinates), the values of y in the preceding expression are : — (47). .2/o=a ; 2/i=a -\- bk -{- ck -\- etc. ; y^=a + 2bk + 4 cfc^ + etc., etc. If a be kmyum, then by subtracting a from the values of y we have a series of equations identical with the above in which — (48) 2/0= ; 2/1 =bk +ch -\- etc. : 2/3 = 2bk -{- ^cJc"^ + etc. We deal first with the cases where a is known and assume that the ordinates 2/1, 2/2> ^te., are the values computed from the axis X, so taken that a = 0. Then the following formulae, in which yi is denoted by i, 2/2 by ii., etc., may be readily deduced : — Formulae when j/g = = a. For- mula. Data. Value of 6. Value of c. (49) Vi ^(1.) (50) j/iandj/s, -^ (4i.-u.) 2^ (—21. + 11.) (51) »i to Vs -gjr- (18i.— 9ii.+2iii.) 25r(— 51.+411.— 111.) 52) !/ito»^ ^ (48i.— 36ii. + 16iu.— 3iv.) 24P (—1041. + 1141i.— 56111. + lllv.) (53) Vi to Vb -^ (3001.— 30011. H-200iii.—75iv. + 12v.) 24j;«(-154i.+214il.-156111. + 611v.-10».) For- mula, Data. Value o{ d. Value of e. (51a) »i to 2/3 ^ (3i.— 3il.-t-Ui.) (52a) !/ito Vt iW' (18i— 2«i.+l«u.— 31v.) 2^. (-«.+«".— 41U.+1V.) (53a) Vi to Vs 2Sk' ( + 71i.-118ii. + 98111 .-411v. + 7v.) 24p(— 141. + 2611.— 24iii. + lliT.—2v.) (53b) Vitove Value of / = f25jfc5 (+51.-1011. + lOUl. — 51v. -I- Iv.) Instead of using the value of the ordinates it is often convenient to form the successive differences, and then the coefficients b to f can be expressed very briefly in terms of the leading differences of the ordinates, corresponding to the values 0, k, 2k, etc., of the abscissa. In the follow- ing results, Z)i, D^, etc., represent the successive leading differences, that is, remembering that y^ = 0; Di = 2/1 ; D^ =y^ — 2yi ; Dg= y.^ - 3^2+3 2/1; etc.; etc. 1 See II., § 6, formulas (9) to (13a). 36 APPEISTDIX A. For- mula. Data, i Value of b. Value of «. (54) (55) (56) (57) !/i &y, i/i to I/s !/ito Vt -2J- C2Ji — i),) 6il; (6i)i — 3Z), + 2D3) (58) j/i to j/s J2J (12iJi— 6i)a+4D3— 3Z).,) gQj (60 2)i—30iJj + 20i),—15D4+ 121)5) 2J2 i'^ 2jr (fl,— jD.,) 2^. (12D,— 12Da + lli).) 2^, (12D,— 127)3 + 112).— lOB,). For- mula. Data. Value of (f Value of e Value of / (56a) VltOJ/3 W »> (57a) Wi to v^ J2P (2D3— 3D,) 24F' •"* (58a) Vx to ;/5 24P (4Z)3— 62), + 72»5) 2jj. (X>4— 2 D.) 120*' ^= Secondly, when a is not known, and the ordinates yi, y^, etc., are distant Ic, 2k, etc., from the Y-asds, we may readily extrapolate a by means of the differences. For the coefficients are simply the numbers of Pascal's triangle (the binomial coefficients) with the first omitted. Thus, the small Roman figures denoting suffixes only, we have — (59). . . .a =2i. — ii. ; or 3i. — 3ii. + iii. ; or 4i. — 6ii. + 4Jii. — iv. ; or 5i. — lOii. + lOiii. - 5iv. + v. ; or 6i. - 15ii. + 20iii. — 15iv. + 6v, — vi. for two, three, etc., ordinates given. When a is found, the problem resolves itself into that for which solutions have already been given, or it may be directly solved. For five ordinates given, not including a, we have, for example : — Formulee. (60) a= 5i. — lOii. + lOiii. - 5iv. + v. (61) ^= life (- '^'^i- + 214"- - 234iii. + 122iv. - 25v.) (62) c = 24P (71i. - 236ii. + 294iii. - 164iv. + 35v.) 1 (63) d= J2P (-71+ 26u. - 36iii. + 22iv. - 5v.) (64) e= ^Ji. - 4ii. + 6iii. - 4iv. + v.) CURVE-CONSTANTS AND INTERMEDIATE VALUES. 37 The values of the coefficients in terms of the leading differences (D) are : — (65) a = 2/1 - Z»i + D, - 2), + 2), (66) ^ = 1^ (12^1 - 18^2 + 22i), - 25Z)J (67) « = 2^-2 (^^^^ - ^^^^ + ^^^^^ (68) ^=1^-3(2^3 -5i)J (69) e = _L D, 3. Evaluation of the differences from the coefficients. — ^When the coefficients of an integral function, viz., one of the form (46), are known, and it is desired to ascertain the values of the ordinates y^, j/i, y^, etc., the common interval between which is k, they may be rapidly computed from differences, viz., from x=Q and y=a, together with the following leading differences :— Factor into numerical coefficient below — Differences, bk + ck^ ^ dk^' -\- ek'^ + fk ^. Di 1 1 1 1 1 D. 2 6 14 30 D. 6 36 150 D. 24 240 D. 120 (70) For equations of less degree than the fifth the table still serves since /, e, etc., may be put equal to 0. 4. Subdivision of intervals. — When the ordinates are to hand for a series of intervals, those for a subdivision of these into m parts may readily be determined by computing a new series of lesser leading differ- ences, d say, using those, D say, of the original intervals, as a basis, as follows : — Differ- D, D^ D, D4 D,, ence. m m' m^ m* m° I- , m--l 2m^-3m+l 6m'-llm'' + 6m^l 2im*~50m^ +35m^-lQm+l d^ = l_ + (71) 2 6 24 120 llm^ — 18m + 7 10m^ — 21m' + Um—3 d^= 1 - (m - 1) + 12 12 3to — 3 7m' — l2m + 5 dj = . . 1 - 2 + 4 d^= ■■ ^ ■■ 1 - 2(m- 1) d.= .." .. 1 38 APPENDIX A. That is, we divide the wth difference by m", and this factor is multiplied into the expression opposite d with the proper suffix. The sum of the terms gives the leading difference in the corresponding d in the first column.'^ When an interval is divided into 2, 4, 8 or 16, etc., parts, the ordinatea may be found by successive " interpolations into the middle."^ 5. Evaluation of constants of periodic fluctuations. — ^The general empirical formula for a periodic curve which may be made to fit given data is — (72).. 2/ (or — )=a+6sin(j8+a;)+csin2 (y +a;)+rf sin 3 (S+a;)+ etc. in which the number of terms to be taken depends upon the given data, and is sufficiently illustrated hereunder. When the values of y are given only for the beginning of the recurring period of the total fluctuation and at the end of the first half period, we have — (73) y =a +bsm(p + x) (74) a = i (2/0 + yi) ; 6 sin j8 = 1(2/0 - Vi) Hence if any definite value be assigned to 6, j8 becomes determinate ; or if to j8, 6 becomes determinate. When there are values of y for the beginning of the total period, and for the instantjS one-third and two-thirds of the period, then we have, writing — y^ - « = rj, ; 2/1 — a = ri ; etc. (75) a = H,y^ + y, -f 2/,) ; tan ;8 = ^^ll-i ri — r^ a and ^ being found, we have — (76) b = r„ cosec ^ Using r„ throughout to denote 2/n — a, where n is 0, 1, 2, etc., we have for four values, viz., at the beginning of a period and at one -fourth, two-fourths, and three -fourths of the period, from the beginning — (77)....a = i(2/„+2/i +2/2 +2/3); tan j8 = ^^ and in the expression for tan ^, we may write r for y. These quantities being found, we then have — (78) b = Tg cosec ^ = rx sec ^. For fifth periods, that is, for equidistant ordinates to 4, the formulae for the constants are : — (79) y=a+b sin {^-\-x)-\-c sin 2 (y+x) 1 See Text Book Iiistitute of Actuaries, Pt. II., Ed. 1902, p. 443. CURVE-CONSTANTS AND INTERMEDIATE VALUES. 39 and the solution gives — (80) a = ts:y. 2 sin 360 [^ (81).. tan /3 = (82).. •2 cos 36«(r., + r,)} r, — r., + 2 cos SB" (r, — rj I _ cosec j8 ! r„ — 2 cos 36° (r^-\- r^ ) ) 3 + 2 cos 72«- (83) tan 2 y = ^ sin 36'']r„(2 + 2 cos 72")+ 2 cos 36" (r,+ r,) ! r,— r^— 2 cos 36" (r^— r J (84). c =cosec 2y. r J — r^ — 2 cos 36*' (r^ — r^ r„(2 + 2 cos 72") + 2 cos 36° (r 3 + 2 cos 72°. The values of sin 36°, cos 36°, sin 72°, and cos 72° are respectively : — i V'(10— 2 V5) = 0.5877853 ; J (V5+1) = 0.8090170 ; J ^7(10+2 V5) = 0.9510565 ; and J (^5 — 1) = 0.3090170. For sixth periods, that is, for equidistant ordinates to 5 the formulae for the constants are : — (85). (86). (87). (88). (89). . tan j8 = V3 {r, - Ia+IJ n + '•2 - 1^4, .tan 2 y 6 = 1 cosec ^ {r^ — r V3 (r, + r,) sec 2 y ■'■4 +'■5) c = (n— J-g +>'4— *■.-,) 2V3 The solution for twelfth periods is sometimes required as, for example, when values are to hand for the beginning of each month. Denoting as before the remainders y„ — a by >•„ we have — (90). •« = 1-2 ^" y- Then making the following additions for brevity of working, viz. — in =»-o +r3 +rg +r9 ; No = r^ ->r r^—r^—Tg '•i + '•4 +'■7 +»'io ; -^1 = '•1 + »"7 — '•t — »"io (91)., ■'0 -i^2 = »"2 + '•s + »■« + '•ii ; N^=r^+r^~-r^—r^ ^0 = '■0 + '■4 + '"s ; -Bo =»'o + ''s— »"6 — '"8 Jlfi = n + r- H- rg ; i^i = ri + t-j — r, — r^ -M'2=r2 + re +r,„; iJ^ = r^ + r^ — r^ — r^„ ■^3 ='■3 +^7 +'■11; -'^s ='"3 + '"5— »"9— '■11 (92). (93). (94). R. VS-Rg 2iZo + V3i?i 6 = tS cosec ^ (2i2o + -/S-R, — -Rj 2^1+ 21^0 --AT, V3i?j) 40 APPENDIX A. (95) " = 173 sec2y(i\ri+ 7\^,) M M (96)....tan3S=fP-|j (97) rf = i cosec 38 (Jl/o — M .) (98) taii4e= y^\ 111 — I12 (99) ^ = i cosec 4eip (101) /= J, cosec 5? (2i?(, — V3i?] — iJa + VS-Rs) 6. Constants of exponential curves. — The case of a curve of the type (102) y = 1 ± ^^i"**"' see equation (20), has already been sufficiently considered : its constants can be found as shewn by formulfe (23) to (30) ; and also that of the type, see equation (32) 1 (103) y =1 + Tji^™±««; see formulEe (35) to (38). In general, curves of this type may be solved by forming the equations y' = y — 1 and taking logarithms when we get such forms as — (104) M = e + log i (± m + ni) and u — e -, ^ — ~ ° ± m ± Mi solutions for which have already been sufficiently indicated. As this process of taking logarithms is the key to many solutions, we now refer more fully to the matter. The essence of this method of solving is that if a series of values on the axis of abscissae be taken in geometrical pro- gression, their logarithms are in arithmetical progression. Thus, ^ being log X, we have — Quantities = x ; kx ; k^x ; k^x ; etc.; Logarithms of same = x '> X+^! X+^fc; ;)^+3fc; etc. Hence the problems of solution are reduced to those of the examples illustrated by formulas (46) to (71). 7. Evaluation of the constants of various curves representing types of fluctuation. — ^The evaluation of the constants of various curves can often be effected by taking suitable ordinates to the curve and solving from their logarithms. This is illustrated in the following series of equations : — ■ (105).... 3J = (Te^^'" = QTe^'i^' = €S^^"' = QTJ^ M' CURVE-CONSTANTS AND INTERMEDIATE VALUES. U We have on taking napierian logarithms — (106) Y = AX"" + O = AM"" + G in which log ^ = F ; log y[ = C ; log Jl. = A ; log Z = a; ; and log M = m. The second curve may be called the first logarithmic homologue of the first, and the first the first anti-logarithmic generatrix of the second. Subsequent curves may be similarly defined as the second logarithmic homologue, etc. Yet again, if G be zero, we have on taking the logarithm of this last expression — (107) y = a + mx, in which log Y =^ y ; and log A= a. This will sufficiently illustrate the matter. Several examples of solution wiU be given of important curves for representing fluctuation. In the curve (108) y = 4 + Bx'^ "If ^ = ; then the solution is found at once from any two values of y and of X. For we have — (109) log y = log B -\- m log x. On Fig. 21 hereinafter, these curves are shewn by thick lines for positive values of m, and by thin lines for negative values. If, however, A be not zero, then we must take three values of y for abscissas of the value x, xk, xk ^, when it may easily be shewn that — (110) . . , . y^-y^ = k-; ovn = ^°g ^y^ - y4^-l°^-^^=:yj-^ 2/2 — yi log k The curve (111) y = B + Ge'"> can be solved by taking the values of y for x, x + k, x -\- 2k, for ,/ . ,/, n [p a(x + 2Jc) g a{x + m ^'-^'^> y^ — y^- c [e "(^ + *) — e«^] Consequently putting 7.,.^^ for the left-hand expression, and writing 2.3025851 for the modulus for changing common into Napierian logarithms (113) a = ^log,o F,,, When a is found the solutions for B and G are obvious. Curves of the equation e^ are shewn by thick Unes on Fig. 22 hereinafter, and those of equation 1 / e^ by thin lines. The exponential curve — (114) y = ^ + .Be"^' can be solved if A be zero, or if A be known, and a new series of y' =y— A be formed. Thus A being zero, (114a) log 2/ = log 5 + nxP log e. 42 APPENDIX A. Hence, as before, taking three values of y for x, xk, xk ^, the solution is — (115) P = A • l°g /[^-^-^ -^^} log A; ^ \log2/2 — logyii (116) ^ = ig^-i"gyi ' xv{ki>—l) log c (117) log B = log 2/1 — wxP log e. These curves are shewn for Fig. 23 hereinafter, for various values of n and p. The curve — (118) 2/ = 4a;™e"^'' is solved by taking four ordinates, viz., for x, xk, xk^, xk^, when the solution becomes 1 — (119) ^ = 1 . log i iogy.-2iogy 3 +iogy4 | log *: " t log 2/1-2 log 2/2 + log 2/3 ) using common logarithms. Then M denoting log e, we have also — (1201 n = (^Qgyi ~ 2 log 2/2 + log 2/3) _ (log 2/2 - 2 log 2/3 +Iogy4 ) ^ '" MxP(kP-l)^ ~ MxPkP{kP - l)^ (121) m = (log ^2 - log yi) - Mn xP (kv - 1). log k There are obviously two other possible formulae for m. (122). . log A = log 2/1 — m logaii + Mnxv the value of M being 0.4342945. Three other formulae are also possible for A. For further formulae see (150) to (153) later ; see also Figs. 21 to 27, hereinafter, for the forms of the curve. 8. Polymoiphic and other fluctuations. — Monomorphic or rather unimodal curves disclose a single maximum (or minimum) value. But there are fluctuations which are polymorphic or multimodal. These may be regarded as compounded of monomorphic curves. PracticaUy their dissection is best effected by the graphic methods of analysis. In general any curve can be represented with great accuracy by either (123) y =a + bxP+ cxi + dx" + etc., or by (224) Y=: ga + bxP + exv + etc. where p, q, r, etc., are not restricted to integral values. The latter curve is reduced to the former by taking the logarithm ; thus, 2/ = log« Y. To solve for the constants we must have six points besides the origin. If the value of a be known, the curve can be reduced to one passing through the origin by subtracting a. Then we take values of y for x, xk, xk ^, xk *, etc. For the case for terms in p and q only, we can proceed as follows : — 1 For a more complete study of the curve, see "Studies in StatriBtical Repre- sentation. On the Nature of the Curve," above given, viz. (118), by G. H. Knibbs Joum. Roy. Soc, Vol. XLIV., pp. 34] -367, 1910. CURVE-CONSTANTS AND INTERMEDIATE VALUES. 43 By writing L for bx'P and M for cx^ , and a for k^ and ^ for k^ ; we have — (125).. yi = L + M; y^ = La+M^; y^ = La^+M^^; y^^La^+M^^ Hence by eliminating L and M from the first three and from the last three equations, we have respectively— (126a). 1 2 ^2 2/1 2/2 = 0; 1 1 2/2 =0. . (1266) a ^ 2/8 a^ ;8'' 2/, Consequently a and ^ are the roots of- (127). 1 2/1 2/2=0 i 2/2 2/3 P 2/3 2/4 Thus the two values of f in the equation — {128)..^^^i{a + p)+ap = ^^yiy^-yl)+i{y,y,-y,y^)+(y2y^-lfl)=0 are the values of kP and /fc«. And since k is known, the solution is to hand by taking logarithms. • The solution for three indices is similar. The six equations can be written — (129) 2/^ + 1 = La™ + ilf;8'» + iVy'» and a, j8, and y ; that is ki>, M, and k^, are the roots of the equation. (130). 1 2/1 2/2 2/3 2/4 2/2 2/3 2/3 2/4 2/4 2/.-. 2/0 2/e = ^ 2, ^ and which may be expanded in the form — (130a) 4iP — 3^2^ + 3^3^ — ^^ = where Ai, ZA^,ZA^ and A^ are the minors respectively of | ^ 1 in the determinant. If the constant a is included in (123) or (124), the solution is more tedious. We must then have seven values of y. Thus — (131) y„ + 1 = o + -Z^a™ + M^"' + By^ (131a) 2/™ + 2 - 2/m + 1 = i'a™ + M'^^ + R'y^ the accented values being L'— L [a — \); M' = M (^ — 1) ; etc. Thus a, j8, and y are the roots of — 1 ^2 - 2/1 2/3 - 2/2 I 2/3-2/2 2/4 - 2/3 P 2/4 - 2/3 2/5 - Vi P 2/.5 - 2/4 2/6 - Vs 2/7-2/6 Writing Fj, Fg' -^i' ^o for the minors of P, ^*, ^, and 1 in the determinant, the equation becomes (133) i'Y,^i^T,+^Yi- y„ = 0. (132). 2/4 - 2/3 2/6 -2/4 2/e - y.. = 44 APPENDIX A. It will be seen from the preceding examples that when, a-s regards their indices, the equations are not restricted to integral values, the Acting power of the curve is enormously increased. To fit seven points with integral indices we should have to have an equation of the sixth degree. 1 Pigures 6 and 7 furnish graphs for simple cases with two indices only. From these graphs, which also are for integral values of the index only, it is immediately evident that the loci of curves with fractional values must he between the curves drawn. The forms of the curves may, of course, be modified also by varying the coefficients : hence the fitting power of expressions of the type considered obviously becomes very great when the limitation imposed by restricting the indices solely to integral values is abandoned.^ 24 '«~ / / ix -1 X *1 -x^ ^+ ex" / / V / g ^ <■ y \ / / ^^ y ^ ^ t / / t- ^ s^> ^--. -^ / / ts- [^ — $A <5 ^ ^ ■^ '' 'V ¥, ^ y ?^ ^ Vs -^ ^ ^ \ \^ ^r* 1 1___^ — /7 \,\ sc- ^ y \, ^ ^ "^ l_ — ^ '^ V \ \ \ • \ -bx -^+ x* i ■x" I_ CZ *1^ \ 'vj \, \ -«4 3 4 1 2 S 4 Fig. 6. 1 See " Studies in Statistical Representation, III., Curves, their logarithmic Homologues," etc., by G. H. Knibbs and F.W. Barford, Joum. Roy. Soc. K.S.Walee, Vol. XL VIII., pp. 473-496 2 The limitations of Jthe fitting power of the curve are discussed in the paper referred to in the preceding footnote. These limitations, in general, are of no moment in statistical results. CURVE -CONSTANTS AND INTERMEDIATE VALUES. 45 A.a. X » X "-J .-t / 7 ^ « / «? « ^ \\\ / / /. ^ .4r / / /• / >- m\ L ^ ^ - - L 6- vVW L^ .1' 1 — . ~^M ^ 7/ V r-- A w) ^ :zz ^- 4" J // \ \ -ii — - J ^^ ■^ ^ =? ^ 1 ^^" hr" T ~a. ^ / 1 / c .D 5- 4- 3- 2 I 0- A N / y / 1 -- "\ - ^ / / / y 1 \ ^ XI :> L^ ,- / ^l t'/ x: < ^ ^ J >- =i-i ?< ^ «^ \ s /f "^ . i ^ __£ ^ i-; 1 ^^ \i ^ \<^j *r / S ; ' »/ \ \s T^ 4 1 i / \ \ - N<^ >^ I / {b\ + j:' ^> ' x-'^ cx ..l ^^ ' - 1 T Fig. 7. Some special cases of fluctuation will now be treated in dealing with problems treating of fluctuating elements that directly or indirectly influence the aggregate or constitution of the population. 9. Projective anamorphosis.— A symmetrical curve of frequency (or symmetrical distribution) may become asymmetrical by the elements being projectively varied by means of different types of projection (plane or other). This change may be called projective anamorphosis. Any character of a population may be regarded as subject to influence acting 46 APPENDIX A. progressively (or retrogressively) with increase of the measiire of the character in question, as for example, if the influence tending to increase weight (or height) acted more or less powerfully with increase of that character. This would lead to an asymmetric or skew frequency. Thus if a normal frequency be denoted by y =(f) {x) ; a specialised frequency conceived to originate therefrom would he given by y' = f («) <^ (a;). This expression may also be skew, dimorphic, polymorphic, or in fact, what we please, according to the character of/ (x). If/ {x)=mx or m/x, a symmetrical curve is converted into a skew curve. If / (x) have a mode such that it is not identical with that of ^ (a;) the latter will be dimorphic . From this it is seen that the ordinates to a dimorphic curve may be the sum or product of the ordinates to two monomorphic curves. It is not proposed to elaborate just here, however, the general theory of anamor- phosis by plane or other projection. It may be easily seen, however, that a skew curve may be readily derived from a symmetrical one, while retaining the general algebraical properties of the latter, by a projection, from a hne parallel to the axis of the given symmetrical curve, through the curve and on to a plane passing through the axis but inclined to the plane of the given curve. This will be more fuUy considered hereinafter. IV.— SPECIAL TYPES OF CURVES AND THEIR CHARACTERISTICS. 1. General. — ^When the characters of a population have a tendency to deviate in either direction equally, and the number of the population is P, the characters wiU be distributed as the coefficients {^ + ^)'» i.e., as the numbers in Pascal's triangle, which, when m is infinite, becomes the curve (134) y = Pe * ; or say Pe * the first form (viz., that when the power w = 2) being the ordinary probability curve, in which k is the modulus. This type of distribution is but one case of the more general expression which, interpreted in a certain way,i has a cusp for the vertex for values of n equal to or less than unity, and a curve convex upwards for aU values greater than unity, the vertex however becoming more flat as n is increased. The curves graphed are a = axis -J -X r~ '/^\\' f Vf\ _^ ///( \\1\ ^yv\ /// i 1/ ^ ^' 1 '^4 .^ '■ n a. ■^ ^ ^. ■ -■ 3: c^ ' a / \ s. ■^ .. — =- ^^ cT -^ . /> _ bJ ;^ --. — :i_J 1_|. X Asymptote. Fig. 8. Asymptote. The curve y = e~*" is coincident with a from the point Y to a point y = 0.3678781 ; it is then parallel to the X axis. All the curves intersect at this point. Such a distribution is symmetrical, and takes the form in the figure hereunder, Pig. 7, in which curve 'a' shews its form for to = ; 'b' for TO = I ; ' c' for w = 1 ; ' d' for to = 2 ; ' e' for to = 4. When the probabilities of distribution are not equal for possible alternatives, and the probabilities of these alternatives are as p and q, the sum of p and q being unity, then the distribution will be the coefficients oi {p H-g)"*. Ifg'and^ are not equal the curve is not symmetrical, but is of a form Uke Fig. 9 hereinafter. Whether results can be made to conform to a particular tjrpe or not depends on the form of the curve, and 1 That is, BO that e-«" and e-«("+*'i are in the same spatial region, or on the same side of an axis, and are not allocated to different regions according to whether the number (n+Sn) is even or odd. 48 APPENDIX A. in particular on the position of its vertex ; on whether its sides meet the axis of the variable more or less sharply or asymptotically, on whether it is monomorphic or polymorphic, or has one " mode" (is unimodal) or more modes than one {is multimodal). Various types of unimodal fluctuations, commencing and ending with zero values or otherwise, have been given by Prof. Pearson. These are intended to reproduce the group-values of statistical data, under appropriate forms of curves, by a method which has been called the method of moments, the forms of the curves being derived from the normal curve of probability. We shall later refer to these, but remark first of all that the critical elements of the curves representing distributions or fluctuations are as follows, viz. : — (a) the value of the ordinate when the variable is zero ; (6) the values of the variable for which the ordinates become zero ;* (c) or, if they do not become zero, the value of the ordinate when the variable is infinity ; (d) the abscissa of the m,ode, or greatest ordinate, and the value of that ordinate ; (e) the abscissa of the ordinate which equally divides the curve area (as, for example, the abscissa which corresponds to the average value, or the centroid vertical) ; (/) the distance between these two ordinates {d) and (e) (the numerator of the quantity defining the skewness) ; (g) the m£an-deviation of the curve (or denominator of the skewness) ; {h) the abscissa of the point where the curvature changes its sign, (point of inflexion) ; [i) the abscissa of the point of most rapid change of direction of the curve. (a)......2/=/(0); (6) f (x) = ; (c) /(^)=A-orO; id) x„ when df (x) / dx = ; and y„ = / (x„) ; (e) Xa when the value of ^ xf (x) dx -^ jf (x) dx for the range of the variable up to x^ is equal to that for x^ onward ; (f) i^a-^m) {g) Wj = ■\/[xJ (x) dx -^J/(x)^*]' in which x is measured from the position of the mean (x^)- (h) Xi whend^f (x) / dx^ = ; (i) Xj, when d^f (x) /dx^ = 0. * The approach of statistical curves to the axis of abscissae or to the axis of ordinates is, in general, not determined by mathematical considerations, but by a knowledge of the nature of the data itself. For example, the terminals of the curve of fertility (discussed hereinafter) deduced from ex-nuptial births, shews a diminution which may be represented very closely by the niunbers 1078, 154, 22, 3J^, for the ages 16, 15, 14 and 13 respectively, i.e., each number is one-seventh of the number preceding it. Merely mathematically, therefore, it is more probable that these should continue for the ages 12, 11, 10, 9, etc., as 0.45 ; 0.064 ; 0.009 ; 0.0013, etc. Even at age there would, of coiu'se, be still a positive value though small. But physiological knowledge indicates that the earliest arrival of puberty is probably over 10 years, hence 11 would be the earliest age for birth, and the ordinate must be zero. SPECIAI, TYPES OF CURVES AND CHARACTERISTICS. 49 2. Curves of generalised probability.— Prof. Pearson proposes to reduce forms of distribution of statistical facts under a series of seven type-forms of curves, representing what may be called curves of generalised probability,! and much work has been reduced on this system. Fig. 9 Type I. (i.). Pig. 10. Type I. (ii.). Pig. 11. Type I. (ii.a). Pig. 12. Type I. (iii.). Pig. 13. Type II. (i.). Pig. 14. Type n. (ii.) His first type (Type I.) is :- (135). ■Vo =2/(1 + «i (1 - „- ) which may take two other fundamental sub-forms, viz., (136). y = 2/0 ( - - 1) (1 ) ' ; and (137). 2/ = 2/0 (1 - „- ) ( 1 + ;f ) which are represented respectively by the forms in Pigs. 9 to 12.^ When V, Oi and 02 are positive the curve meets the X axis at the distances Oj and a^, see the figures. The abscissa of the mode is and the curve is skew. 1 See his " Contributions to the Mathematical Theory of Evolution." Phil. Trans., Vol. 185 (1894) A, pp. 71-110; Vol. 186 (1895) A, pp. 343-414; Vol. 187 (1896) A, pp. 253-318; Vol. 191 (1898) A, pp. 229-311; Vol. 192 (1898) A, pp. 169-244; Vol. 192 (1899) A, pp. 257-330; Vol. 195 (1900) A, pp. 1-47; Vol. 195 (1900) A, pp. 79-150 ; Vol. 197 (1901) A, pp. 285-379 ; Vol. 197 (1901) A, pp. 443-459. See Phil. Trans., Vol. 186 A, pp. 364-5. 50 APPENDIX A. If, in the formula for Tj^e I., oj be made equal to a^, then the formula becomes that of Type IT./ shewn by Figs. 13 and 14, viz. — (138). 3/ = 2/o (1 ;) the basic form of which, when y^, is unity, is an elUpse with semiaxes a and 1. The figure becomes a circle when i^ is ^ and a is 1. In general, any form can be deduced from the basic form which, when va is unity, is a parabola (the quantity within the brackets) in (138). If this quantity be infinite and positive the figure becomes X' P' Y P X : see Pig. 13. If positive and greater than unity, it is the curve r'r ; if unity it is the parabola s's ; if less than unity, the curve t't in Fig. 13. The abscissa of the mode is 0, and the curve is of course symmetrical. If v be made negative in (138) the formula becomes 1 (139) y = 2/0 X^\ a, and is shewn by Fig. 14. The abscissa of the mode (of mediocrity) is at the origin. If in the second sub-form of Type I. we make a^ infinity, then (140). ■y = Voi- - 1)' the form of which is shewn in Fig. 11 ; that is, the curve is asymptotic to the ordinate whose abscissa is distant + a from the origin, and asymp- totic also to the axis OX. Fig. 15. Type in. Fig. 16. Type IV. Fig. 17. TypeV. ?. -?i ^^y^V'^ola Asffmpit,te Jlsympcaee ?' > 7; Asi/mpCo/^ Fig. 18. Type VI. Fig. 19. Type VII. Fig. 20. Various. 1 Op. cit., pp. 364.5, SPECIAL TYPES OF CURVES AND CHARACTERISTICS. 51 When in formula (135) a^ is infinity, then its form becomes Type III., viz., (141) y = y, (1 +^re-"' and is of the form shewn in Mg. 15. The abscissa of the mode is at the origin, and the curve is skew. Type IV. is of the form shewn in Fig. 16 ; its equation being : — 1 + -A e-" to«-^*/<» ; or = 2/o cos*"' Q.t~'^ being the angle the tangent of which is x/a. The curve is asymptotic to the X axis on both branches ; its mode is at the distance —va/2m from the origin, and it is skew : see Fig. 16. Tjrpe v., is of the form shewn in Fig. 17, and its equation is : — (143) y = 2/0 ^^e~ " The curve is limited on one side at the axis X, i.e., for a; = 0, and is asymptotic thereto at the other ; its mode being at the distance y/f. The curve is skew. The mean is at the distance y / {p —2) from the origin. Type VI. is of the form shewn in Fig. 18. Its equation is : — (144) y = y^{x — a)«^ a;~«i The curve is limited on one side only, viz., when a; = a. The mode is at 0^1/(9-1 -g-z)- Type VII. is the ordinary probability curve: see Fig. 19, viz. : — ?! (145) y = j/o e-" the mode being at the origin and the curve unhmited in either direction, and of course symmetrical. Curves a to e. Fig. 20, are typical forms of the following character- istics in a population, viz. : — (a) Infantile mortality, income, probates, value of houses, etc. ; (6) MortaUty from scarlet fever, diphtheria, etc. ; (c) Pauper frequency, divorce frequency with respect to duration of marriage, frequency of scarlet fever with age, of typhus, etc. ; (d) Senile mortality, mortality from enteric at different ages, fre- quency of marriage of wives corresponding to age of husbands at marriage, etc. ; (e) Height, weight, strength frequency, anthropometric measure- ments, etc, 52 APPENDIX A. 3. The method of evaluating the constants of the curves of generalised piobabihty. — Two things are requisite in. using the Pearson curves, viz. (i.) to select the appropriate type of curve ; and (ii.) to evaluate the constants of the selected curve. The selection of a curve which can be made to fit the given group-data depends upon relationships among the moments calculated about the mean. These relationships determine three criteria, which, after the necessary computations have been made, indicate the appropriate selection.^ Solutions can also be effected by means of a combination of graphical and numerical methods. The numerical solutions can be effected by taking logarithms, that is, (146) log 2/ = log 2/o + log / (x). The process in detail can readily be followed from the examples in III. (See in particular § 7). In general the solution must be tentative, and it is important to notice that the type-curve selected is not valid if the data have to be altered larger amounts than they are probably in error. The principle which should be employed is the following : — ^The adoption of a type-curve can be regarded as satisfactory only when it represents the data within the limits of their probable errors. In other words the geometric form and the algebraic processes should be subordinated to the data and not vice versa. 4. Flexible curves. — Although the type-curves just considered fulfil their general purpose fairly well, experience shews that their "fitting pmoer" is somewhat limited. To overcome this, other types are necessary, the " fitting power" of which is greater. In order to embrace as many forms as possible under cover of a single formula a curve may be so taken that its limiting forms shall include all parabolas, all hyperbolas (or parabolas with negative indices), all exponentials with positive or negative indices, and all curves of the normal probabihty type. Such a curve wUl necessarily include all intermediate forms. I have called this type of curve a, flexible curve. Formula (149) in the next section is a curve of the type in question. Its graph depends fundamentally upon the values of the indices m, n, and p, and its vertical scale depends upon the constant A . The mode of solving to determine its constants depends upon taking a series of values of the abscissa in geometrical ratio, and is indicated in the next section. 1 See the article by Professor Pearson already referred to, also " Frequency Curves and Correlation," by W. Palin Elderton (C and E. Layton, London, 1906) ; and " Statistical Methods with special reference to Biological Variation," by C. B. Davenport (Chapman and Hall Ltd., London, 1904). The curves indicated on p. 57 and p. 81 of Mr. W. Palin Elderton'a work do not satisfactorily represent the data, forasmuch as the curves chosen were in- sufficiently flexible. SPECIAL TYPES OF CURVES ANDXCHARACTERISTICS. 53 5. Determination of the constants of a flexible curve.— The probability curve, see (134) hereinbefore, viz., (147) 2/ = Ce *" or, ~k^ + <' in which c = log« G, may be put in a more general form, viz.- (148) y = e /W + Fix) + that is, its modulus k and constant G may be assumed to be functions of x. If we suppose that F(x) = a +^ log {±x) ; / (a;) = ya:« ; c = ; and write p = 2 — ,s ; w = -1/y ; log ^4 = a ; m = j8, the expression (148) can be written (149) y = Ax'^e"^ see (118) in III. 7. This curve can fit a great variety of forms, viz., such as are shewn on Figs. 21 to 27, referred to later. In practice it is not quite satisfactory to depend on four points. A better fit can be secured by taking several, say r, series of ordinates for values of the abscissa x^, k^x^ kl Xg, x„ k^x^ ¥r Xf. Each set will give a value for p, say p^, p^, etc., and a mean (geometric, arithmetic, or other) can be taken, p say. Or writing 7^,,, for log Vp- ^ log y« + log yr, we have (150) i) log {k,.k, k,) = log {n[{ Y,j r,,3)}; ill *■ denoting the product of r different sets of the quantities in the brackets. The use of this mean value of p, being inconsistent with each set of four ordinates, gives for each set two solutions for n, three for m, and four for A, that is in aU 2r, 3r and 4r solutions respectively for these constants. Having found the mean value for p we use it, in solving for a mean value of n, thus :— ^ , ni (log yi - log y2 - log ys + lo g y^} n[{MxP{k^P - 1) (k^ -1) (151).. r log n= log rTiii^^v^i.2v WTTv TVl ' °' (151a) r ^ri»g iyiy^/y2ys) Zl\MxV{kv +1) (kP - 1)2| 1 By comparing this with (120) it will be seen that the mean is taken of two quantities each of which give n, on the principle that if a/b=c/d approximately {a+c)/{b+d) is sensibly the arithmetic mean, or having two equations which give n, we assign an equal weight to each. The geometric mean, however, is taken in obtaining a mean result from the difierent sets. Of course {n-^ + nf)/r would also be a satisfactory value, n here denoting the value obtained by using the mean value of p. Although the two formulae are not identical, practically there is no cogent reason for preferring one to the other. 54 APPENDIX A. Adopting the mean values, thus found, for p and n, we have three different values for m given by each set. Reverting to formula (121), if we give double weight to the value found from the intermediate term \ye get^ (152) ,„_ -S[(}ogy^+ logy2)+iyfiogys+logyi)~Mni:J,xP{k^P -IW+I)] 4i7^ log k. Mean values for p, n and m being to hand, we have for A four values from each set of ordinates, see (122) hereinbefore, the general formula being (153) log ^ = log y^ - log (p-i x) - Mn (k"-^ xf . hence for a mean of 4r values of A we have 1 3 ( 153a) . . log ^ = j^ ; 27;; log (2/1 2/2 y%y^) - 4m2^ (log a; + g log ^) - Mn i:[ [xv (F*" + k^f 4- AP 4- 1)]) M denotes throughout 0.434. .etc., if common logarithms are employed, or imity if Napierian. Ignoring the coefficient A the first and second derivatives of the curve (149) are respectively (154) dy / dx = x-^-'^ e^' (m + npx'); and (155). .dhf/dx^ = x'^-^e^ {m{m—\)+npx'i' {2m-\-p—\)+ n^^x^*\ hence the mode (maximum or minimum value) is given by J. (156) X = {-m/npY which becomes, for p = 1, simply— m/w. The point of inflection is given by solving the equation (157) P2 + p (2m +^ - 1) +m (m - 1) = in which F denotes npx'^ ; this gives : a58^ ^. _ ! 2m + y- 1 ± V[4mff + (p - 1)"] ^ which, when m = 1 gives the value /I -1- \ - - (159) ^i = - (-^j ^ and also x^ -(1 / np) " for the abscissa of the mode. 1 The principle indicated in the preceding note applies, viz., if (a+6+c)/d equals (a -\- ? -^ y)/S approximately, then (« + n -f 6 + j3 + c + 7) / (d + S) is sensibly the aritlimetic mean. t$ SPECIAL TYPES OF CURVES AND CHARACTERISTICS. 56 The integral of a curve can take a number of forms as follows, viz : — (160). .fydx =\x'^e^ dx = x^ I nxP (m + 1) ■nTal^v (m + 1) 1 m+l 1 "^(m+jj+l)!! +" ' + (m+r^+l) r ! +• -etc J; or (161) 5!^e»4l ^P^^ (^^^)^ '••m+l ( m+^ + 1 "f"(m+39+l)(m+2i)+l) ■"•• , (-1)^ [np xoy , , I + "7 i 1 1 / 7 — 7 r-TT- ± etc. \ ; or ^ (m+p + l)....(m + r^ +1) =^ j' (162) '^"'"''"^' e«^/l - ^-P+^ + (m-j)+l)(m-2j>+l) _ ■«:P [ 7vpx!P {npx^)^ +{-lY ("^-y + l)----(m-r3> + l) ^ ^tc. 1 Between the limits and oo the integral may be put into the forms of the second Eulerian integral, and is (163). ...... f>">e -»»="«««= — !^ ^ ^ {pn P ) which, when m = o, gives (164) j:^-'"^dx = r(^)/(pr^) The abscissa, Xc say, of the centroid vertical, or mean of the distribution. IS jx«^+ie-'^dx \ p J ^^®®^ ^'^ Ja:'»e-"*'rfa; "" ^/m + l\ i It is sometimes necessary to make the definite integral (163) when multi- pHed by the coefficient A, equal to unity. In such a case we must have the value of this constant the reciprocal of that given in the value of the integral mentioned, viz. (163) ; that is (166) A =pn P /^(^) Simplioations of these general formulae are often possible. ^ 1 For a fuUer study of this curve, see " Studies in Statistical Representation," by G. H. Knibbs. Jour. Roy. Soo. N.S.W., Vol. XLIV., pp. 341-367 ; 1910. 56 APPENDIX A. The forms of the curves are as shewn on the Figs. 21 to 27. If w in e"^ be zero, the curve degrades to Ax^, and we have the forms in Fig. 21, in which the capital letters shew the curves when m is positive, and the small letters when m is negative. Fig. 21. Fig. 22. If m be zero, x^ will be unity, and if f also be unity, the curves be- come e"^, the forms of which are shewn onFig.22,the upper hues denoting the values when w. is posi - tive and the lower when n is negative. If p, however, be not unity, and p and n be positive, we shall have such forms as A, B, and G on Fig. 23. If p be negative and n positive, the forms become those shewn by the curves D, E, and F in the same figure. \\ b / / M / / 1 c\ sM [/. c_^ ^- ,/- /7 v^ ~. — ^. / ^ ■-^ ar-- t / / / c/ /^ / / y ,b ^ t^ ( ) z Fig. 23. Fig. 24. If n be negative and p be positive, the forms become a, b, and c, the reciprocals respectively of curves A, B, and C ; and if both n and p be negative, the curves are such as d, e, and f, viz., the reciprocals re- spectively of curves D, E and Fin the same figure, viz., Fig. 23. Figs. 24 to 27 give values of the curves when both m, w, and p have values other than zero, the Ught fines denoting the reciprocals of the curves shewn by the heavy fines, and the curves being the following, viz.: Fig. 24 Fig. 25 Values of— VAIiUES OF — m n V m n p .. A = \ -\ \ Fig. 26 . . A = 1 -i 6 .. B = \ -\ 3 )j . B = _6 -\ -6 .. C = \ -1 \ 3J . C = 6 -i 6 .. A = \ -2 5J . D = 6 -i 1 .. B = \ -1 Fig. 27 . . A = -1 i 6 .. C = \ -\ j» . B = -6 J 6 .. D = 2 -1 'J . C = 1-11 .. E = 2 -2 )J . D = 1-21 SPECIAL TYPES OF CURVES AND CHARACTERISTICS. 57 In the reciprocal curves, viz., a, b, c, d, etc., the signs of m and n are changed, but not that of p?- These wUl sufficiently illustrate the possible forms of the curve. Fig. 26. Fig. 25. Fig. 27. 6. Generalised probability curves derived from projections of normal curve.=ln Fig. 28 let'bYa denote a normal "error" (or probability) curve, the ordinates of b and a being denoted by corresponding suffixes. If a Une be drawn the distance I above OY and parallel thereto (and parallel therefore also to the plane of the curve), it may be represented by the point 0' in any plane at right angles to the plane of the curve. 1 It may be mentioned that H. P61abou, in dealing with the influence of temperature on ohemioal reactions, developed a relation in the form log y = a-'rb/x + c log x ; which, ofjoourse, may be written in the form y=ah~'' x", which is merely a simple case of formula (149). See M6m. d.l. Soc. des Sciences physiques et naturelles de Bordeaux [5]. 3, pp. 141.257; 1898: Compt. Rend. 124, pp. 35, 360, 686 ; 1897. 58 APPENDIX A. Let a line be drawn from any point, on the curve, viz., a, at right angles to 0'. This will be the Une O'Q, which, when produced to q on a line VOqU, making the angle 6 with the line PO, gives the point corres- ponding to a. The abscissa then may be taken either as Oq or as its orthogonal projection on OP. The latter is more simple. If it be pro- duced to q' on a plane making the angle 6' with the axis OY, it will give a result of greater skewness, see the points a^ and aa^ in the figure. The scheme of projection will be obvious from the figure, and need not be described in detail. Let ^ denote any abscissa on the curve derived by projection, and X the corresponding abscissa on the original curve. Then by similar triangles we have at once the relation of x and £ in terms of I and 9, inasmuch as (167) x/l = ii ~x} / ita.n9. This gives, on writing m for (tan 9) / I, (168) i = x/{l -mx); x = $ / {1 + m^) from which it is at once evident that the same result may be obtained by any values of I and 9 whatsoever, which give the same value of m. Thus, the point S with the projecting height 00" = T gives the point U, the ortho- gonal projection of which Q" is identical with the result with the projecting height 00' = I, viz., R, as is evident from the figure. Fig. 29 shews by a heavy fine the curve derived from the curve in Fig. 28 by projection on to the plane VOU, and by a thin line the curve similarly derived by projection on to the plane WOR in that figure. Fig. 29. SPECIAL TYPES OF CURVES AND CHARACTERISTICS. 59 Hence, if for x in the probability-curve equation, we substitute its numerical equivalent, we obtain (169) y = i/g-' + ^omi+cf =i/e„(.+^)2. K in the second expression being cm^, and jj, being the reciprocal of m. The curve is asymmetric, since the denominator differs in value according as ^ is negative or positive. Incidentally we notice that if I be relatively large or d relatively small, m is small, and the asymmetry is not marked ; and when I is infinite or d zero, the asymmetry vanishes, as is seen by the projection. In this last expression when ^ is negative and equal to /x, 2/ = 0, so that there is a terminal of the curve on the negative side cor- responding to a; = — 00 . When tan 6 = l/x, then mx = 1, and | is infinity ; that is to say, the projecting line is parallel to the plane through the axis. When — | is one-half of —x, then the point with the same ordinate on the positive side is at infinity. This can also be seen on the figure.^ This indicates the limitation of the method of projecting onto a plane, namely, that if there is to be a corresponding point at a finite distance on one side of the axis, the abscissae on the other side cannot be reduced to a greater amount than one-half. This, however, can be overcome by projection on a curved surface. Thus, if projected from the intersections with an equilateral hyperbola orthogonally on to the X axis, from a line parallel to and distant the height I from the Y axis, the Y axis 0"H of the hyperbola being the distance p on the negative side, and the X axes, being identical (see Fig. 30) we have (I'O) ^ = i-A^ + ^) ;°^^ = ^-rTii A denoting •p'^/l- Hence , substituting the former expression in the ordinary [)robability curve equation, we obtain (171) y = l/e-'Ci- xfK + f)"-! This gives a terminal to the curve on one side, and an asymptotic relation to the axis on the other, and may be made as skew as we please, as is evident from Fig. 30 and from Fig. 32 giving a projection so derived. A similar scheme of projection using a surface whose right section is a parabola, the abscissa of whose vertex is p (from the origin), and whose equation is tj = gf (f — i>)^ gives the result in which y denotes g/l : see Fig. 31. The value of ^, therefore, is (173) I = p +2^1^± ^^^y^" {p-x) + l]\ 1 That is, when OQ' is one-half of OP', the corresponding point on the positive side is at infinity. 60 APPENDIX A. This gives terminals for both branches of the curve, viz. : — Since both p and y may be arbitrarily determined, the position of the terminals of the curve, in relation to the mode, may be made whatsoever we please. Although this leads to a somewhat complicated expression for I, it discloses the character of the curve obtained by projection. Its equation is (175) y = l/e'ti + Tfr p)«]« the asymmetry of which is evident. Fig. 30 illustrates the projection on to a surface whose right section is an equilateral hyperbola, and the type of resultant curve with one as5maptote is shewn on Fig. 32 : see curve %, bi,. ..a^, bj.. ., thereon in a thin firm Hne, the thick curve A, B,...A', B'... etc., being the probabihty curve from which it is derived. Mg. 31 is similarly an example of a projection on to a surface whose right section is a parabola, and is shewn on Fig. 32 by a broken line: see curve a2, h^.-.a,'^, b^... etc. The scheme of projection is sufficiently evident from the figures. Reverting to projection on a plane, it may be noted also the pro- jections may be varied by making I a function of y instead of a constant, as, for example, I = ky'^, which, writing k for (tan d)/k, would give (176) y = l/e<'0- + Ki/yy This does not lead, however, to any simple expression for y in terms of ^ only. We may notice that since Z = for y =0, both branches are unlimited (that is to say, the asymptotic relation of the basic curve remains) and the curve is more distant from the X axis than is the basic curve ; the curve most closely approaches the type of that with I constant if w be less than unity. If n be negative and numerically greater than unity, we shaU have ^ sensibly equal to x for very small values of y, or X = ^{\ — K-2/") approximately, and the branches are unlimited. These projections shew that though initially a frequency may be distributed according to the ordinary probabihty curve, yet the final circumstances may be such that the " frequency is altered in several of its characters," viz., its symmetry, asymptotic relations, etc. SPECIAL TYPES OF CURVES AND CHARACTERISTICS. 61 Kg. 30. Fig. 31. j&if,- i^- Pig. 32. I, J^* qa 1, qt (5f„ H, "P, P. -^; ft Pi Mg. 33. 7. Development of type-curves. — ^A consideration of the form of the equations derived from projections shews that if we put as the funda- mental form (177). .y, or l/y = ^oA *'^ + */*""" '^ - */«" we may include all cases by variations of p, m, k, a and 6. When a; = 0, the value is «/o> 't^t is to say, the mode is at the axis. If a and 6, each supposed to be positive, are finite, then for a negative value of x equal to a, or a positive value equal to 6, we have y = 0, that is the branches of the curve terminate .at the axis of abscissa for the negative value of a; = a ; and for the positive value of a; = 6. If 6 be infinite, the curve, which is skew, becomes (178). ■y = yo/e * (1 + x/a)"" « .ttta M—tnx and if a be infinite and b finite, the curve is skew, and its equation is (179) y = yo/e**"''^-*/*)'" 62 APPENDIX A. If both a and b are infinite, then the preceding curves (178) (179) become (180) y=yo/e * and is symmetric, but if b (or a) be negative, then the curve is (181) y =yo/c he'" This curve is asymmetric^ and both branches are asymptotic to the axis. The reciprocals of these curves give the other forms required. 8. Evaluation of the constants of the preceding type-curves. — ^The value of 2/q is assumed to be derived from the data. When all the quantities are divided by the ordinate of the mode, viz., by yQ, we have a series of redMced values of the ordinates, r) say. Then, as a rule, by taking the logarithm twice we can obtain the necessary solution. Thus — (182). .Tj =e-^'*); hence log rj ='?'=/(a;); andlogT}'=log/(a;) which gives a linear equation. Thus, with the necessary number of values of the ordinate and the corresponding values of the abscissa, a solution of the constants is to hand. If more than the necessary number are given, the least-square method of forming normal equations may be employed. This method wiU not solve, however (177), (178), or (179), where (183)..logrj'=j3loga; -jlogA; + malog(l+^\ + mb\og{l + ^\ These, however, are very readily solved by expanding the logarithms, and sometimes a and b can be estimated from the graph of the curve. 9. To determine the surface on which the projection of a normal probability-curve will result in a given skew-curve. — ^From what has preceded, and from I^s. 28 to 32, it is evident that the form and equation of the curved surface, on which the projection of a normal probability- curve will furnish any given skew curve, may readily be determined. The problem more generally stated is : — Given two curves to find the surface on which the projection of one will furnish the other. On.Eig. 33 let Y. .Pd and Y. .Qa be the branches of a normal probability curve, and YQa Q^, and YPa P^ be the branches of a skew-curve, the axis OY being identical for each. Draw radial lines from Y to the orthogonal projections on to the X axis of various points on the normal probabihty curve, viz., to the points qa, qb, etc., and Pa, Pb, etc., and from the points Qa, etc.. Pa, etc., whose ordinates to the skew-curve are identical with those of the corresponding points on the normal curve ; and draw lines parallel to the axis OY. Then the intersections a, b, etc., a', b', etc., are points on the projection surface. Reference to the figure ^ p is to be understood merely as on operator raising the number in numerioaj value, but not afieoting its sign. SPECIAL TYPES OF CURVES AND CHARACTERISTICS. 6S makes the proposition obvious. Thus, the equation to the normal curve being known, that of the skew-curve can be found in the form y = l/ef(^\ as soon as the equation of the curve of the projection surface is ascertained. In finding an equation to fit any series of groups the skew-curve may, in practice, be drawn freehand : a suitable normal probability- curve may then be drawn with the same mode and vertical height : the points on the surface found by the method indicated. In general, this wiU give a somewhat irregular projection-surface, which, however, may ordinarily be so modified as to conform to some geometrical form easily expressible algebraically, from which the requisite formula may then be found. From Figs. 30, 31, and 33 it will be evident how the equation may be ascertained. 10. Reciprocals of curves of the probability-type. — ^The curve -q = l/y, also of type of practical importance, may similarly be derived by pro- jection from the normal probabihty-curve : thus (184) 7] = l/y = e " , or more generally, rj =e'' that is, its logarithmic homologue is the parabola 17 ' = yua;'', in which rj '= log rj, and /x = l/k^. Thus in Fig. 33 the reciprocal of the normal probability-curve (curve 1) is shewn by the curve marked 1', 1', while the curve 2', 2', is the reciprocal of the curve marked 2, 2. The lateral scale in the figure, however, for curve 2, is four times greater than for curve 1. It wiU be seen that the type is somewhat similar to the curve of instantaneous rate of mortality according to age. 11. Dissection of multimodal fluctuations into a series of miimodal elements. — ^It is obvious that any multimodal fluctuation may be analysed into a series of unimodal elements ; for example, a series of the form (185) y= ^0 + ^6*''=-''^'/'' + ...4,6^'^-''^'/'=' + .. may, with a sufficient number of terms, be made to fit any continuous curve whatsoever to any assigned degree of accuracy.^ There is no complete general solution of the problem, however, of dissection. We have already shewn that a dimorphic curve may be the sum or product of two monomorphic curves (see III., § 9, Projective anamorphosis). The difficulties of dissection, however, are not unduly great with graphic methods. ^ See " Contributions to the Mathematical Theory of Evolution" (on th« dissection of Asjmunetrical and Symmetrical frequency curves, etc.). Prof. Karl Pearson, Phil. Trans., Vol. 185-A, pp. 71-110; 1894. " Sui massime delle curve dimorfiohe," Dr. F. de Helguero, Biometrika, Vol. III., pp. 84-98, 1904 ; and also his " Per la risoluzione delle curve dimorfiche," Biometrika, Vol. IV., pp. 230, 231 ; 1905-6. " Sulla statura degli Italiani," R. Livi, Firenze, 1883. " Die natarliohe Auslese beim Menschen," O. Ammon, Jena, 1893, v.— GROUP-VALUES, THEIR ADJUSTMENT AND ANALYSIS. 1. Group-values and their limitations. — ^The data of population statistic are ordinarily given in the form of group-vcdues. For example, in the age-distribution of a population the data are ordinarily in the form of the numbers of persons between the ages x and x-\-k, x-\-k and x-\-2k, and so on, where k may be a month, a year, 5 years, 10 years, etc. Hence, when the number for any group of smaller limits is required, some curve must be assumed which will give the same group-values if the latter are to be regarded as correct. In other words, if we suppose the numbers between the ages x and X -\- dxto be P{x) dx, then the number in the group between the ages X and a; -f ^ is (186) ^N,+,^Pj^-^''ix)dx in which, if P denote the total population of aU ages, the value of the integral between the limits and the end of Ufe, say 105 (or c») is neces- sarily unity. This is the fundamental conception of the use of group- values. Thus, omitting the coefficient P, the value of the integral between any Umits, when its total value is unity, is the proportion of the whole population which lies between the limits in question. When grov/p-values are known to be subject to error, each group can be modified in amount so as to conform to some distribution regarded as more probable than that furnished by the crude data. Thus, if in the numbers according to age a census return gave for " ages last birthday " 29, 30 and 31, the numbers 20,000 ; 24,000 ; 18,000 ; we should know ordinarily that the number 24,000 was in excess, since the numbers must fall off as the ages increase unless immigration prevent. We deal primarily with the case where the groups are assumed to be correct ; having either been corrected, or having been taken accurately. 2. Adjustment of group-values. — ^In cases where group-values are properly regarded as subject to appreciable error, they should either be first adjusted before the constants of mathematical formulae representing them are determined, or the computation should be so effected as to automatically make the adjustment a minimum. The Hmitations under which group-results are obtained are of two kinds. The results furnished may be either — (a) actually subject to large errors ; or (6) insufficient in number to furnish a truly representative example. For example, misapprehensions as to one's exact age must necessarily have the effect of causing numbers of persons to be attributed to the wrong age-group, thus diminishing some groups and increasing others. GROUP VALUES : ADJUSTMENT AND ANALYSIS. 65 A certain tendency to misstatement is confirmed by census-results, which reveal the fact that ages ending in are characterised by excessively large numbers, and that the numbers for ages ending in 5 are also some- what excessive, while the numbers for the adjoining years are in defect. In the other case, hmitations in the numbers available prevent one knowing exactly what would have been given had the numbers been indefinitely large. In these latter cases, however, it is often possible to surmise what the curve would have been had the numbers been large, and the actual data may be redistributed so as to conform therewith. In both instances the principle to be followed is that some groups should be so increased, while others should be so diminished as to conform to the most probable distribution which may, for convenience, be called the " ideal distribution." In effecting these changes in the numbers furnished by the data for individual groups, the alterations should not only be as small as possible, but also the accumulation of the alterations (that is, their algebraic sum) should be alternately plus and minus, and should never become large in amount. Various considerations may serve as a guide in effecting the altera- tion : for example, excluding the consideration of dehberate misstate- ment of age and tendency to uniform error in one direction, the number of cases in which the misstatement of age is one year only is, in general, larger than the number in which the misstatement is two years ; and so on. Experience shews also that large positive errors are Ukely to be made for ages ending in ; for example, 30, 40, 50, etc. ; and lesser positive errors are Ukely to be made for ages ending in 5 ; for example, 35, 45, 55, etc., while errors of defect are to be expected in ages 29 and 31, etc., and 34 and 36, etc. Adjustments are, as a rule, preferably made in the light of a full consideration of aU the circumstances affecting the case, and not merely by piirely mechanical or merely arithmetical methods. A redistribution of values may be regarded as excellent when the curve giving the values of the groups is, in the nature of the case, probable, and when at all points it deviates from the successive values of the groups in such wise that the deviation is always relatively small, and the aggre- gate alternately plus and minus. 3. Representation of group-values by equations with integral indices. — Any curve representing a series of statistical data may be represented by the following expression, viz., — (187) y = a -\- bx^ -\- cx^ + dx"" + etc. and, if p, q, r, etc., be not necessarily integral, with a small number of terms. Integrating this we shall have (188) jydx =x{A + BxP+ Gx' + Dx' + etc.) 6 „ c d in which A = a; B= — ^ ; = ^-py ; L> = ^-^;p^ ; etc. 66 APPENDIX A. When p, q, r, etc., are the successive integers 1, 2, 3, we have for x=0, k, 2k, 3k, etc. Q Range of Factors into numbers below ^' the Abscissae, a ^bk ^ck'' (189). I. II. III. IV. V. k 2k 3k 4:k k = k(l 2k = k{l 3k = k(l U = k(l 5k = k(l 1 7 19 37 61 Idk^ 1 15 65 175 369 1) 31) 211) 781) 2101) It is easily seen that with integral indices, the above expression of w+l groups can be fitted by an arbitrary equation of the nth degree. Denot- ing the heights of the groups by the small Roman letters i. to v., the heights being found by dividing the group-values by the base k, and the successive differences of height by h^, hi, etc., the simplest scheme of solution is to hand in the following series of equations, which are readily obtained by differencing and substitution. 4. Formulae depending on successive differences of gioup-heights. — We give first formulae depending merely on the difference of heights, viz., the differences i.— ; ii. — i. ; iii. — ii. ; etc. ; that is, if we denote the successive heights of the groups by (190) ^0 ; ^0 + ^1 ; ^0 + ^1 + ^ ; etc., the successive differences of height will be (191) hg ; hi; h^ ; h^ ; etc. hg = i., denoting the height of the first group from the X axis, see Pig. 36.^ • + \ 1 + ^ M a 1 Curve begins at "a' from A o ■ ■a .^ A cQ A V \ / \ 1 V > V X k i V. k * Fig. 34. 1 These (191), are the first column of differences if the groups be divided by their base-values viz., by fc. GROUP VALUES : ADJUSTMENT AND ANALYSIS. 67 The following, as convenient formulae for the coefi&cients a, b, etc.. in equation (187), can be deduced, viz. : — For three groups : For four groups : — (192).. a = feo -^(5^1 -2/^2); b = \(2hi-h2); c=^^{-h^+h2] (193) a =K-^ {Uhi-lOh+^h) (193a) .... 6 = j^ (35A.1 - 34^2 +11^3) (193b).... c= -^A-5hi + 8h2-3hs) (193c).... d= A-^ih-^h+h)- 6k For five groups :- / (194) ....a=Ao-^^i + ^(- ilh + 86^ - 51^3 + 12^*) (194a) .... b = n ^1 + i^ (^^^1 - ^*^2 + 4:1^3 - 10^4 ) (194b). ... c= ^, (- 17^ + 37/^2 - 27^3 + 'Jh) (194c).... d= ^^(3hi-8hz+'7hs-2hi) (194d)....e= 25^^" ^+^^2 -3^3+^4)- If instead of heights we use group-values, the quantities found, say a', b', c', etc., will be k times those above given, and must be reduced -accordingly 5. Formulse depending on the group-heights themselves. — ^Instead of using the difference of the group-heights, the coefficients of the equation may be expressed in terms of the successive group-heights themselves, found by dividing the group-numbers by the value of the common interval along the abscissa ; that is, by dividing the integrals between the successive Hmits having a common interval k, by that quantity. It will be sufficient to give the results for from three to five groups. These results are : — For three groups : — (195) a = -g- (Hi. - 7ii. -f 2iii.) (195a) b = -j^ ( - 2i. + 3ii. - iii.) (195b) c =p2(J- - 2ii- +in-) 68 APPENDIX A For four groups (196) (196a). (196b). (196c). a = Jo (25i- - 23ii. + 13iii. - 3iv.) .6 = 12fc (_ 35i. + 69ii. - 45iii. + lliv.) c = jp (5i. - 13ii. + lliii. - 3iv.) d =gi-3(-i. +3ii. -Siii. +iv.) For five groups : — (197) a -- (197a) b -- (197c). (197d). 60 J_ 12k (1371. - 163ii. + 137m. - 63iv. + 12v,) (_ 45i. 4- I09ii. - 105iii. +51iv. - lOv.) (197b) c = gp (17i. - 54ii. + 64iii. - 34iv. + 7v.) 6P (_3i. 4- iiii. _ I5iii. + 9iv. - 2v.) ^ =24F(^ - 4ii. + 6iii. - 4iv. + v.) If the aggregate numbers or group^alv£s are used, instead of the heights, the denominators will be 1/fc, 1/k'^ 1/k^ instead of those above. 6. Formulae depending upon the leading differences in the groups or in group-hefehts. — It is often convenient in practice to work with differences instead of the group-values or of heights. In the latter case the coefficients are similarly given by the following equations : — The coefficients of equation (187) expressed in t«rms of successive leading difEerences of the group-heights are : — (198).. (198a). (198b). (198c) (198d). .a = \ (i. Di i(A + 6 A - D. -IDs\+1 D,j 11 12 + ^oi>3 C = d = e = Ds f(+ j^3 ->^ + F(2i ^* GROUP VALUES : ADJUSTMENT AND ANALYSIS. 69 In the above Di, D^, -D3 , and D^ are the leading differences of the heights only, viz., of i., ii v. As before, if the group values are subtracted, without first dividing by k, the denominators shovild be l/k, l/fc^ l/k^, instead of those above given. Formulae (198) to (198d) are correct for any number of groups up to five, the division lines on the right hand side shewing the results for two, three, four and five groups. 7. Determination of differences for the construction of curves. — When the equation of the curve is to hand, it is often required to find values of the ordinates corresponding to a series of values of the abscissa. This is most conveniently effected by obtaining the successive leading differences : from these the required values can be obtained. These are : — (199) f (x) = a + bx + cx^ + dx^ + ex^ (199a).... Di/(0) = .. b +c +d +e (199b). . . . Z)2 / (0) = 2c +M + Me (199c). . . . D3 / (0) = 6d + 36e (199d). . . . Z>4 /(O) = Me It may be remarked that when k=l these difference values become (200) A /(O) = Ai. - I Ai. + I Dsi. -Id,L (200a) Dzfm = Dzi. - | I>3i- + I D^i. (200b) -D3 /(O) = Ai- - 2 -Oii- (200c) i>4/(0) = i>4i- in which the symbol D\i., D^i-, etc., denotes the leading differences derived from the series from i., ii., iii., etc. 8. Cases where position of curve on axis of ordinates has a fixed value. In the equation (187) it may happen that the curve is required to pass through the intersection of the axes OX, OY; or at a fixed distance therefrom on the Y-axis. In this instance the solutions given are invalid, inasmuch as a is initially given, not determined from the group- values. The most convenient procedure is to subtract this value a from the heights i., ii., iii., etc., of the ordinates, or the value ka from the APPENDIX A. group-values (or areas) I., II., III., etc. This procedure gives new values, viz., y' = y~a, and the solution required is then of the successive in- tegrals (group values) divided by Ic. (201) ^ fy'dx = ^ f{bx + cx^ + etc.) dx that is, oi\bx -\- \cx^ -{■ etc. It is obvious that in this instance n groups will require an equation of the wth degree, instead of, as before, of the [n — l)th, the imposed condition of a fixed value for a involving this limitation. The following formulae give the value of the constants in terms of the heights. For two groups, curve passing through origin, (202) 6=4 (7i - ii) ; c = jj-, ( - 91 + 3ii) For three groups, curve passing through origin, J_ 9F (203) ^ = iP (85i - 23ii + 4iu) (203a) c = —^ (- lOi + 5ii - iii) (203b) d = ^3 (Hi - 7ii + 2iii) For four groups, curve passing through origin, (204) b = ^^ (415i - leiii + 55iii - 9iv) (204a) c = ggp ( - 755i + 493ii - 191iii + 33iv) (204b) d = ^ (119i - 97ii + 47iii - 9iv) (204c) e =^^{- 125i + 115ii - 65iii + 15iv) For five groups, curve passing through origin, (205).. . Jb = jg^^ (120191- 598111 + 3019111 - 981iv + 144v) (205a). .c = g^ (- 343i + 273ii - 155iii + 53iv - 8v) (205b). .d = g^3 (2149i - 211111 + 1429iii - 531iv + 84v) (205c). ..e = -ggp- (- 133i + 147ii - llSiii + 47iv - 8v) (205d). . ./ = J200P (13'^i - 163ii + 137iii - 63iv + 12v.) i GROUP VALUES : ADJUSTME2SrT AJSTD ANALYSIS. 71 The constants in the terms of the leading differences of the heights are : — ^ For two groups, curve passing through origin, (206) 6 = ^(3i- ^D,i) (206a) ....c=^(-|-i+ Ad^I) For three groups, curve passing through origin, (207a)....c = ^ (-3i+|-Dii- ^ D^i^ (207b).... rf = ^ (li ~ ~D,i+ |-^2i) For four groups, curve passing through origin, i/2'i 1^ 7 1 \ (208) 6=^(-gi_ j2Ai+i8i>2i--8i)Bi) (208a) ....c = J,(-f i + gi),i - l^i + |-^Z)3i) (208b). ...(?= y ( |i- |i)ii+|i)2i - l^i) (2080 . . . .e =^ (-|i + I Ai - 4i),i + |z)3i ) For five groups, curve passing through origin, 1 / 1S7 77 47 9 2 (209)....6=-^(^i- ggDii+ggD,i-^i)3i+ 2gi>4i (209a) ••c = -p-( -"81+ jgAi- -g -021+ 32-^31- ^Dii 1/17 17 17 13 7 (209b) ..<«= p(^ 6" i- 12^1+ i8^2i- 24-^31+ -30-041 (209c) ••e = ^(-|-i+i^Ai-2^Ai+3l^3i- j^Ai (209d) ../=^( ^ i_ ^Ai+ g^Ai- g^i)3i+ 4i)4i ^ i denotes the height of the first group-result ; D^i = ii — i ; D^i = iii — 2ii — i ; D^i = iv — 3iii + 3iii — i ; etc.; that is, they are the leading differences. 72 APPENDIX A. 9. Determination of group-values when constants are knovm. — When the equation is in the form (187), jp, q, r, etc., being 1, 2, 3, etc., the most ready way to compute a series of values of groups to k,h to 2k, 2k to Sk, etc., is to form the leading differences, and from these the successive values of the groups can be readily formed. The following formulae give the required result : — ^ (210).... 1. =ak +^bk^ +^ck^ + ^^dki + ^^ek^ + ^fk^ (210a) . . Dil. = bk^ +2ck^ + 3idk* + 6ek^ + 10J/A;« (210b) . . Dgl- = 2c/fc3 + Mjfc* + 30ek^ + 90//fc« (210c) . .Dgl. = 6dk^ + 48ek^ + 260fk^ (210d) ..Dil. = 24e^s _^ 200fk« (210e) . .D5I. = 120//fc« When the equation is of a less degree than the fifth, zeros can be substituted for the coefficients ; thus for a fourth degree, /=0 ; for a third degree /=0 and e =0 ; and so on ; and the formulae stiU hold good. 10. Curve of group-totals for equal intervals of the variable expressed as an integral function of the central value of the interval.^If we have a series of group-totals for equal intervals of the abscissa, as, for example, for to k,k to 2k, etc., and if those values divided by the common interval are represented by the ordinates at ^k, l^k, 2\k, etc., to a curve the equation of which is an integral function of the type of formula (187), then, whatever be the value of x in this equation, the ordinate for the point X wiU give very approximately the group-total for x — \kto x-\-\k. That is to say, denoting the ordinate to the curve representing the groups a; ± JA; by Y, and that to the curve representing the original function by y, if (211) Y = F{x +\k) =yf^+''dx=f^+''f{x)dx for the values x=0, 1, 2, etc., then it follows that very approximately (212) Fix+lk+q) = f^' + 'flx) dx provided that the forms of F and/ are the same, that is, that they are both integral functions of a single variable. This result is important, and may be estabhshed by the following consideration. If we compute F {x) = /^f{x)dx so that the two are in agreement for x=\k, \\k and 2|i, in the first function, with the limits to ifc. A; to 2k, and 2k to 3fc in the second, then it is easy to establish that if the original • D^l., D.2I; etc., denote the series of leadinc/ differences, viz. (II.— I)- (III. - 211. + I.) ; (IV. - 3III. + 3III. - 1.) ; etc. ' GROUP VALUES : ADJUSTMENT AND ANALYSIS. 73 equation be a-\-hx-\-cx^, and if the equation for the group-total, divided by the common interval, be A-\-Bx-\-Cx^, when x is the value of the abscissa for the middle of the interval, then (213) A=a + ^ck^; B = b ; C = c. If we extend the solution to the third power of x, that is, extend the limits to S^k and 3fc to 4fc respectively, we have (214) A =a + ^ck^; B=b +-^dk^; G =c; D = d. If we further extend the solution to the fourth power of x, and the hmits to 4JA; and 4A to 5k respectively, we obtain {215)..A=a+^ck^+^^ek*; B=b+^dk^; C=G+-^ek; D=d; E=e. If the fifth power of the variable be included, that is, the hmits be 5\k and 5k to Qk respectively, then (216)..^=a+^cfc2+^eA*; B=b + \dk^+ ^^fk*; C=c+\ek^; D=d+~fk^; E = e; F=f. It will be observed that up to the second power of the variable, the effect is that A differs from a only by a constant, consequently the function F gives rigorously the correct result, viz., that given by integrat- ing the function /. For powers higher than the second, the result is true only for k=^, 1^, etc., in F, and for any other values is more or less in error. This error cannot, in general, however, attain appreciable magni- tude, because it is repeatedly reduced to zero at intervals of k, viz., at the values of the abscissa, ^k, l^k, etc. In practical statistical examples the coefficients b, c, d, e, f, etc., are generally in diminishing order of magnitude, and we see from the equations (213) to (216) that the corresponding numerical factors also rapidly diminish ; hence the difference between the rigorous value \f{x)dx and the approximate value F {x) must generally be very smaU, and, by the formulae given, can be readily tested in any numerical examples. 11. Average values of groups. — An average value y^ of a group is the quantity (217) yr = -^j::ydx in which y denotes the value of the ordinate, and Xx to X2, the range of the variable. Reverting-to formulae (187) and (188), and retaining the same meaning for the constants, the mean value of the range x to X -\- kis (218)..?/, =A + \_[B{{x+k)J> + -^-xP + '>']+ (7{(a;+A!)«+i-a;«+i}H-etc.J APPENDIX A. which takes a simpler form if j)> q> >". etc., are 1, 2, 3, etc. Where x has a series of values 0, k, 2k, etc., as in (189) the averages are given by omitting the factor k in the formulae. More generally, that is, for any value of X and k we have (219) yr = a+b (x+l^ +c(x^ +xk + jk^) + d {x^ + lix% ^xk" + ^ k") + e{x* + 2x^k + 2x^k^ + xk^ +"5 **) For groups bounded by curves of the exponential type we may note that (220) a' = e-r log a == e"*^ Thus, the rate of change at any point of the curve y = we™^ is '-) t' and the mean rate y^ is (221) / = <^ (we'»^)/da; = mwe" (222) y^ = mne^''. mk that is, this is the mean ordinate to the curve. If the ordinates for the beginning, middle, and end of any range of values of the abscissa, that is, if the ordinates corresponding to the values X, x-\-\k, and x-\-k, are to hand, and the group-values are the integral of an equation of the type (199), then the value of y, is 1 1 fl 1 2^ (223). .2/,=g(y.+42/^+2/:»+*)-24**|5e+/(x+2A)+3? {x^+xk+^'^)+Q\^. The negative term (in braces) is absolutely negative, x being positive, if e, / and g are positive, and it is usually so small as to be neghgible. When a;=0 and k=\, the value of (223) takes the very simple form 1 1 / 1 1 23 \ (224) yr = -Qiyo + ^ym + yk) ~ 2i\'5^ '^ J f ^ 28 ^) 2/ot denotes, of course, the middle ordinate. This result is important, because it shews that group-values can be calculated with considerable precision by the " prismoidal formula" if we have middle as weU as terminal instantaneous values of each group. VI.— SUMMATION AND INTEGRATION FOR STATISTICAL AGGREGATES. 1 . General. — ^In effecting statistical summations, regard is to be had to two elements, viz.: — (i.) Order of accuracy significant in the case in point ; (ii.) Arithmetical consistency of results. Curves drawn freehand among data, that represent either groups or instantaneous results, and which shew visible variations, can, for some purposes, be integrated with sufficient precision by careful graphing and the use of a planimeter.'- When arithmetical smoothing has followed graphic, in order to enhance the accuracy, numerical calculations are virtually required as being of corresponding precision. As a rule group values (or the total area between any ordinates, the curve, and the axis of abscissae) can, if the ordinates are relatively near each other, be computed by means of the prismoidal, Simpson's, Weddle's and similar rules. Finally, for work of the highest precision, actual integrations by the method of the infinitesimal calculus are required. In general, however, the precision then far transcends that of the data. The extension of implied precision far beyond that of the data is seen in all actuarial tables : this matter is referred to later, since the year change in probabihty of life is a quantity of a much larger order than that to which results are expressed. 2. Area! and volumetric summation formulae. — Statistics relating to population involve both areal and volumetric summMions. The latter can, however, always be represented by an areal graph. If the curve represent instantaneous and not group-values^ about a particular value of the variable, then the areal value can be computed without computing the equation of the curve and integrating it. It has been shewn* that if an axis be equally divided, that is, if x=0, k, 2k. . . .nk, and the curve passing through the terminals of the ordinates (y) from these points is assumed to be represented by an integral function of x, then suitable multipliers or weights may be deter- mined, which, appMed to the ordinates, will give the area. If there be an * Amsler's Integrator will cover a considerable area, and gives in the one operation (on four cylinders and discs) the values of following integrals, viz.: — fydx; ^JyHx; ify'dx; ^fy'dx that is, the area, the statical moment, the moment of inertia, and the cubic moment about the axis x. No mechanical integrator, however, can possibly approximate to the precision attainable by arithmetic. " That is, represents the frequency y, for a given value x of the variable and not the group-mean for x—^k to x + ^k. See V., 10.— -Curves representing group- totals, formulae (211) to (216). * See " Voliuues of solids as related to transverse sections," by G. H. Knibba, Joum. Roy. Soc. N.S.W., Vol. XXXIV., pp. 36-71, 1900. See Prop. (O), p. 70. 70 APPE2SrDIX A. odd. number of equidistant ordinates the curve may be of the same degree as the number of ordinates, viz., (w+1) ; if the number of ordinates be even, the degree of the curve must be one less than that number {n). It has been shewn ako that if the curve bounding the area is of a less degree than that satisfied by the number of ordinates, then there is one-fold, two-fold, .... k-iolA infinity of multipliers which will exactly give the area, according as the degree of the curve is 1, 2, .... fc less than the number of ordinates .^ The formulae can be readily constructed, and are exhibited in the table hereunder.^ The significance of this table may be indicated as follows : — When n-\-\ equidistant ordinates are given for a curve of the wth degree, there is only one system of weights that will give the integral correctly between the limits and n. In the table this system is in- dicated in each case above by an asterisk (*). Further, when n is even, the unique series of weights, applicable to n-\-\ equidistant ordinates, is also applicable to a curve of the (w4-l)th degree, but this is not true when n is odd. When M+2 equidistant ordinates are given for a curve of the wth degree, any value whatever may be assigned to one of the weights (say Wg ), and the corresponding values of the other weights may be expressed in terms of Wq. In this case there is evidently an infinite number of possible systems of weights, each of which wiU give the integral accurately for a curve of the nth. degree. In the foregoing table the systems of this nature are indicated by a dagger (f), the coefficient (i.e., 1) of the arbit- rarily selected weight being shewn in heavy type. As an example, there may be taken the case in which seven equidistant ordinates of a fifth degree curve are given. Here the weightings shewn by the table are ti)g = Wq-, Wx = 3.3 — 6wo; w^ = — 4.2 + ISw,,; w^ = 7.8 — 2Qwg; Wi= — 4.2 + 15wq; w^ = 3.3 — Gw^; w^ = Wq. If Wq be given the value 0.3 this series becomes —(1, 5, 1, 6, 1, 5, Ij , which wiU be recognised as Weddle's rule . Similarly, when w+3 equidistant ordinates are given for a curve of wth degree, two weights may be arbitrarily selected and the remaining n-{-\ may be computed in terms thereof, thus admitting of a two-fold infinity of systems of weighting. In the foregoing table systems of this nature are indicated by a double dagger {%). Similarly, when w+4 ordinates are given for a curve of the wth degree there is a three-fold infinity of systems, when n-\-5 ordinates are given, a four-fold infinity, or, in general, when r ordinates are given for a curve of the wth degree there is an (r— w— l)-fold infinity of systenjs of weighting. ' Ibid, § 16, pp. 60-71. Examples of the development of fc-fold infinity of multipliers are given on pp. 64-67. 2 Prepared by Mr. C. H. Wiokens, A.I.A. SUMMATION AND INTEGRATION. 77 .a a c3 ■a « ,a g s-g ** S s < S3 ^ • I . .4 O -*» Jl SI u O § tn 3 c « C3 O a a .a '3 1 1 CM o X A ' 1 -^ to 1> C O -H ■* -H (N t> Ol -c ■* c(N rt ■* ■* CO 1 1 f \ I> ■ CO —1 CO f \ m lo o o io» rt t~ K5 >o i> .-H 1 rt U50 O lO -1 1-^7 1 1 (M M M ir- CO ^ TO 1 p4 ■* -O T* -H 1 1 (D rt ■* -^ ■* rt 1 1 M OO >0 T(l 1 ■§ rH :« * CO CO ■ ^ Tj< O -* i-H ^- 1 1 rf CO +4- 1 1 -IS 1 1 p4 O lO to O ' IM ■* CO — 1 1 1 1 1 % X -f -* rt X r-i CO « •-' X 00 Til 00 1 X A / ^ (N (M t>. lO (M CO 0<» X T* lO to o» oq ^ CO 1 1 1 1 P4 CO CO ^ 1 1 r^ CO 00 CO O 1 »H CO CO -1 1 1 o o »H CO CO rt IH CO CO rt 1 1 ^ O 00 CO 1 1 r4 ;o ooco 1 1 r* O >0 CO ^^ 1 p4 O >0 to 7-1 SS2 * -H |co H- 1 1 ++ 1 1 -Ico 1 1 1 X X X X X rt ■* rt t — — \ OS • CO OtO 00 t- O 00 1 1 — * — \ 1> to lO IMOO -H 1 iP 1 O H « ^ & ^ S g & & ^ ^ 1 ^ ^ ^ & * CQ 1 = H d :i -T< W * ^ ^ & ^ ^ 1 ■8(5 J O -a O o to I> 78 APPEXDIX A. 3. The value of groups in terms of ordinates. — ^It is often convenient to ascertain the value of groups between certain limits of a variable. If the ordinates be supposed to conform to the equation a+bz ; or a-\-bx-\-cx^, etc., etc., we can construct a series of equations which are rigorously true under the particular supposition, and may be regarded as approximations in the general case. By comparing the expression for the integral between assigned hmits ^^'ith the values of the ordinates, we deduce the following expressions for the heights of the groups in terms of the ordinates. TABLE Vn. Values of Group Heights for Difierent Ranges of the Variable in Terms of the Ordinates to the Curve. 1st Approximation. Formulae (225) to (228). Ranges of Integral 0-\ \-\ l-\\ l\-2 Semi-group-heights ^(82/0 + 2/1); ^(^o+Sj/i); |(-yo+%i); ^(-Syo + Tyi) 2nd Approximation. Pormulse (229) to (232). Ranges of Integral 0- J \-\ Semi- group-heights ^ (Sy, -\-5y^-y^); — {2y^ ^Wy^^y^) Ranges of Integral 1-1 J l|-2 Semi-group-heights -jg ( - 2/o + 1 l2/i + 22/2 ) ; ^ ( - 2/o + Sj/i + 82/2 ) 3rd Approximation. Formulae (233) to (236). Ranges of Integral 0-^ ^-1 ^Td£tr^"l§2(^^^^«+^^'^2/i -431/2+92/3); ~(252/„+1972/i -373/2-1-7^3) Ranges of Integral 1-1^ 1^-2 ^^^g?tr^"l^(-^2/o+1552/i+532/2-7y3);j^(-72/„+532/i+155y2-92/3) 4th Approximation. Formulae (237) to (240). Ranges of Integral. Semi-group-heights. (237) 0-1 = 28^ (16942/0 + 1969yi - II9I2/2 -f 499i/3 - 91^^ ) (238) i-1 = 28^ (3142/, + 31992/1- 9212/2 -F 3492/3 -6I2/4) (239) l-H = 2^ ( -9I2/0 + 21492/1 + 10592/2 - 281^3 + 44^/^ ) (240) lJ-2 = ~ (-612/0+6192/1 + 25892/2 - 3II2/3 + ^y^) SUMMATION AND INTEGRATION. 79 « — 1st Approximation. Pormulse (241) to (243). Ranges of Integral 0-1 h~^i 1~^ Group-heights -^ (Vo + yi) Vi ji -Vo +3^/1 ) 2nd Approximation. Formulse (244) to (246). Ranges of Integral 0-1 |-1| 1-2 Group-heights j^ (5t/o +82/1 -2/2 ); ^ (Vo +222/i +y2 Y, 12 (-^^o +§2/1 +^yz ) 3rd Approximation. Formulse (247) to (249). Ranges of Integral. Group-heights. (247) 0-1= ^ (%o + 19J/1 - 52/2 + 2/3 ) • (248) i-l|= ^ (2/0 + 222/1 + 2/2 + O2/3 ) (249) 1-2= 2^" 2/0+132/1+132/2-2/3)- 4th Approximation. Formulse (250) to (252). Ranges of Integral. Group -heights. (250) ^-1 = W ^^^^^° + ^^^^^ ~ ^^^^^ + ^*^^^' ~ ^^^' ^ (251) ^1*= 5^ (2232/0 + 53482/1 + I382/2 + 682/3 - 172/4 ) (252) li-2 = ^ ( - 192/0 + 3462/1 +456t/2 - 74ys + II2/4 ) In applying these formulse the actual common-range of the interval on the axis of abscissse is immaterial ; that is, we may read throughout to P ; ^kto k etc.; instead of to J ; ^ to 1 ; etc.; the ordinates 2/0, 2/1 > etc., being taken of course 0, k, 2k, 3k, etc. By these formulse, therefore, we may halve groups. It wiU be noticed that the coefficients are always symmetrically opposed for semi-groups standing in the same relation to the ordinates ; for example, with two ordinates, to | is the same form as | to 1 ; with three, to ^ agrees with 1| to 2, and ^ to 1 with 1 to 1^ ; with four ordinates, the only symmetrically opposed pair are 1 to 1^ and 1^ to 2. From this it is evident that, for the third and fourth approximations the formula for the remaining group-heights within the limits of the ordinates 80 APPENDIX A. used can be written down by inspection. Thus for the 3rd and 4th approximations the group-heights of the various semi -groups are as follow : — 3rd Approximation. The ordinates for the semi-group 2| to 3 are the inverse of those for to | 2 to2J „ „ , ^tol Hto2 „ „ ., ltol| (as already given). 4th Approximation. The ordinates for the semi-group 3| to 4 are the inverse of those for to ^ 3 to 3^ „ „ „ |tol 2^ to 3 „ „ „ 1 to 1^ „ ' „ 2 to 2^ „ „ „ lJto2 4. The value of group-subdivisions in terms of groups. — ^It is often required to divide a group. Practically we may always halve a group and halve again it necessary. If we divide groups with a common interval (Jc) on the axis of abscissas we may, with advantage, use the growp-Jieight (g) instead of the group number G ; that is, we may use g=0/k. Then we obtain the following series of formulae, which, like the last, are rigorously accurate if the groups are given by the integrals of the equation a-\-hx ; a-\-bx-\-cx^ ; etc.; etc. They may therefore be regarded, as in the previous instance, either as a series of approximations, or as rigorously accurate, according as they represent exactly or approximately the sub- divisions of groups given by the integral equations referred to. TABLE Vm. Values of Gteoup-heights foi different half-ranges of the variable in terms of the heights of successive whole groups. 1st Approximation. Formulae (253) to (256). Ranges of integral 0-J ^1 1-1^ 1^2 Semi-group-heights -J {5gi -g^); -^ (3gi +g2); -^ {gi -f 3^2 ); -^ ( -^i +5gz ) 2nd Approximation. Formulae (257) to (260). Ranges of integral 0-^ ^1 Semi-group-heights -g (ll?i — 4^2 + fl^s ) ; g- (5^i + 4g'2 — ffs ) Ranges of integral 1-1| 1^2 Semi-group-heights -^ (gi + ^2 - ffs) ' ^ i- ffi + ^9z + ff?)- SUMMATION AND INTEGRATION. 81 3rd Approximation. Formulae (261) to (264). Ranges of integral 0-J ^ to 1 Ranges of integral 1-1^ lJ-2 4tli Approximation. Formulae (265) to (268). Ranges of integral. Semi-group-heights. (265) 0-i = j|g (193gri - I22g2 + 88^3 - 88^4 + Ig^ ) (266) i-1 = j^ (689-1 + 122^2 - 889-3 + 389-4 - 7g^ ) (267) 1-^=118 ^'^^' + ^^^^' " ^^^' + ^^^* " ^^' ^ (268) l|-2 = jig ( - 79ri + gSgr^ +52^3 - 189r4 + 8^5 ) The opposite symmetry of the coefficients for semi-groups in S3mi- metrically opposed positions, having regard to the total number of groups in question, is obvious, as in the case for ordinates. The same remarks apply, mutatis mutandis, as those made regarding the coefficients of the ordinates. 5. Approximate computation of various moments.^In connection with the application of the method of moments in statistical investigations of distribution (population and other) it is often necessary to compute moments from available data. This can also be done from the available ordinates in the following manner : — It. is obvious that the curved boundary of any group, covering a limited range of the variable, can be represented with considerable pre- cision by a curve of the second degree : see V., § 11, formulae (217) to (224). Let the group-height be denoted by g, that is, let 9- denote the group-area divided by k, that is, the group-range on the axis of abscissae. If y' and y" are the ordinates to the curve for a—^k and a-\-^k respec- tively, and «/(, be the central ordinate, viz., at the distance a from the intersection of the axes, and if h be the distance of the mean of the terminals y' and y" from the terminal of this central ordinate, that is, if M=2/o— i iy'+y")' then the group-height is given by the equation (269) 9=^{y'+y") +j^=-Qiy' +^ya + y") and the equation to the curve is — (270) .... y=t/a + ^-^-^ (^ -«)+ p (*-«)' = ^«+ ^ ('^-'*) +"<*- *)* APPENDIX A. the origin being at the distance a from the ordinate y^. This curve is regarded as vaUd only for the group to which it applies, and not for adjoining groups. From this last equation we can compute the successive moments, Jf q denoting the area, M^ the statical moment, M2, the moment of inertia, and M^ the moment of the fourth order. It is important to attend to the signs of h and c. If y"— «/' is positive, that is, if the ordinate is increasing in the direction of a-\-\k., then b is plus ; and c is plus if the curve is convex upward : that is, if h is positive. Thus the several moments are : — (271) M^=lc{ya+Y2 "^'^ = ^ ^^/^ + \ ^) (272) Ml -aMo = ~bk^ = ^^k^ (y" - y') (273) M^ - 2aM, + a^, =^ *' (%« + 3^) (274) Ms- iaM^ + 3aWi - a^^ = §^ ** (y"-y') and may be very readily computed from these formulae, which are rigorously exact on the supposition made, and will be sensibly correct generally. 6. Statistical integrations. — Ordinarily, statistical data are subject to considerable error and uncertainty, and meticulous precision in regard thereto is, therefore, usually unmeaning. The approximations of statis- tical technique itself, should, however, aim at a somewhat higher order of accuracy than that characteristic of the data, in order that the error should not prejudicially accumulate through mere computational vitia- tion. The great majority of cases of integration occurring in ordinary statistical practice will be found to have been solved. Valuable tables of integrals are available.' 1 (i.) Sammlung von Formeln der reinen und angewondten Mathematik W. LAska, Braunschweig, 1888-1894, pp. 1-1071. (ii.) Tafeln unbestimmter Integrale. G. Petit-Bois, Leipzig, 1906. (iii.) Een Aanhangsel tot de Tafels van onbepaalde Integraleu. D. Bierens de Haan. (iv.) BxpoB6 de la th^orie des propri6t^, des formules de transformation, et des m6thodes d'6valuation des Int^grales dSfinies, partie 1, pp. 1-82 • partie 2, pp. 83-181 ; partie 3, pp. 183-698. Bierens de Haan. Amster- dam, 1860. (v.) Nouvelles Tables d'int6grales d6finies. Bierens de Haan, parties pp. 1-733, Engels, Leide, 1867, ' i # SUMMATION AND INTEGRATION. 83 The integrals of curves of the type of (20), II., § 19, are sometimes required : that is, — (275) /a(6a;)±"'±"''-^ dx = '^f y±"'±''y dy = Ajy'^e^v^^svdy = Afe^^™^'^y^^"svdy in which A= a/b ; and y= bx. This last form may be expressed by an exponential series. Or {2'7G).. fyo+nvdy =/2/™ Jl + ny log y + | (««/ log y) " +....]dy which may be integrated term by term. Again (277)../x-c^x=.|l-2,+-3^-^+..;+-^|2-y^+-43----| w ^a; 3 (log x) 2 Jl nx n^x^ 2 ! 3 42 ^ 53 ■■■} + ..Ketc. Similarly, forms of the type of formula (32a), see II., § 23 1 log X (278) ./a;<±'"±"""'da; =/e<±"'±™)''da; can, if m and n be regarded as positive, be put in the form ^ '■ ■■•Jy ^ {m^nxy ^ 2! {m^nxfv ^'-^ h\ (m+nxfv ^' ' j* which can be integrated term by term. The integrals, however, are tedious. For example : — /- log x , log X 1 f 1 1 1 1 1 1 (p—^)m4>v-^ "'"(p-4)m20J'-4 '^■■"'"2wP-Y^ ^ wP-^(f>] 1 J « "'"(^-l)mJ'-i» °^ .^ ^ denoting (m+wa;)''. If ^ = 1, and n is positive, this takes the simpler form — (281) . . . .ya;^^+^* = - log a; log {m+nx) - ^ (log nx)^ ^ ■m.a m* , ^ Owing to the very great elaboration of the terms of many of the integrals, practically it is preferable to compute a sufficient number of ordinates, and integrate by any suitable summation-formula (given hereinbefore). 84 APPENDIX A 7. The Eulerian integrals or Beta and Gamma functions. — ^The Beta and Gamma functions are of special importance in statistical integrations. They are : — (282) J ^x ^i X) ax~/^z ^i z) — /^, (1 + ^)'+"' - » (l+y)«+'« that is, in the more brief notation — (283) B (I, m) = B (m, I) = ^p^~^~^^ Further — (284:)..C e'-'x^-^dx = (^ 0°^-)" dy=(n-l)r e'"^ x"-^dx = r{n) respectively, from which it is evident that : — ^ (285) r(l) = l; r(w + l)=w! =nr(n) Thus, in order to calculate F (n) we have, if it be an integer, it is equal to (n—1) ! , if not an integer, it can be readily found, since its logarithms have been tabulated for the range 1 to 2 to two places of decimals and to 9 places of figures.^ Thus — (286) n {n + 1) {n+2)....{n + k- 1) T {n) =r{n + k) which, logarithmically, is perfectly convenient to use. By putting kz = x, in (284), it becomes obvious that (287) ./J'^e -"^ a;»-i dx = ^-^ (288) ^-^ W = ^'(^) =/o'"e-^a;"-i log a: dx. Examples of the application of these formulae have already been given: see IV., § 5, formulge (150) to (166). 8. Table of indefinite and definite integrals and limits. — ^In an addendum small tables are given, for convenience, of indefinite integrals ; of definite integrals, for example, between hmits such as zero and unity ; zero and infinity ; etc., and of limiting values. These embrace those which more frequently occur in statistical investigations. ' r(i) = ^/n. ' Traits des Fonctions EUiptiques, Legendre, Paris 1825-8 (logarithms to 12 places). Sammlung von Formeki, W. LAska, pp. 290-1. Braunschweig (logarithms to 9 places). Biometrika, J. H. Duffell, Vol. VII., 1909-10, pp. 43-7 (logarithms to 7 places). Vn.— THE PLACE OF GRAPHICS AND SMOOTHING, IN THE ANALYSIS OF POPULATION-STATISTICS. 1. General. — Graphs of the data are necessary in any analysis of population-statistics purporting to aim at thoroughness. A graph indicates not only the general trend of the data, but also whether the individual items conform with great exactitude to that trend, or whether they deviate considerably therefrom. The criticism of deviations ordinarily depends upon whether numbers or ratios are being analysed. Where figures are of the nature of ratios, if, on the working-graphs the numbers be written, it is possible to see at a glance whether changes in any part of the graph of the crude data are significant or otherwise. Thus a ratio resulting from 30,000 divided by 10,000 would be materially changed so far as the numerical data are concerned by an alteration, say, of one- thousandth. To change the ratio say from 3 to 2.997 would mean an alteration of 30 in the numerator or of 10 in the denominator ; whereas, if the original data were the numbers 3 and 1 , an alteration of a single unit would greatly disturb the ratio. In general, we are concerned with two kinds of alteration ; one may be called the " redistribution of the data without alteration of their aggregate ;" and the other may be called the " alteration of data to coincide with what is deemed the most probable result," having regard to all the facts. It is, for example, sometimes desirable to keep the aggregate of the smoothed results identical with that of the data. In other cases this is less essential, and it may be said that probably m^uch time is often wasted in making re-distributed data agree with the original as to the aggregate of units represented. As to general method it may be noted that when the original facts have been plotted, a curve may be drawn freehand by anyone familiar with the characteristics of the various type-curves, and especially those of probabiUty-curves. By means of sets of curves, French curves, and sphnes of various kinds,* the freehand curves may then be improved so as to be really smooth and conform to what might be called the probable indication of the data. When the numbers represented are large, limitations of scale may operate to Hmit the smoothness as deduced by scaled values, from the graphs, but a little simple differencing wiU suggest necessary adjustments, or the differences may be graphed. The adjust- ments having been made, the aggregate can be formed by adding together the scaled or properly differenced ordinates thus adjusted. If this operation has been weU done the total will be so nearly in agreement with the original data that a common factor of correction can be used throughout, that is, all the ordinates may be increased or dimin- ished in the same ratio, and the finally deduced ordinates will then agree * Splines of transparent celluloid are most convenient. 86 APPENDIX A. with the data, and at the same time form a smooth curve. If the data when plotted are visibly irregular, meticulous precision in adjustment is obviously but a waste of time. For this reason one of the great merits of the graphic method is that, not only can the analyst see at a glance the conformity or otherwise of the data to a particular type of curve, but he can also judge whether the data yield results of a high order of precision. It has already been mentioned (see IV., § 1) that the initial and terminal characters of the curve and its mode (maximum and minimum) are important. It may be added, that if the curve is not drawn as uni- modal in type, the reason for the adoption ctf a particular form must really depend on the character of the data, and may not be decided merely upon mathematical considerations. 2. The theory of smoothing statistical data. — ^It may often be known a priori that phenomena should exhibit a regular progression, and that data, when graphed, shewing as zig-zag hnes, do not really represent the ideal fact, owing either to the paucity of the data, or to unavoidable error therein. In a series of group^alues, i.e., totals or aggregates between a series of limits of a variable, it is important to bear in mind that — ^assuming the counts on which they depend to be correct — ^what is known is merely the series of aggregates themselves : the probable distribution yielding these aggregates has to be conjectured. When the totals or aggregates are themselves regarded as subject to error, then the distribution may be modified within the Umits of probable uncertainty, some groups being diminished and others, particularly adjoining ones, increased. There are four principal classes of data to which the process of curve- smoothing is appUcable. These may be indicated as foUows : — (i.) Frequencies of a phenomenon at successive epochs or during successive periods of time ; as, for example, population estimates at given dates and numbers of deaths occurring during successive years, (ii.) Rates of occurrence of a phenomenon per unit of reference during successive periods ; as, for example, birth-rates per thousand of population per annum for successive years, (iii.) Frequencies in respect of successive values of characters capable of continuous variation ; as, for example, the number of persons at each age recorded at a given census, (iv.) Rates of occurrence of a phenomenon per unit of reference in respect of successive values of characters susceptible of con- tinuous variation ; as, for example, rates of mortahty per unit per annum during a given decennium in respect of each age. In all these cases the characteristic of continuous variation^ is assumed to exist either actually or virtually. Where statistical results are discontinuoiia such a process is, strictly speaking, inapphcable ; as for 1 See I., § 9. ~ GRAPHICS AND SMOOTHING IN POPULATION STATISTICS. 87 example, in the tabulation of census ]Dopulation according to birthplace, occupation, or reUgion. In some cases, however, although the data are strictly speaking discontinuous, the principle may be applied partially ; for example, in the case of a tabulation of dweUings according to number of rooms or according to number of inmates. In such cases the character possessed is progressive without being continuous ; nevertheless, with proper qualifications, the smoothing principle may be applied even to these. Another example, more nearly approaching but not attaining con- tinuous variation, is the representation of dwellings according to rental value. 3. Object of smoothing. — ^From the foregoing it wiU be seen that the data to which the smoothing process is strictly appUcable are those which may be regarded as functions of a continuous variable. But whether such functions are readily expressible by means of algebraic formulae or not, is, of course, reaUy immaterial. The essence of the matter is that in any instance the data are in the main such as admit of representation by means of a continuous hne, or a continuous surface or sohd in relation to continuous units of reference. When such representation has been made of the crude results of observation, it is ordinarily found that the line surface or solid exhibits evidences of marked irregularities as between adjacent points or series of points, their general trend, however, suggesting an underlying basis of orderly progression. This progression is, of course, afEected by minor influences operating at individual points, and is more or less masked by the paucity of the data on which the repre- sentation has been based ; thus, suggesting further that were it possible to obtain data of unlimited extent, these irregularities would become negligible. For this reason the object of the smoothing process may be said to be that of removing these apparently accidental irregularities, and of thus disclosing the basic or ideal uniformity which may be presumed to represent the facts in aU their generahty. 4. Justification for smoothing process. — ^The justifications for the smoothing process may thus be said to be : — (a) That the irregularity does not represent the phenomenon in its generality, since much of the observed irregularity is known a priori to be due only to paucity of data ; (6) or that it is known that the phenomenon subject to observation is reaUy regular ; (c) or, again, that the observed data suggest that regularity of trend wiU not efficiently represent them. It has been objected that any system of smoothing is, strictly speak- ing, unwarrantable, since such a process virtually attempts to make the facts accord with more or less questionable preconceptions regarding them. To this view it may be rejoined that if the process were such as to produce results which, though smooth, differed systematically and materi- ally in their distribution from the original observations, the objection would be valid. Where, however, due consideration is given to the 88 APPENDIX A. relative magnitudes of the original data, and the smoothed results accord therewith as closely as the data will allow when these exhibit a general trend, then the only preconception that can be regarded as operative is the justifiable one that ordinarily natural phenomena do not progress per saltum. In this connection it must be noted that where there is distinct evidence at any stage of a cataclysmic disturbance of results, the smoothing process for such points or periods will usually be invaUd or not properly applicable. Examples of such cataclysmic disturbances of statistical data are war, famine, pestilence, earthquake, etc. Even in these cases, however, it appears admissible under certain circumstances to apply a smoothing process ; as, for example, in cases where the disturbances referred to are of more or less frequent occurrence, and are not merely isolated instances. One of the most cogent justifications for the smoothing process has its warrant in the fact that the recorded results of any statistical observa- tions are necessarily approximative, and hence that the value of the function recorded for any given value of the variable is probably not usually more accurate than an estimate based on the recorded values in respect of preceding and succeeding values of the variable. This con- sideration suggests the idea of weighting successive observations to obtain most probable values, which idea forms the basis of one of the leading methods of adjustment. Again, where the results of the observations are to be employed as guides to future action, it is clear that these results should, as far as practicable, be freed from all fluctuations which may be considered merely accidental, and thus unlikely to be reproduced in future experience. This is of considerable importance in connection with the construction of mortality and sickness, superannuation, and similar tables to be used in the computation of rates of premium, and for the conduct of valuations. 5. Mode of application of smoothing processes. — ^It has already been indicated that one of the main objects of the smoothing process is the discovery of a smooth series which presumably underKes the irregular data furnished by a limited number of observations, and it has been implied that a process to be justifiable must, in addition to smoothness, be characterised also by what has been called " goodness of fit"; that is, within reasonable limits it must reproduce the characteristic features of the original data. The methods of applying the smoothing process vhich have up to the present been employed, may conveniently be grouped in three classes, viz. : — (a) Graphic Methods ; (6) Summation Methods ; and (c) Methods of Functional Conformity. These methods have been employed in connection with observations in many fields of research ; as, for example, general statistics, actuarial science, physics and chemistry, astronomy, tidal theory, biology, etc. In the actuarial field, an extensive and systematic use of the process has been made, and a most detailed examination of the underlying principles has been carried out. GRAPHICS AND SMOOTHING IN POPULATION STATISTICS. 89 (a) Graphic method. — As its name indicates, this method is based on the attainment of the desired smoothness by means of a graphical representation and adjustment of the observed data. For example, the subject of observation being the infantile mortality e;x:perienced in a community during a given period, and the periods of observation being calendar years, a base line is taken and divided into equal parts, each of which represents a year. On these parts as bases a series of rectangles is constructed, the area of each rectangle being proportional to the rate of infantile mortality averaged for the corresponding year. The upper parts of these rectangles will present in the case supposed the appearance of flights of steps with uniform treads and unequal rises. The necessary smoothing may be effected by drawing a continuous free-hand curve through the upper portions of these rectangles in such a manner as to include between certain limits the same area approximately as is contained in the rectangles covering the same range.^ The area enclosed by the part of the base hne relating to any year, the ordinates drawn from the extremities of this part, and the portion of the curve between these ordinates will represent the smoothed result for the year under review. Whether, as in the example just given, the data should be represented by areas, or, as is sometimes more suitable, by ordinates, is a matter which is determined agreeably to the appropriate interpretation of the result to be attained. It may be noted that the method of representation by rectangular areas is specially applicable to cases where the data are functions not of single values of the variable, but of ranges of such values. For instance, in the above example, the rate of infantile mortaUty stated for any year is a function not of any one point of time in that year, but of the range of values representing the whole of the year. In most cases, however, the system of representation by means of ordinates would be equally valid, and sometimes more convenient.^ Referring again to the above example, from a point on the base hne representing the end of each year an ordinate could be drawn representing the rate of infantile mor- tality for that year, and a free-hand curve being drawn amongst the upper points of these ordinates, the ordinate to any point on the curve would represent the rate of infantile mortahty for the year ending on the date corresponding to the foot of the ordinate. Similarly, the ordinate for smoothing" might be drawn from the beginning or the middle of the line for each year, or, indeed, from any point uniformly selected in each, and a corresponding interpretation of any point taken on the curves drawn amongst the upper points of such ordinates would be apphcable. 6. On smoothing by differencing. — ^A curve continually convex (or continually concave) upward might possibly be drawn with a single difference. We have, by the theory of differentiation — (289). .dy/dx=d(a+bxP+cx^+eto.)/dx=pbxP-^+qcxi-^ + etc.; ^ In practical examples it is rarely possible to make the curve such that the adjusted areas are continually identical with the rectangles on the same base. « See, however, V., § 10, formulae (212) to (216). 90 APPENDIX A. hence, Up, org, etc., should happen to be integers, at some stage of differ- entiation, this particular term of the expression wiU be x''=l, and hence that difference wOl vanish. Probably in no case are population-statistical results actually representable by integral values oip, q, etc., hence, strictly, there is no limit to the series of differences. These, however, ultimately become high negative powers of x, and consequently when x is large their value is small : they must ultimately become of negUgible amount. Again, statistical data often involve exponential forms, particularly those of the type ae~"*, the differential of which is — nae~'^'', from which it is evident the successive differences are interminable. Since, however, de ~^/dx =l/e*, the higher differences for large values of x become insensible. Hence, we shall always be justified in taking differences only to the stage where they are appreciable. Thus if at any stage of smoothing we make the second difference a constant, we are making the curve one which the equation y^a-\-bx-\-cx^ wUl reproduce ; if we go on then with a constant third difference, we add a stretch of a new curve, viz., y'=a'-f-6'a;+c'a;^ -{-d'x^; and so on. Such methods are unobjectionable when the tangents to the curve at the point of junction may be regarded as sensibly identical. 7. Effect of changing the magnitude of the differences. — ^It is often useful to be able to recognise instantly the consequence of changing the magnitude of a difference. This can be indicated at once by a table. Table IX. — Efiect on the value of a function of a change of a unit in a leading difference. Difference in which the change takes place. Effect on the value of y where its suffix is- 1 2 ! 3 6 8 9 10 1st difference 2nd difference 3rd difference 4th difference 5th difference 1 2 1 1 ! I j ! ' 5 10 10 5 1 6 7 8 9 15 21 28 36 20 15 35 I 56 35 1 70 21 i 56 84 126 126 10 45 120 210 252 It will be recognised that these are the figures of Pascal's triangle taken diagonally, or the diagonal series in this are the figures of Pascal's triangle taken vertically. By means of such a table one can see at a glance the effect on any value of the function of changing a leading difference. GRAPHICS .\ND SMOOTHING IN POPULATION STATISTICS. 91 8. Smoothing, by operations on factors. — ^The smoothing of a suc- cession of ordinates or of group-values may often advantageously be effected not by operating upon these numbers themselves, but upon their ratios to each other. This may be called factorial smoothing. Let A, B, G, D, etc., be the series of quantities to be smoothed. The ratios B/ A, C/B, D/0, etc., are formed, and denoted by b, c, d, etc. These are graphed and smoothed by any process. '^ The smoothed values, denoted hyb',c',d', etc., are then used to form a new series of quantities ; thus A = A, Ab' = B"; B"0' = G", etc. The sum of these is then made equal to the sum of the original series of quantities by a common factor k, thus — (. (290).. /fc A + B + C+D+ etc. ^ ]l+b'\l+c\l+d(l + ..)\\f- ' A+ Ab+ Abc+ Abcd+eto. ^ |i^^/ J^^' ; l-Fd(l-f . I then the smoothed values A', B', etc., are A'=kA; B'= kAb' ; C'=kAb'c'; D'= kAb'c'd' ; etc. Sometimes, on taking out the ratios, it becomes evident that they should have a common value, since they shew no systematic progression. In such a case, let m denote the mean value, then the denominator A-\~Ab + Abe + etc. in (290) becomes A + Am + Am^ -\- etc. Smoothing of this kind is serviceable for initial and terminal values. 9. Logarithmic smoothing. — ^In a similar manner quantities may sometimes be advantageously smoothed by smoothing their logarithms. In this connection we bear in mind that it a series of numbers are in geometrical progression their logarithms are in arithmetical progression. Let log A, log B, etc., be denoted by a, ^, etc., which are graphed, and when smoothed denoted by a', j8', etc. If the sum of A", B", etc., corresponding to the smoothed values, do not agree with that of the original values, k will be the factor of correction, and may be found as before, that is, by (290). This process may be called logarithmic smoothing, and like factorial smoothing, is often useful for initial and terminal values. 10. On the difference between instantaneous and grouped results.^— When instantaneous results are smoothed the resulting smooth curve represents the equation which reproduces the values of y corresponding to given values of the abscissas. When, however, group-results are smoothed by differencing, the resultant curve strictly represents the value of a group of the same base (supposed, of course, constant) with any central value throughout the range smoothed: see V., § 10. When, however, group results are few in number (that is, have relatively large bases) the graph must be drawn upon a different principle, viz., it must, as far as the probabOities of the case wiU admit, make the areas between bounded by the curve, the abscissae, and the ordinates identical with the ' Arithmetically, i.e., by difference, or mechanically, by splines, etc. 92 .-UTENDIX A. area of the group, or, in other words, the mean height of all the ordinates to the curve in any given range of the abscissa must be equal to the height of the group. That is, if /i is the height of the group, then : — (291) h = ~-\_ r'f(x)dx. X2 — Xi Jxi / (.r) denoting the smoothed curve drawn. This method may be called " the method of equivalent grov/p-values," and it will, in general, either not depend on differencing at all, or depend thereon to a less extent than when the bases are relatively smooth and the groups numerous. 11. Determination of the exact position and height of the mode. — It is often desirable to ascertain with such precision as is possible the abscissa and height of the mode. Two approximate solutions are de- sirable, viz. : — (a) when the graph shews that three groups should be taken into consideration; and (6) when/owr groups. In the former case (a) the formulse are extremely simple ; in the latter (6) they are much less so. If more than four groups are to be taken into consideration it is better to determine the general equation of the curve and solve to obtain that value of X which makes dy/dx=0. As an approximate solution will be available from the graph, there is usually very httle difficulty in obtaining an exact value of x. Then the corresponding value of y can be found from the equation: see V., §§ 3 to 7. Case (a). In Fig. 35 let K denote the mean of the heights of the groups on either side of the maximum group and the height of this last, and let k be half the difference of the height of the groups on either side. Let also a denote the difference of the height of one group and the greatest group, and |3 similarly the difference of the height of the other group and the greatest group. Then (292) K =~ (a+/8); and A = 1 (a . ^). Then a second degree curve, giving the same group values, gives the abscissa of the mode: — (293) M = -^5 ; aici ^,' = -J-^ and the height A, of the mode, above the maximum group is (294) ^=r2^ + lT If /, g, and h denote the heights of the rectangles we should have for the constants of the curve — (295) « =~{nf + 2h-lg) (295a) . . . .b = 3g - 2f - h (295b) ....c =^{h+f)-g the base of the curve being considered unity. GRAPHICS AND SMOOTHING IN POPULATION STATISTICS. 93 In the case (6), differences of height being as shewn in Fig. 36 the constants of the curve which must now contain dx^ will be {29Q)..a=-^-^{y+y'); 6=^(15/3-8'); c 12 (y + y')' a being reckoned from the point K, half-way between A and B to the point L, that is, to the curve. The value of the abscissa of the mode is given by (297).. x„ y +y' 2(3^8 ' 1 I ./fl I 2 (15|8-8')(3^-§') - The sign of the term under the radical can readily be determined in a practical example. The general expression for y,^ is lengthy. In cases practically occurring we may compute it from x„ when that value is found : that is, it is (298). .y„ = ±^^y + y'(l -3a;;)+ (15^ - 8'):«„-2(3^-§')<| the ordinate being reckoned from the line parallel to the axis of abscissae and half-way between the points A and B in Fig. 36, i.e., the line MJ in the figure. IP 'a T PG 0C+/3 » Fig. 35 Fig 36. The formulae (293) and (294) and (297) and (298) are not quite satisfactory, and in general it is better to compute the coefficients of the equation which fits a considerable stretch of the curve, and find the position of the maximum by dy/dx = 0, if very great precision be required. 94 APPENDIX A. 12. The testing of smoothed or graphic results. — ^When smoothed graphed resxdts are obtained they will, in general, need, as already indi- cated, to be arithmetically tested. The fundamentals of arithmetical testing are the foUowing : — (i.) The sum of the graphed results should be sensibly (or exactly) equal to the sum of the original data ; (ii.) The deviations, positive and negative, between the aggregate of the smoothed results and the data up to each given value of the argument should, consistently with the type of curve adopted, be a minimum ; (iii.) The position and ordinate of the mode should be carefully fixed, and as well as the data will permit ; (iv.) The position of the terminals should conform to the probabiUties of the type of data so far as that can be determined.^ ■^ In general, they cannot be determined mathematically. For example, the frequency of births of given ages, so far as mathematical relations are concerned, might be continued to start at the age 0, but in view of physiological considerations we shoTild not be justified in starting at 0, but at, say, the age 11 ; similarly in regard to the terminal, which may be made to meet the axis of abscissae for age 60 (or such later age as may be indicated as occurring, should satisfactory information be to hand). Vm.— CONSPECTUS OF POPULATION-CHARACTERS. 1 . GeneraL — Thus far the consideration of the theory of population has been concerned only with its numerical aspect, and with the mathe- matical form of expressions under which it may be necessary to subsume the facts. These constitute an essential preKminary only. It remains now to consider in detail some of the various characters of importance. Not only are population-statistics, in the narrower sense, signi- ficant both (i.) in themselves, and (ii.) in comparison, but so also are all facts that may properly be regarded as expressions of the various char- acteristics of a population. Following the nomenclature of biology, these may be called more briefly its characters. Such characters may relate to — (a) Vital phenomena, that is, to birth, life and death, to repro- duction in all its aspects, to disease and all the modes of its incidence ; (6) Anthropometry, that is, may relate merely to the human form and its variations, or to its growth and decrepitude , (c) Anthropology, that is, they may refer to man's general evolu- tion, both physical and psychical ; (d) Sociology, that is, they may concern man in respect of his social life, an important element in which is his economic evolution, and they may concern also the reaction of this upon his numbers and the density of his aggregation. (e) Migration, aggregation, segregation, or wide dispersion, colonis- ation, etc., that is, the direction and velocity of movement of populations, the tendency to Uve in more or less dense groups (large cities or villages) or to spread over the earth, etc. All these have significance in regard to the rate of development of the world's people. It is well to bear in mind, also, that population- characters may be in two forms, viz., either actual or potential. The importance of the subject is seen in the impossibility of maintain- ing the present rate of increase for any great length of time (see II., § 34) ; and its range of subjects is best seen through a conspectus. Characters may be simple or complex, their manifestation may be instan- taneous or durational ; and the evidence of their nature direct or de- rivative. The greatness of the range of population-characters, and the number of significant relations subsisting among them is so vast that no statistical presentation of them can be exhaustive. Thus important questions are continually arising involving demands for new statistical compilation, for human affairs can be properly analysed only with the aid of a well-founded and technically satisfactory statistic. The simplest population-characters are expressible in regard to units, as, for example, the numbers in a population ; the wealth possessed, etc. The complex 96 APPENDIX A. are those which involve multiple fields of comparison, for example, the number of one sex, who, being between given limits of age, and belonging to a given occupation, die of a particular disease. That the number of comparisons possible is very great is obvious from the fact that n things considered in their mutual instantaneous relations, that is, n things considered each in relation to 1 n—l other things, are 2"— 1. The following table will shew the number possible up to w=10. TABLE X. No. of Elements in Combination Elements of Original Statistical Data. 1 2 3 4 5 6 7 8 9 10 4 , 5 6 10 4 1 10 5 1 6 15 20 15 6 1 7 21 35 35 21 7 1 8 9 10 28 36 45 56 84 120 70 126 210 56 126 252 28 84 210 8 36 120 1 9 45 1 10 1 Total possible combinations of elements 1 15 31 63 127 255 511 I 1023 The total possible for 12 is 4095, for 20 is 1,048,575. There were, for example, 17 main questions to be answered in the Australian Census ; thus there would be 2^' — 1 (viz., 131,071) possible tables by combinations of these results, and a considerable proportion of these would be of real significance. 2. Characters directly given or derivative. — ^Important characters are not always immediately yielded by the data : they are often to be ascertained only by analysis. Thus, as in the case of statistics generally, population statistics may be either A. Direct, viz.:- A (i.) Instantane- ous (numbers at a given moment). (Examples) : No. of persons living ; wealth possessed by them at a particular instant ; etc., or A (ii.) Durational (or number of occurrences dur- ing a unit of time) (Examples): Num- ber of persons bom, married, or deceased diu'ing a day, month, or year ; etc. or) B B (i.) Instantane- ous (nvimbers de- duced represent- ing a state of things for a given moment or epoch). (Examples) : Mas- cuUnity at birth, or at a census ; wealth possessed, per individual ; expectation of life ; etc., or Derivative, viz.: — B (ii.) Diu-ational (numbers deduced of occurrences dur- ing a imit of time). (Examples) : Birth, marriage, or death- rates per day, month, or year ; average wealth de- duced from probate returns ; etc. The above indication of the nature of population statistic reveals the reason of its extent, which is much greater than is implied in the CONSPECTUS OF POPULATION CHARACTERS. 97 number of mere combinations of different fields of statistic considered in their instantaneous relations alone. 3. Characters in their instantaneous and progressive relations. — The characters of a population are fully studied only when examined both in their instantaneous relations, and in the progression of these with time. Suppose, for example, that characters A and B both vary with time, and that such variation can be expressed by rational integral functions thereof ; then the constant relation of the characters is given by f2q9^ - = l^ = a2(l + b^t + c^t^ + etc.) ^ ' ' A ~ Fi{t) ^ ai(l + bit + Cit^ + etc.) = ^ [1 +(62-6i)i-.|6i(62-6i)-{C2-Ci);-, = - ix.t\ then it will follow that the ratio f for persons will be (305). .v = (I + |)m+ (-i - |)/= ^ N+/) + 4 /^ (»*-/) ^ 8av, ordinarily at least to « = 100. 98 APPENDIX A. that is, it will be the mean weighted according to the relative numbers of males and females. The result may at once be written out from (302) and (303), and re-expressed is {30Q)..pt = '^j[w(, + /o+/^o("*o-/o)]+'»Wo i6m(l + A^o) + /^o^'I + /o {cf (1 - /Lto) - f^oibf ^'+ Y')\]t'+ etcj From this it is obviously impossible to secure consistency among formulae for persons, males, and females, where the variation with time of those for the two last is not identical, without complexity of expression. Moreover, when variations with time have to be considered, as well as many fields of comparison, not only do general formulcB become too involved to be of practical value, but also the number of relations neces- sary to exhaust the statistic becomes hopelessly large. For this reason it is often desirable to compute the coefiBcients for males, females and persons independently : if this be done with care the involved incon- sistency may be regarded as negligible. 4. Conspectus of the population-characters with which the ordinary census is concerned. — ^In Section 1 of Chapter II. of the general Census Report, a classified statement and a brief review of the objects and uses of a census are given. These present, however, only one aspect of some of the leading characters of population. In the following conspectus a somewhat different and more extensive sketch of such of these characters as are capable of statistical measurement, and which constitute normal bases for comparisons, is furnished : — A. — ^Numerical constitution of population at a given epoch in regard to (i.) Sex, and (ii.) age ; (iii.) birthplace ; and (iv.) length of residence in country of enumeration ; (v.) nationaUty ; and (vi.) race ; (vii.) conjugal condition ; (vui.) duration of marriage ; and (ix.) size of family ; (x.) infirmity ; (xi.) degree of education ; and (xii.) school attendance ; (xiii.) rehgion ; (xiv.) occupation — {a) designation ; and (6) grade ; (XV.) dwellings — (a) material ; and (6) number of rooms ; (c) mode of occupancy ; and (d) rental ; (xvi.) localisation. In each case the statistical data initially represent the number of persons possessing the character or group of characters specified, as, for example, the number of persons having a family of a given size, the number of persons having a given duration of marriage, CONSPECTUS OF POPULATION CHARACTERS. 99 In the case of dwellings the enumeration is twofold, and comprises, for example, the number of dwellings of a given material, as well as the number of persons Hving in dwellings of a given material. B. — ^Relative constitution of population in respect of characters enumerated in A. In this section are comprised the ratios of the numbers possessing a given character or group of characters to the numbers possessing a wider range of such characters, as, for example, the ratio of males under 21 years of age to the total population of all ages and of both sexes. C. — Variations of population at different epochs. This may involve merely variations in aggregate population, or may comprise variations in the numbers possessing any combination of the characters enumerated in A, or in the relative constitutions deduced under B. D. — Mean population at a given period. As in the case of C, this may involve merely the aggregate population or may comprise the mean population possessing any combination of the characters enumerated in A. The mean population for any unit of time represents the number of such units of human life hved by the population or section thereof under observation. E. — ^Fluctuations of population during a given period. These arise from : — (i.) Births (see F) ; (ii.) deaths (see G) ; (iii.) migration (see H). F.— Births, (a) The statistical data initially represent the number of births classed according to the following categories, taken either singly or in combination. (i.) Whether live or still birth ; (ii.) sex of child ; (iii.) whether born in wedlock or not ; (iv.) age of father ; and (v.) age of mother ; (vi.) birthplace of father ; and (vii.) birthplace of mother ; (viii.) occupation of father ; (ix.) duration of parents' marriage (see I.) ; (x.) locality ; and (xi.) date of birth ; (xii.) date of registration ; and (xiii.) position of child in family, i.e., whether first, second, etc. (xiv.) single or multiple birth. (6) The derivative statistical results comprise, inter alia, particulars concerning the relations between (i.) Live and stUI births ; and (ii.) nuptial and ex-nuptial births ; (iii.) male and female births ; (iv.) number of births and population from which derived, UBRARY SEP 17 1945 DEer. OF AGRIC, ECON. 100 APPENDIX A. These may involve merely the relation between total births and total population, or the relation between the number of births possessing any character or group of characters enumerated in F (a) and the appro- priate subdivision of population from which derived. In the one case the result would be the crude birth-rate, or ratio of total births to total population, in the other it would comprise such results as, say, the nuptial birth-rate in a given area amongst fathers of a given age, birthplace, and occupation, who had been married for a given period. Similarly (i.), (ii.) and (iii.) may involve merely totals possessing the characters specified, or may relate to subdivisions possessing any character or group of characters enumerated in E : as, for example, the relation between live and still births amongst the nuptial male births of women of a given age and birthplace, who had been married for a given period. G. — ^Deaths. (a) The statistical data initially represent the number of deaths classed according to the following categories, taken either singly or in combination : — (i.) Sex of deceased ; (ii.) age ; and (iii.) birthplace ; (iv.) cause of death, (a) primary, and (6) secondary ; (v.) occupation; (vi.) length of residence; and (vii.) locahty; (viii.) age at marriage and re-marriage ; (ix.) number of issue, according to sex, and whether hving or dead ; (x.) date of registration. (6) The derivative statistical results consist mainly of particulars concerning the relations between the number of deaths possessing any character or group of characters enumerated in G (a) and the appropriate subdivision of population from which derived, such, for example, as the death rate from a specified cause in a given locality amongst males of a given age, birthplace and occupation. (c) As derivative results of the second degree may be classed such particulars as (i.) Index of mortahty ; and (ii.) corrected death-rates ; (iii.) expectation of fife ; and (iv.) detailed mortahty tables. H.— Migration. Complete statistical data would initially represent an enumeration of migrants classed according to the characters specified in A, with the exception of (xi.) length of residence ; and (xv.) dwellings. Such detail is quite impracticable, and the main characters available in Australia are : — (a) For traf&c by sea : — (i.) Sex ; and (ii.) whether adult or child, or preferably exact age; (iii.) port of embarkation ; and (iv.) port of disembarkation ; (v.) nationaUty or race ; and (vi.) date of migratioi), CONSPECTUS OF POPULATION CHARACTERS. 101 (6) For land-trafific by rail : — (i.) Sex ; (ii.) state in which arrived ; and (iii.) from which departed ; ■ (iv.) date of migration, (c) For land-traffic by road : — Similar details as in (6).i I. — ^IVIamage. (a) The statistical data initially represent the number of marriages granted in a given period classed according to the following categories taken either singly or in combination : — (i.) Age of bridegroom ; and (ii.) of bride ; (iii.) conjugal condition of bridegroom ; and (iv.) of bride ; (v.) birthplace of bridegroom ; and (vi.) of bride ; (vii.) occupation of bridegroom ; (viii.) locaUty ; and (ix.) date of registration ; (x.) by whom celebrated ; (xi.) ability of bridegroom to sign register ; and (xii.) of bride. (6) The principal derivative statistical results are those concerning the relations between the number of persons married during a given period and possessing any character or group of characters enumerated in I (a) and the appropriate subdivision of the population from which derived, such, for example, as the marriage rate amongst bachelors of a given age, birthplace and occupation. J. — ^Divorce. ^ (a) Satisfactory statistical data would initially represent the number of divorces granted in a given period classed according to the following categories taken either singly or in combination : — (i.) Age of husband ; and (ii.) of wife ; (iii.) duration ; and (iv.) issue of marriage (a) males'; (b) females ; (v.) locality ; and (vi.) birthplace of husband ; and (vii.) of wife ; (viii.) occupation of husband ; (ix.) sex of petitioner ; and (x.) cause of petition ; (xi.) date of rule nisi ; (xii.) and of making rule absolute ; (xiii.) by whom marriage was celebrated. (b) The principal statistical results derivative from the foregoing would be relations between the numbers of persons divorced during a given period and possessing any character or group of characters enumer- ated in J (a), and the appropriate subdivision of the population from which derived, as, for example, the proportion of husbands of a given age, birthplace and occupation, who had been petitioners in granted divorce cases. ' In Australia thia last information ia not available. ^ Complete statistics not available in Australia, 102 APPENDIX A. K. — Sickness and Accident.^ (a) Satisfactory statistical data initially represent the cases of dis- ablement by sickness or accident occurring in a given period classed according to the following categories taken singly or in combination : — (i.) Sex ; (ii.) age ; and (ui.) birthplace of person disabled ; ( iv.) cause of disablement ; (v,) occupation ; and (vi.) locality ; (vii.) date ; and (viii.) duration of disablement ; (ix.) conjugal condition of person disabled ; and (x.) number of issue : (xi.) whether or not disablement terminated by death. (6) Derivative statistical results would consist mainly of relations between : — (i.) cases and appropriate population ; (ii.) cases of deaths, (c) Derivative results of a second degree consist of sickness tables constructed from initial data. 5. The range of the wider theory of population. — ^The conspectus just given has obviously been hmited to matters with which the census and ordinary vital statistics are more directly concerned. In a wide consideration of population, however, the characters of importance include a much larger range, embracing what has already been indicated in § 1, hereinbefore, viz., the anthropometric, anthropological, and sociological, including the economic. This has already been referred to : see I., § 6, iii. and iv., and II., §§ 13-18. Because of this fact, a complete theory of population must take account of (a) the reactions of eugenic facts and arrangements upon the numbers and mode of growth of the population of the entire world and of its constituent peoples, and (b) even of the reactions thereupon of all economic and social conditions, including those arising from mobility. This is seen when one contemplates the part played by modern facUities in transport and communication. Nor are the physical and psychical characters of the population less foreign to a complete theory. For the same reason there are aspects of subjects not directly enumerable as population facts, which have immediate touch therewith ; such, for example, as national, munici- pal and private wealth and their fluctuation, concentration and dis- persion ; the productivity of such wealth, the economics of national and municipal revenues, expenditures, and administrations ; the productivity of private wealth, and, indeed, of wealth of all kinds ; the correlations between population- fluctuations and such financial characters as national UabiUties ; the quantity and velocity of the circulation of currency ; the relations between primary and secondary production and population development ; the growth of institutions expressive of a deepening recognition of social solidarity in co-operative effort, and in the national- isation of the greater pubUc services, etc. And finally, it may be said ^ Complete statistics are not available in Australia. CONSPECTUS OF POPULATION CHARACTERS. 103 that all facts which throw any light whatever on the possibility of world- production of food supplies and the fluctuations of population with abundance or want belong to tlie wider theory of population, and demand appropriate mathematical investigation. These wider facts are, of course, beyond the range of the narrow limits of ordinary official statistic, but no comprehensive view of the significance of a study of population is possible, which excludes the study of the reaction of material, psychical, or social conditions upon its growth and fluctuation. 6. The creation of norms. — The significance of statistical results is fully recognised only by comparisons with the similar results for other populations. Such comparisons are effected in the most general way by the creation of norms for each population-character. The principle which governs the constitution of a norm is that it shall represent the character selected on the widest possible basis. Thus, if statistical data existed for every population in the world, world-norms would be possible for every character statistically recorded. Western civiHsation is fairly homogeneous and statistical data are available for many characters. Thus it should be practicable in the near future to create a series of norms for the greater part of the western world. These might be regarded as the normal or usual value of any character in question, with which the same character in any particular population may be compared. It is obvious in order to compare a series of populations the best basis is the average value of any character : furthermore, if a compared character is affected by the deviation of any other from the average the value of the norm and of the deviation therefrom furnish the best basis for necessary corrections. The essential nature of a norm is perhaps best seen by regarding it as representing the characters of aU the populations included, considered as a single population. Thus the deviation of the characters and any particular population about the secular changes therein of this great aggregate gives the most informative presentment of the position of the population in question, that can possibly be had : in short, it makes the scheme of comparison as broad as is possible. 7. Homogeneity as regards populations. — ^Two communities may be said to be homogeneous with regard to any series of characters, when those characters are identical. In comparisons between communities in regard to any one character, it is necessary, in order that the com- parison should be a just one, that aU other characters which have any influence thereupon should be identical ; or, to put this more generally, the comparisons of any selected characters in a community are legitimate only when these communities are homogeneous with respect to all other characters which may have any influence on the comparison. For ex- ample, the birth-rates of two communities are immediately comparable if the relative numbers of married and single at each age are the same, because the birth-rate then (presumably) reveals the fertility under identical physiological conditions. 104 APPENDIX A. Since, however, different communities are more or less heterogeneous, appropriate schemes must be developed through which rigorous com- parisons can be effected. Thus, for example, corrections may be applied in such a way that any character compared or contrasted wiU not be affected by difference of other characters. The most convenient way of securing such a result is to adopt, as a basis for aU comparisons, a population so characterised as to represent all others to be compared as nearly as possible. Such a population may be caUed a " normal " or a " standard" population, and any character in regard to which it has been standardised may be called a " norm." 8. Population norms. — In order that any character of a number of populations or communities may be conveniently compared, it will be necessary that whatever population be adopted as basis, it shall represent each as nearly as possible. It is easy to see that, in regard to any character under review, such a basis must be a weighted mean, so that the character adopted as basic shall be the character of the population formed by aggregating all populations which may have to enter into comparison. Thus if P, Q, B, etc., be populations, and p, q, r, etc., be the values of some one character in each, then the best basis of comparison is : — Pp+ Qq+ Rr+etc . _ Ss _ ^'^^^' P+ Q+ B + etc. - S - " S being the sum of P -|- Q + -R + etc., and s the norm. It is immediately obvious that, in general, the secular changes of norms will be less marked than the secular changes in respect of the same character of the individual populations from which the norm is determined. For this reason it will be necessary for the progress of exact statistic to estabhsh a series of norms for all elements the comparison of which are important. That is, we must adopt a standard or normal population of definite characters, or, in other words, create a series of population norms to serve as a basis for comparisons. The scheme then of com parison is to apply the ascertained attributes of each existing population to the standard population. This process will reveal what would have been manifested had each population been similarly constituted to the standard population. 9. Variations of norms. — ^Inasmuch as, in the present development of statistics, norms have not been created, except perhaps as regards the constitution of population of each sex according to age, it will suffice to indicate the outhnes of a general method of studying the variation of norms. Since necessarily they can vary only slowly, a decennial determin- ation will be probably always sufficient, and when a number of decennial changes are to hand, the investigation of their variation will become possible. Whether such variation wiU reveal any sign of periodicity or not it is at present impossible to say. It is not unKkely that periodic elements of variation will be found superimposed upon slow secular changes. This, however, must be left for the future to determine, and the appropriate method of analysis will depend upon the character of the data. CONSPECTUS OF POPULATION CHARACTERS. 105 10. Norm representing constitution of population according to age. — A norm for males and one for females of European race is of importance for properly comparing death, marriage, birth and other rates. The use of such a norm was proposed by Dr. Ogle at the meeting of the " Institut International de Statistique, " in Vienna, 1891, and the index of mortality at present used is based upon such a norm, though not a properly constituted one. The aggregation of the populations of a con- siderable group of countries between which also there is migration, removes the speciahsing influence of this latter element, and secures the general advantages of large numbers. The following results were obtained from combining the populations of England and Wales, Scotland, Ireland, the United States, the German Empire, Norway, Sweden, Italy, Canada, Australia, and Newfoundland generally for the censuses of 1900 or 1901.1 The numbers are given in each age-group, and above a given age :— TABLE XI. Population Norms for 1900. European (1900). India (1901). European (1900). Numbers in Age- Numbers in Age- Numbers at and Group in total of Group in total of above age indie- Age. 10,000 10,000 Age. oated Fe- Per- Fe- Per- Fe- Per- Males. males. sons. Males. males. sons. Males. males. sons. 270 263 266 266 276 271 10,000 10,000 10,000 1-4 971 953 962 988 1,063 1,025 1 9,730 9,737 9,734 5-9 1,139 1,119 1,129 1,394 1,382 1,388 6 8,759 8,784 8,772 10-14 1,057 1,038 1,047 1,264 1,081 1,174 10 7,620 7,665 7,643 15-19 975 980 977 866 835 861 16 6,563 6,627 6,596 20-24 915 931 923 787 892 838 20 5,588 5,647 5,619 23-29 808 813 810 879 894 887 25 4,673 4,716 4,696 30-34 715 705 710 848 851 850 30 3,865 3,903 3,886 35-39 640 624 632 609 657 583 35 3,150 3,198 3,176 40-44 563 550 557 648 652 650 40 2,610 2,574 2,544 45^49 470 463 467 370 339 356 46 1,947 2,024 1,987 50-54 413 417 415 437 452 445 50 1,477 1,661 1,520 65-59 331 344 337 177 169 173 55 1,064 1,144 1,105 60-64 272 290 281 254 303 278 60 733 800 768 65-69 197 212 205 66 79 72 65 461 510 487 70-74 136 150 143 76 91 84 70 264 298 282 75-79 79 88 84 27 32 29 76 128 148 139 80-84 36 43 39 30 35 33 80 49 60 55 85-89 10 13 12 5 6 5 85 13 17 16 90-94 3 3 3 6 7 6 90 3 4 4 95- 1 1 1 3 4 3 95 1 1 1 Total . . 10,000 10,000 10,000 10,000 10,000 10,000 10,000 10,000 10,000 10,000 ^ See "The determination and uses of population norms representing the oon- Btitution of populations according to age and sex, and also according to age only." By G. H. Knibbs, and C. H. Wickens, Trans. 15th, Int. Congr. Hygiene and Demo- graphy, Washington. Vol. VI., pp. 352-378. 106 APPENDIX A. 11. Mean age of population. — ^The mean age, x^, of a population is given by the formula /"xl-dx 1 if xL (308) x^ = -^—1- = 4 + %rr^ • approximately. Zj, denoting the relative frequency at the age x, co the greatest age attained or considered, and L^ the number of age x last birthday, it being assumed that this number may, on the average, be regarded as of age « + J. Omitting the J, this last expression really gives the correct mean age last birthday. The mean age next birthday, x^ of a population under the age n is rQnQ\ ^ - ' ^^n-i+ (w - 1) L„_2 + + Lq y'^'J'') •'■n — r I r I IT ^n-l -r -L'ji-2 + + -^0 From this formula it is evident that, with a table giving the number at and above each integral age, aU that is requisite to obtain the mean age next birthday is to divide the total population into the sum of the num- bers from the youngest to the oldest ages. Deducting ^ gives the usual approximation to the mean exact age, while a deduction of unity gives the mean age last birthday. The mean age in years of the normal or standard population is, for 1901 :— Males. Females. Persons. 26.934 years. 27.341. years. 27.148. years. This mean age is, of course, not what is known actuarially as the expectation of life at age 0, but is the average age of aU persons hving at a given moment, or, in other words, it is the average past Hfetime of the population at a given moment. On the other hand, the expectation of hfe at age is the average future hfetime of all persons born. In the case of a stationary population, however, with rates of mortaUty varying with age, but remaining constant for each age through a great length of time, the average past hfetime of the population at a given moment is equal to its average future hfetime, that is, the average age of the popula- tion is equal to the average " expectation of life" of the population as a whole.^ Thus for the population of Europe in 1901 persons had Uved on ^ The expectation of life e° of the Ix dx persons of the exact age x is the future lifetime T ^ of these, divided by their number, that is — "■l = jx^dx/lx = Tx/ h and consequently the total future lifetime of these Ix dx persons is ex Ix dx - Ix dx Tx / Ix = Tx dx Hence the total fvitiire lifetime of the whole existing population between and w is r e°. Ix dx = r Tx dx aud as a whole existing population is J Tx dx, the average future lifetime or expecta- tation of life of the whole existing population is J Tx dx / j Ix dx, which may be shewn to be equivalent to J xlx dx/ j^ Ix dx, or the mean age. CONSPECTUS OF POPULATION CHARACTERS. 107 the average about 27 years. The expectation of hfe changes with the lapse of time, and is appreciably lengthening. Thus the secular change of the norm will be the weighted average of the changes of the constituent populations. 12. Population norm as a function of age. — ^The number of persons, Y, at and above the age x may be closely represented by (310). . . . y=;fca«'|8»' = 52674 (0.99961 )i'"8o8' (O.lsggS)!"*"* which is a development of the Gompertz-Makeham type of formula. The constants indicated fit very closely the values of the norm given in pre- ceding table.' This matter will be dealt with more fully hereinafter. ^ For solution, vide op. oit. pp. 364-7. IX.— POPULATION m THE AGGREGATE, AND ITS DISTRIBUTION ACCORDING TO SEX AND AGE. 1. A Census and its results. — ^A well-conducted Census furnishes results which are substantially correct so far as the aggregate number of persons and the aggregate number of each sex is concerned. That is, if p, m and/ denote the errors of the numbers of persons, males and females respectively, and P, M and F their respective aggregates, thenp/P, m/M and f/F are all extremely small quantities, which can have no important bearing upon the general theory, or upon any deductions flowing from it. Unfortunately this is not true regarding the numbers of either sex between given age-limits. In Chapter X of the Census Report, it has been shewn that for Aus- tralia the Census results bear intrinsic evidence of great improvement in regard to accuracy of statement respecting age ; see §§3 and 5. The nature of this is shewn in the tables given of numbers and percentages for the ages 28, 29 . . 32, and 48, 49 ... . 52. The exces sive statements, for example, for the ages 30 and 50, became markedly less. The results were as follow : — Census Age. 1891. 1901. 1911. 1911(adjusted). Percentage of age- \ , quinquennium in- [ 30 23.35 22.98 20.90 19.96 eluding two years 50 29.06 25.77 21.75 20.16 on either side A glance at Figs. 37 and 38 hereinafter will shew that the curves of numbers according to age for ages 30 and 50 do not depart very much from a straight line. For the former age the curves are concave upward ; for the latter, convex upward. Hence at 30 the mean should be somewhat less, and at 50 somewhat more, than 20 per cent. The ratio determined from the smoothed results are shewn in the final column. We shall consider the question of smoothing the results later. For each it is seen that the numbers for the ages in question were excessive, enormously so for 50 years of age, in the 1891 Census. The error, however, was diminished for the Census of 1911, probably largely in POPULATION AGGREGATES AND SEX DISTRIBUTION. 109 consequence of a special attempt to ensure the population appreciating the necessity for accuracy.^ It may he said, however, that statements of age leave much to be desired. 2. Causes of misstatement of age. — ^Many people are so indifferent as regards their age that they are really unaware what it is, and for this reason tend to assign round numbers (viz., ages ending with the figure or the figure 5), as roughly expressing about their ages. In the case of persons approaching 21 years of age, what may be called " matrimonial reasons" exist for an overstatement, and this may continue to operate for a year or two. In the case of females the tendency to overstate the age is, on the whole, negative for a considerable period of life.^ For the older ages, however, there is probably a distinct tendency in the opposite direction.* 3. Theory of error of statement of age. — ^Assuming both a tendency to express in round numbers ending in and 5, an age not accurately known, and also particularly in the case of females some tendency to under- state age, except for ages above, say, 60, we ought in general to find the following characters in the crude results of a Census, viz. : — (i.) In smoothing the crude results so as to conform to the general trend, the results for ages ending in have to be considerably reduced ; while those ending in 5 have to be reduced a somewhat smaller amount. (ii.) The amounts of the corrections for ages above and below the round numbers on the whole shew some asymmetry, though at the same time, owing to the masking effect operating in ages so close as a; + and X -\- 5, this character is not definite. (ui.) The curves for males and females exhibit systematic differ- ences of form due to systematic misstatement. Figs. 37 and 38 shew the graphs of the numbers for each year from to 100, for the Australian Commonwealth. It will be seen from these that, for a population profoundly affected by migration, no systematic difference of form actually exists of sufficient magnitude to unmistak- ably indicate systematic misstatement of age. The marked tendency to give ages ending with the figure is, however, very evident, so also that to give ages ending with the figure 5 is also fairly clear. 1 Where the official admimstration of a commnnity is sufficiently systematic to reqtiire every one to keep a card of identification, it is easy to get correct answers to this and similar questions. The public appreciation of the importance of correct answers is regrettably deficient. ' For matrimonial and economic reasons, and even reasons not entirely dis- associated with personal vanity ; the two latter reasons also operate in the case of males, but to an appreciably lesser extent. ' Certain investigations shew that vanity concerning longevity is not whoUy absent in either sex. 110 APPENDIX A. AUSTRALIA, 1911. Fig. 37. Coxumencing points of age-groups of one year at i AUSTRALIA, 1911. i indicated. 50 .0 30 20 10 ^ FEM AT.F.S i ■s \ N V \ \ 1 V \ ^ \ O v tn \ S s, \ _ B a so i( Fig. 38. Commencing points of age-groups of one year at age indicated. POPULATION AGGREGATES AND SEX DISTRIBUTION. Ill The curves in Figs. 37 and 38 are interpreted in the following way, viz. : — ^The ordinate or vertical distance to the curve at any point repre- sents in thousands the number of males (or females) in the age-group of one year, commencing at the age in question. The zig-zag line denotes the results furnished immediately by the Census, and the curve the smoothed (and more probably correct) results. 4. Characteristics of accidental misstatements, and their fluctua- tions. — ^The Censuses of the various States of Australia never having been combined, it was desirable to compile the three preceding Censuses, viz., those for 1881, 1891, 1901, in order to deal thoroughly with that of 1911. The results were not in age-groups for single years for 1881, but were for the later Censuses. In doing this it was found on inspecting the graphs for 1891, 1901 and 1911, of the numbers enumerated for each age, that in the statements of age there were tendencies to concentrate on certain ages, and to avoid, certain others. In order to definitely examine these tendencies a tabulation was made of the data in respect of the unit figure in the year of age stated in Australia at the Censuses of 1891, 1901 and 1911. To enable an estimate to be made of the degree of error involved in these statements of age, the smoothed results were similarly tabulated according to the unit figure in the year of age, and the ratio of the former set of results to the latter was obtained for each sex and each unit figure. The results should, of course, be unity if the errors balanced, or had no tendency in any direction. « The ratios so obtained are as follows : — Table XII. — ^Ratio of Number Recorded to Adjusted Number, Censuses 1891, 1901, 1911, Australia. Year Unit Figure in Age Last Birthday — OF [Census 1 2 3 4 5 6 i 7 8 ; 9 1891 1901 1911 MALES. 1.1388 .9167 1.0088 .9545 .9969 1.0366 1.0207 .9513 1.1044 .9369 1.0072 .9677 .9809 1.0343 1.0134 .9636 1.0485 .9956 .9944 .9787 .9990 1.0085 1.0097 .9691 1.0055; .9532 1.0144J .9667 1.01911 .9695 FEMALES. 1891 1901 1911 1.1251 1.0926 1.0367 .9288 .9270 .9895 .9978 1.0039 .9935 .9848 .9861 .9895 .9943 .9979 1.0056 1.0077 1.0106 1.0050 1.0117 1.0128 1.0066 .9640 .9708 .9770 1.0125 1.0165 1.0148 .9558 .9738 .9760 112 APPENDIX A. The outstanding indications furnished by this table are for both sexes (i.) A marked tendency to concentrate on ages ending in 0. (ii.) A less marked but persistent tendency to concentrate on ages ending in 5, 6 and 8. (iii.) A marked tendency to avoid ages ending in 1, 3, 7 and 9. (iv.) A tendency to state ages ending in 2 and 4 with fair accuracy, concentrations and avoidances being in evidence, but relatively small in respect of these ages. The table also furnishes an indication of the increasing accuracy of statement of age at successive Censuses, the excess at ages ending in having fallen from 13.88 per cent, in 1891, to 4.85 per cent, in 1911, iu the case of males, and from 12.51 per cent, in 1891, to 3.67 per cent, in 1911, in the case of females. Another interesting feature of the results is the evidence furnished that inaccuracy of statement is more marked amongst mules than amongst females. Thus, for the Census of 1891 the mean deviation from unity (irrespective of sign) of the above ratios was .0438 for males, as against .0332 for females. The correspond- ing figures in 1901 were .0358 for males, as against .0281 for females, and in 1911 they were .0181 for males, as against .0143 for females. ENGLAND AND WALES, 1911. 1 7 6 5 N - s ">« \ K 3 2 1 ( \ \ \ L \ »i \ ^^ u 3 23 30 4 ) 5 a CO 7 8 90 for " persons" at ages 50 and ; see Fig. 39. Another remarkable feature, worthy of attention, in the population-graphs for Australia, as compared with those of England and Wales, is the similarity Fig. 39. of the features for ages 37, 38, 39, and 40, viz., in the graphs for " males" and for " females" of the former, with that for the latter country. There is also some similarity 60, due to excessive numbers for the ages ending in 5. Characteristics of systematic misstatement. — ^It having been ascertained that in some cases the ages given in the Census cards were not correct, notwithstanding the exphcit directions, persons who made mis- statements were invited to send in corrections. Out of over 7000 re- ceived, 1660, containing definite information as to the age given and the amount of misstatement of age in the case of females, were tabulated in age-groups, and according to the number of years the age had been mis- stated. Of these, one-half (830) were for the State of Victoria, and the balance of 830 for the State of New South Wales. The tabidated results were as follows : — POPULATION AGGREGATES AND SEX DISTRIBUTION 113 Table XIII. — Analysis of 1660 Cases of Misstatements of Age at Census of 1911, Australia. CoKUECT Age. No. PER 1000. cokrbctios is Years. Un- der 20 21 to 2-, 26 to 30 31 to 35 36 to 40 41 t.1 45 46 to 50 51 tio 55 56 to 60 61 to 70 Ov- er 70 Total. /o Crude. Smooth- ed. ca a Over 5 . 4 . 3 . 2 . I 1 . 5 2 1 3 5 4 5 I 3 6 1 2 1 8 1 1 1 4 5 1 1 1 3 2 1 2 1 5 1 1 1 1 4 2 2 1 1 1 2 1 — ■ 7 4 7 17 14 40 79 45 79 191 157 449 19 64 96 146 226 449 Total Smootlied 20 5 10 17 12 18 1! 6 11 11 8 8 6 f. 4 3 3 1 ' — 2 ; 1 89 89 1000 -53.6 1000 S ■a B , 1 . f 2 . 3 . 4 . 5 . 6 . I: 11-15 Over 15 '.'. 10 2 1 55 21 4 80 62 36 18 8 5 1 1- 56 62 48 26 30 13 7 6 2 6 1 72 87 45 49 23 21 10 7 4 13 2 49 48 37 23 26 20 8 8 4 16 2 2 41 54 27 19 23 13 9 8 1 9 5 02 22 17 11 9 11 4 3 5 10 3 9 5 11 9 3 6 1 4 2 3 1 1 6 9 5 9 4 3 1 2 6 1 1 1 3 1 1 401 372 231 165 126 95 42 37 20 63 14 5 255 237 147 105 80 60 27 24 13 40 9 3 255 193 + 37* 145 107 79 58 41 27 18 11 + 25* 3 1 Total Smoothed 13 13 80 77 211 168 257 284 333 337 243 268 209 189 117 120 55 64 46 44 7 7 1571 1571 1000 = 946.4 1000 Grand Total Smoothed 33 18 90 94 223 186 269 298 339 348 254 276 217 195 123 124 58 67 47 46 7 8 1660 1660 = 1000 * The abnormality is about 37 in the one case, and 25 in the other. The 193 and 11 \(Ould be the normal values in a total of 1000 — 37 — 25 = 938. 35 40 True Age of Females. Belative frequency of Overstatements (A) and Understatements (B) of age with females according to true age. Kg. 40. In the above table, the results of which are shewn in Figs. 40.. 41 and 42, the " smoothed" figures for the aggregate number of overstatements according to age probably very closely represent the tendency in general : the results, however, for under 20 years of age appear to be unduly large. The smoothed results for the aggregate of understatements according to age indicate the probable tendency in general. The smoothed result for the total number of misstatements (over and under) according to age are merely the sum of the preceding. The crosses, squares and circles 114 APPENDIX A. represent the age-group aggregates for overstatements, understatements and total misstatements, respectively. These results are shewn re- spectively by curves A, B and C in Fig. 40. The smoothed results of the aggregate number of overstatements according to the amount of overstatement (see the vertical column at the right hand side of the table) probably represent the distribution, but the aggregate 89 is so small that it can be regarded only as a rough indication. The graph of this is curve A of Mg. 41. AUSTRALIAN CENSUS, 1911. S c3 So la \ ei \ \ A B \ \ s \ \ ■^v \ N ( ) a? si < i« > ^ a a ho 3 o c- O 80 3 4 6 6 7 a 9 10 11 13 13 14 16 16 17 Misstatement of age in years. Curve A denotes overstatement ; cui.'ve B denotes understatement. Fig. 41. The smoothed result of the number of understatements according to the amount of understatement, is probably represented by the final column in the table. In this, however, the abnormality of understatements of 2 and 10 years is very striking. The graph is curve B of Fig. 41, and the abnormal position for 2 and 10 year understatements is shewn by the small squares with circles surrounding. This abnormality is probably on the whole real ; that is to say, misstatements of 2 and 10 years had a real predominance over the number which might have been expected according to a probable law of frequency based upon misstatements of other amounts (say, a frequency varying inversely as some power of the magnitude of the misstatement).'^ At the same time it is also possible that in part it repre- sents defects in the allegation as the amount of misstatement. 6. Distribution of misstatement according to amount and age of persons. — ^By forming a series of 10-year groups from Table XIII., with the central ages 20, 25, 30, etc. (completed years), and plotting these as ordinates, some idea is obtained of the form of the function representing the relative frequency of misstatement according to both age and magni- tude of misstatement. Curves are then drawn among these positions, the results shewn on Fig. 42 being thus obtained. The families of curves are obviously fairly regular, and are skew. The positions of the ordinate - terminals, obtained as described, are shewn in the following way. The 1 In a Census the frequency is for integral amounts of misstatement only. POPULATION AGGREGATES AND SEX DISTRIBUTION. 115 ^-^^ character of the mark denoting the terminal of the ordinate for a mis- statement of 1 year is a dot ; for 2 years a vertical cross ; 3, a square ; 4, a slanting cross ; 5, a circle and vertical line ; 6, a lozenge ; 7, a circle and horizontal line ; and 8, a slanting cross. After the age 55 the results are rather irregular. The broken lines for understatements of 2 years and 10 years shew what may be regarded as the "normal" positions. That is, had there been no peculiar predominance in the adoption of ages differing by these amounts from the true age, the frequency curve would have been found in about the position of these broken lines. They are numbered with light-faced figures. The frequency of misstatement according to age, as indicated in Table XIII. and Fig. 42, refers to the number actually existing in the age-groups, for which Table 18 of Part I. of the Australian Census may be consulted (pp. 32-33). To ascertain the frequency for equal numbers of females a cor- rection is necessary, viz., division of each result by the number in the age- group to which it refers. Although over 7000 acknowledgments of mis- statements of age were received, mostly from women, the proportion these bore to the aggregate number of misstatements was not ascertainable, and after a study of other errors revealed by the zig- zag character of the enumerated age-groups, it was decided to regard the characteristic misstate- ment as sensibly negligible. The absolute scale of the frequency is not known, since the total number of misstatements could not be inferred. Neverthe- less its form is important as throwing light upon the relative frequency of misstatements of different amounts by women of different ages. The result may be summed up as follows : — 40 50 Correct Ages. The figures on the curves denote the amount of misstate- ments in years. Fig. 42. 116 APPENDIX A. The analysis of acknowledged misstatements shewn in the table gives the following indications (of course for females only) : — (i.) Understatement of age constitutes 94.64 per cent., and over- statement 5.36 per cent, of the aggregate cases of misstate- ment, (ii.) Excepting in the case of understatements of 2 years and 10 years, which are evidently abnormal, the frequency of mis- statement diminishes with the number of years misstated, at first very rapidly and later more slowly. (iii.) The greatest frequency of understatement of all amounts corresponds to the age of about 37J years. (iv.) The age corresponding to the greatest frequency of understate- ment of a given number of years increases with the amount of understatement approximately in the ratio of about 1 J years for every year of understatement, except in the case of 2 and 10 years. (v.) The frequency of understatements of 2 years is about 1.2 times that which would accord with the general tendency to under- statement ; and the maximum is for the age of about 35 years . (vi.) The frequency of understatement of 10 years is about 3.3 times that which would accord with the general tendency to under- statement ; and its maximum is for the age of about 30 years . While these indications, being based upon only 1660 investigated cases, have limited validity, they are probably substantially correct. An insufficient number of returns were received from males to draw any deductions as to the frequency of misstatement according to age and amount of misstatement. For cmrection purposes misstatements regarding age are best tabu- lated according to the age declared ; on the other hand, for the expression of the measurement of misstatement they are better tabulated according to the true age. Since probably by far the greater number of persons give their age correctly, it is probably desirable to regard the curves for over- statement and understatement as discontinuous at the value zero. 7. The smoothing of enumerated populations in age-groups. — ^The generalities of smoothing have been partially dealt with in VII., herein- before ; see particularly §§ 1-9. Figs. 37 and 38 shew the graphs of the enumeration in age-groups of the Australian Census of 1911 ; obviously these are not the true results. It is obvious that the " smoothed" curve must be of higher accuracy than the zig-zag results, since there are strong reasons for believing that the numbers are sufficiently large to give a " smooth curve." The following principles may be taken as a guide in smoothing : — (i.) Any smoothed curve so drawn as to equalise the zig-zag results (doubtless) better represents the facts than the original data. POPULATION AGGREGATES AND SEX DISTRIBUTION. 117 (ii.) The drawing of the smoothed curve can be assisted by arith- metical and algebraic devices. (iii.) The adoption of a particular position for the smoothed curve must be governed not only by mathematical considerations, but by the probabilities of each particular case. (iv.) If arithmetic or algebraic methods are employed, they should be such as do not involve systematic error. (v.) The accumulations of error at all ages should be as small as possible, and therefore should frequently change in sign, and the grand total should be approximately (or exactly) the enumerated total. ^ The method of smoothing by drawing a curve fulfilling the con- ditions indicated is known as the graphic method. Before considering it further, we shall examine the essential character of smoothing by grouping, and the limitations of smoothing by grouping methods. First, we consider the error introduced by mere means of aggregates. 8. The error of linear grouping. — ^If a series of points lie on a curve say, convex upwards, their mean, weighted or otherwise, will obviously lie below the curve, that is, x'^, y'^, denoting the mean of the co-ordin- ates, and w the weight assigned to any point, the point having these co-ordinates, viz. : — will, in the case supposed, be below the curve. If the original points lie on a straight line, the point wUl, of course, be on that line. Graphically, the point may be determined for equal weights thus : — Let P, Q, R, S, etc., be any points : the point midway between P and Q is the mean of P, Q ; the point one-third of the distance of this mean from R, towards R, is the mean of P,Q,R ; and, similarly, that one-fourth of the distance of this last toward S, is the mean of P,Q,R,S ; and, in general, the mean of n points is 1/wth of the distance of the mean of {n — 1) points towards the wth point. It follows from this that when n values are taken of any quantities, which, being'graphed, are found to lie, not upon a straight, but upon a curved line, then the mean of the independent variable (or argument) does not correspond to the mean of the dependent variable (or value of the function) unless the points representing them are all symmetrically situated about the middle point. Thus, if we have the numbers in a population at, say, ages 50 to 55, the mean does not correspond to the age 52. We proceed to consider the magnitude of the systematic error involved. 1 Exact correspondence is neither essential nor extremely desirable, but as it is easy to secure, there is no reason why it should not be insisted upon. A simple way of securing it is to multiply each group-result by a correcting factor, viz., in VII., § 7, herein. 118 APPENDIX A. If we suppose the results to be representable by the equation y= A + Bx-{-Cx^-}- etc., a,nd take points on either side of the middle so that the correct value of «/ is A, we readily derive the following ex- pressions shewing the errors of ternary, quinary, and larger groupings: — (312).... ^IJy=A+ f Gk^+ I Ek^ + etc. (ternary). (313) -^ Sy =A-{- 2 Gk^+ &% Ek^ + etc. (quinary). (314) ^ Zy =A+ i Gk^+ 28 Ek^ + etc. (septenary) (315). ... ^2y=A + 6^Gk^ + 18^ Ek* + etc. (nonary). (316). ... ^Zy =A + lOGk^ + 178 Ek* + etc. (undecenary). If the number of terms in the groups be denoted by n, the law of increase in the numerical coefficients, y say, of G and e of ^ is as shewn hereunder : — (317) yG = ^(n^-l)G. (318)..e^=.[-l(n-l)+i(«-l)2+l(,.-l)3+i(«-l)4ii7 The latter may be put in the more concise form in (319) hereunder. Hence the error of a simple mean is shewn in the most general form by the following expression, viz. : — (319).. -^^=^+1 (wa-l)(7F+ -L .[(w^- 1) (3^2 _ 71 .BA* + etc. The values of Gk^, Ek*, etc., can be very readily expressed in terms of the ordinates to say the roughly smoothed curve. Thus, using accents to denote ordinates symmetrically situated on either side of the middle (unaccented) ordinate, we have — (320) J [7], -27] + 7j') = Gk^+ Ek*+ etc. (321) -g {v„+V,-^+v'+i')=^Gf^^+^ I Ek*+ etc. We may therefore from the above equations obtain the value of y, free from the systematic error due to curvature. Thus (322) 2/0= ^ {■Sy-iv, -^ + v')} and from (313) and (321) (323) yo = j {Ey -(,,„ + ^ - 4^, +,,' +n")] for ternary and quinary groupings respectively. These correction-terms in the inner brackets are, as a rule, very small.^ 1 To reduce the arithmetical work any one number may be taken from each of the values of -q. POPULATION AGGREGATES AND SEX DISTRIBUTION. 119 The repeated application of any system of grouping leads to more highly smoothed results, but is unobjectionable only it freed from syste- matic error. It, however, even then, never wholly removes the vitiating influence of a value which is seriously defective or excessive. It is easy to build up from the preceding formulae a system of coefficients by means of which the repeated groupings can be performed in one operation. Thus, each ordinate being assumed to have equal weight, we have for repetitions of ternary groupings — Table XIV.— Coefficients for Repeated Grouping. No. of Repeti- tions. Factor. Resiolting Grouping. Weights toibe Applied to Co-ordinates. 1 3 Ternary 1 1 1 1 1 9 Quinary 12 3 2 1 2 1 27 Septenary 13 6 7 6 3 1 3 1 81 Nonary 1 4 10 16 19 16 10 4 1 4 1 243 Undecenary 1 5 15 30 45 51 45 30 15 6 1 The scheme of deriving these is evident.^ In the same way it is necessary to buUd up also the scheme of corrections from (314), (316), etc. 9. Graphic process of eliminating systematic error. — A simple approximate method of graphically eliminating the systematic error indicated in the preceding section is based on the fact that the distance k between the mean of a series of n ordinates on a parabolic curve and the vertex of the curve is given in Table XV hereunder. Table XV. — ^Position of Mean of n Points.- Number {n) of points on curve . . n = 3 Proportional distance of mean of j k = the ordinates from centre of j chord towards vertex of curve I , _ 4 4 J^ -Q^ -2^ 15' 9 7 11 9'' T%^ T^ .ZZh Aih .50h .53h .55h .58h .60h the height h being the distance from the middle of the chord to the vertex. Thus, if a series of means of n ordinates are plotted, and a curve be drawn through them, this series can be taken to give an approximate guide to 1 Thus, 1.2.3 1 120 APPENDIX A. the shape of the true curve. A section of double the stretch being then taken^ the interval between the chord and curve along the ordinate is assumed to be four times the similar distance for the central ordinate of the original stretch. Hence in this case the points defined by the means should be moved the following amounts, viz., those in Table XVT. Table XVI. — Distance oJ Vertex from Mean oJ n Points. Number of ordinates for which a mean is taken n = 3 4 5 6 7 9 11 Proportion of vertex-dis- lr._]_Tj^Tj^TT '17 ^w ^ jr ^ tj tance of the doiMe I*" 6 36 8 60 36 48 lO stretch to be taken as 1 a correction ( k = .167H .139H .125H .117H .111 H .104H .lOOH H denoting the height of the vertex above the chord double stretch. This correction will eliminate the greater part of the systematic error, but not the whole, inasmuch as the curve has been flattened by taking the series of means : hence the corrections having been applied to the mean points a new curve may be drawn, and the process repeated if necessary. A smooth curve is then drawn among the points ultimately defined. This process, however, yields resultswhich, after aU, are but little better than a direct attempt to draw a smooth curve among the points given by the ordinate- terminals ; it is tedious, and its probabihtyis but little greater than that obtained by directly drawing the smoothed curve and correcting it by arithmetical (or algebraic) methods (" hand polishing"). To avoid its tedium of drawing and hand-polishing, what are called summation methods have been used. In these a weighted mean is obtained, the weight factors having opposite signs in order to eliminate the systematic error indicated in formulae (312) to (316). 10. Summation methods. — Summation methods in so far as they are rigorous, eliminate the systematic error involved in weighted means where the weights have no change of sign. Rigorously devised algorithms, applied to a series of ordinates strictly conforming to a curve of the wth degree, will reconstitute the given ordinates, whereas mere means of a series of ordinates wiU not only not do so, but wUl increase the error with every repetition of the grouping. The taking of the means of a series of ordinates is therefore vahd only where the general trend is either linear, or so nearly linear as to make the corrections referred to negligible. Suppose, then, we have a series of ordinates, the terminals of which 0, P, Q, R . . . . Z, are to be smoothed. Evidently we can draw an 1 That is, if w -|- 1 be the nvunber of ordinates, a curve defined by 2 w -f- 1 ordinates is taken ; thus, if 3 points are originally taken, the curve of double stretch will be that defined by five points. POPULATION AGGREGATES AND SEX DISTRIBUTION. 121 integral curve of the nth degree through any n-\-l such points. Geo- metrically, the summation smoothing process is the following :^Draw a curve of the nth degree through the points 0, O+i; 0+2i; 0+m : a similar curve through the points P P+wi : a third through the points Q, Q+ni ; and so on.^ This will give a series of curves of the nth degree, usually close to one another, and sometimes intersecting. The mean position of their inter- sections on the ordinates (or ordinates produced) is the smoothed curve required. The flexibility, or fitting power, of the curve depends, other things being equal (a) on the degree of the curve ; and (6) on the nearness of the points 0, 0+i, etc.; and consequently of P, P-fi, etc., to each other. It may readily be demonstrated, graphically or otherwise, that as the value of i is increased, minor fluctuations are more and more obliterated. The whole range being limited, the larger the value of n the more points on the curve are fitted by one stretch : hence the smaller i will be ; and the fitting power will consequently be increased. Since the mean position of the intersection of the curves and the ordinates defines the position of their terminals at the smoothed curve ; and since each point O, P, etc., is the start of one of the component curves, any abnormality in its position (i.e., deviation from the general trend) is reflected in the mean result ; that is, it produces a deviation of a smaller amount in the direction of the abnormal point. The defect of all summation methods is seen, from their geometrical representation, to be the following : — (i.) The degree of obliteration of minor fluctuations is quite arbitrary and depends upon the character of the summation-system. (ii.) The result is vitiated by all abnormalities: the method, in fact, does not lead to real smoothing, but to the reduction of the magnitude of the oscillations of the curve. This may be shewn analytically in the following way. We observe first that if there are q-\-\ points in the total range of q intervals of any component curve taken, then in a complete'^ series there wiU be g+l intersection-points on the ordinates. The mean of these is to be taken. The first complete term arranged according to the powers of the common distance (k) between the ordinates, and the second term wiU be re- spectively : — (324) y^ = -^ {K +«a-i + - •«o) + (^-i+268-2+36«-3 + - OA; + (c^-2+2%g-3+3^g-4 + -)^^+K-S+2"(^,-4+3^<^g-5 + -)fc»+-} ^ Where h is the common interval on the axis of abscissae between ordinates, the comjmon interval i between the points wiU always be an integral multiple of Ic greater than 1 ; that is i = 2k, or 3fc, or 4k, etc. ^ It is, of course, not essential that the series should be what has been called here complete, and in Woolliouse's method it was not complete. A complete series may be defined as one where, q + I being the number of points including the terminal ones ranged over by any curve, the initial point of the {q + l)th curve is on the same ordinate as the final point of the initial range, viz., the zero (or first) curve. 122 APPENDIX A. (325) 2/,+i = ^ <(a,+i + ..a,)+(l>,+ --)k+(Ct-i + ---)k^+ Thus the coefficients of the powers of x are changing every term, and con- sequently the equation of the smoothed curve of, say, s+ 1 points will be of the degree s, that is, it has no relationship whatever to the degree of the originating equations of the wth degree passed through the points ; +i; +2i, etc. It is thus seen that results of a " smoothing" by " summation" methods are in principle toto coelo different from those obtained by methods which ensure conformity to some function adopted for considera- tions of the nature of the case.^ Numerous papers on the summation method have appeared from time to time in the Journal of the Institute of Actuaries by various investigators, of whom the principal are the following : — J. A. Higham, W. S. B. Woolhouse, G. F. Hardy, J. Spencer, T. G. Ackland, G. J. Lidstone, G. King, R. Todhunter. Some of these have contributed several papers on the subject. A specially valuable one, on " The rationale of formula for graduation by summation," by G. J. Lidstone, appeared in the Journal of the Institute of Actuaries, Vol. XLI., pp. 348 et seq., and XLII., pp. 106 et seq. An important paper on the subject by Dr. J. Karup wUl also be found in the Transactions of the Second Actuarial C!ongress, p. 31 et seq. The subject of graduation of summation has also quite recently been re-examined by Mr. C. H. Wickens,^ and formulse based on ranges of three determined points (0, +», and +2i) and four determined points {i.e., including also +3i) are discussed for the developments of quinary formulse and formulse other than quinary, the adjective denoting the number of spaces into which i is divided. That is, if i =rk then the formula derived is an r-ary formula. It is shewn that there are great advantages in making the series complete, and that in taking the mean it is advantageous to allow only haK-weight to the terminal points of intersection on any ordinate.* The following weights (Table XVII.) have been deduced by Mr. Wickens for the different ordinates about the middle ordinates, th 1 Prof. Karl Peajson's scheme, adopted by many biometrioians, is to resolve the data under a suitable type-form derived from a generalised theory of probability, certain criteria being used to decide which form should be preferred. A single Pearsonian curve, however, will not apply to population-enumerations, although the population-curve may be empirically considered to be a combination either of Pearsonian or of other curves. 2 An extension of the principle underlying Woolhouse's method of graduation, read 30th October, 1911, Trans. Act. Soc, N.S.W., Session 1912, pp. 243-7. ' There are many physical analogies for this process. For example, if a physical property be measured at equidistant points along a line including the terminals the mean value is {a + 2b + 2c -\- ... -}- 2y -\- z)/2N, where N is the number of spaces into which the points divide the line. POPULATION AGGREGATES AND SEX DISTRIBUTION. 123 marked (3) and (4) being deduced from curves passed through 3 points and 4 points respectively. The similarity is obvious. Other formute may be obtained from the paper in question. Table XVn. — Summation-formnla-coefficients to be applied to a Series of rOrdinates Deduced on the Basis ot (3), and on a Basis oi (4) Determined Points. Binary Ternary Quarternary Quinary Senary Ordinates i=U i=Zk t=4J; i= = 5fc i=6fc 9+1 (x/k) ( 3) ( i) (3) (4) (3) (4) (3) (4) (3) (4) - 12 - 11 -5 -3f - 10 . , 6 6 -8 -7* - 9 -2 — 1^ -9 -9 - 8 6 6 -3 — 2^ -8 -8f - 7 -3 -2* -3 -3^ -5 -6J - 6 6 6 -4 -4 -2 -2t — 6 -1 -f -3 -U 29 25f - 4 -1 -% 12 10§ 56 53i - 3 - 1 — 1 19 17i 23 22| 81 81 - 2 7 63 36 36 33 33? 104 106| - 1 9 9 13 13^ 51 52J 42 43^ 125 128A 1 6 1 6 18 18 64 64 50 50 144 144 1 9 9 13 131, 51 m. 42 43* 125 128i 2 J 7 &i, 36 36 33 33f 104 106J 3 - 1 — 1 19 17* 23 22| 81 81 4 -1 -1* 12 lOf 56 53i 5 — 1 -t -3 -'H 29 25f 6 —4 -4 -2 -2* 7 -3 -2* -3 -3J -5 -6# 8 -3 -2* -8 -8f 9 -2 -1^ -9 -9 10 -8 -7* 11 -5 -3S 12 Sum of Co- efficients 3 2 3 2 54 54 256 256 250 250 864 864 For the mode of obtaining the values given by these formulae by processes of summation, reference should be made to the paper, in which also the smoothing coefficient is given as follows : — Table aviu. — Smoothing Coefficients. Interval No. of Terms or Ordinates Series (3) •v/(7«» + l)/4«" Series (4) s V7(««-l) {s^+5) + 36s/6s* 4s-l 2 7 .1683 .1683 3 11 .0741 .0615 4 15 .0415 .0316 5 19 .0265 .0193 6 23 .0184 .0130 7 27 .0135 .0094 8 31 .0103 .0071 9 35 .0082 .0056 10 39 .0066 .0045 124 APPENDIX A. The smallness of the smoothing coefficient is a measure of the efficiency in smoothing.^ 11. Advantages of graphic smoothing over summation and other methods. — ^This graphing of the group-results of an enumeration (numbers according to years of age in the instance immediately under review) yields a succession of rectangles, or, if we prefer, points denoting their heights. Smoothing in such a case consists essentially in transferring numbers of. those who alleged they were a given age to some other nearly identical age, the reason for this transfer being that it is judged a priori (and justly so) that the irregular distribution indicated by the data does not accord with the real facts. To do this there is no better way than to draw among the tops of the rectangles (or the points representing them) a smoothed curve following every variation of their general trend, which, in the judgment of the analyst,^ is regarded as probably conforming to the facts. This can be done, and the result scaled and smoothed arith- metically, that is, by differencing. The aggregates as by enumeration and by the smoothed curve can be formed, and the accumulated differ- ences examined to see that they are kept within probable limits ; that is, are alternately positive and negative, and are never great (see VII., §12). The initial curve can then be amended whenever improvement seems possible ; thus in its final form the grand total can be made identical with the enumeration, and the difference between the enumerated and smoothed aggregates up to any value of the variable (age) can be made the least possible for the form of curve deemed to he best on examining the graph of the enumerated results.^ The logic of this process has been admirably expressed by Whewell, and before him again by Sir John Herschel, in the following passages : — ' ' This curve once drawn must represent .... the law .... much better than the individual raw observations can possibly .... do The series of lines joining the consecutive points . . . cannot possibly repre- sent reality If, however, we thus take the whole mass of the facts .... by making the curve which expresses the supposed observations regular and smooth .... we are put in possession .... of something more true than any (one) fact by itself." — Sir J. Herschel, Trans. Astr. Soc, Vol. v., pp. 1-4. 1 See G. F. Hardy, Journ. Inst. Act., Vol. xxxii., p. 376. ^ Any attempt to dispense with the element of judgment is really illusive. The adoption, for example, of a summation method will yield appreciably different results according to the range taken. Thus » real undulation in a population curve may be virtually obliterated by the process. ' There is a tendency to forget that technical processes are but instruments in the hands of the user, and formulae employed confer no validity to the elements depending upon judgment. POPULATION AGGREaATES AND SEX DISTRIBUTION. 125 " The peculiar efficacy of the Method of Curves depends upon this . . . that order and regularity are more clearly recognised when thus exhibited to the eye as a picture (and) not only enables us to obtain laws of Nature from good observations, but .... from observations which are very imperfect, .... We draw our main regular curve not through the points given by .... observations, but among them." — ^WhewelU, Novum Organon Re- novatum, Bk. III., Chap, vii., p. 204, 3rd Edit., 1858. Finally, it may be remarked that by adopting the graphic method of smoothing, minor and unmeaning fluctuations are avoided. The invalidity of merely mechanically applying various summation formulae has been shewn by G. J. Lidstone ; he has indicated how, by the summa- tion method, unmeaning fluctuations are introduced into what may be known a priori to be a straight line.^ 12. Graphs of Australian population distributed according to age and sex for various Censuses. — ^Adopting the principles indicated, the graphs of the enumerated population of Australia for the Census of 1911 distributed according to age, shewed that, both for females and for males, the adoption of any function to which the results should be conformed was out of the question. It was evident also that a " summation method" was quite unsuitable. In the results for 1911 there was a sharp increase in the numbers for ages 13 to 18 ; then a zig-zag result up to age 22 before a decided decrease appeared. It was thus evident that results must be examined, and the smoothing based upon considerations as to the possibility of misstatement. The data therefore were simply graphically smoothed by drawing first a freehand curve among them, the changes of direction of this curve being made a minimum, so far as that was possible, while following all fluctuations deemed to represent the actual facts. This curve was then carefully drawn with the aid of splines, French curves, etc., the ordinates' scaled off and adjusted arithmetically.* The result of this smoothing is shewn on Figs. 37 and 38 As has been shewn in § 10 and formulae (324) and (325) hereinbefore, this is obvious from either geometrical or analytical considerations. For that reason the graphic process has been preferred to summation processes, which latter are regarded as theoretically invalid for the reasons indicated.* 1 See also T. B. Sprague, Journ. Inst. Act., Vol. XXX., pp. 161-3, 1892 ; James Sorley, Journ. Inst. Act., Vol. XXII., pp. 309-340, in particular 3 : The Graphical Method, pp. 321-8 ; T. B. Sprague's works on " The Graphic Method, etc.," Journ. Inst. Act., Vol. XLI., p. 182. ' On the rationale of the Formulae for graduation by summation. Joiu-n. Inst. Act., Vol. XLI., 1907, p. 360, and diagrams A, B and C. ^ Identical methods were also applied to the data of the earlier Censuses. * In the summation methods, as we have seen, fluctuations are introduced into curves in order to conform-to a convenient algorithm, rationally deduced. But a little re flection will convince any mathematician that the minute oscillations in the directions of the tangents, involved in the process, would be better eliminated, when that can conveniently be done ; and in any case, in the presence of large departures of individual results from the smoothed curve, these small fluctuations have neither real significance nor validity. 126 APPENDIX A. Graphs shewing the distribution according to sex and age have been prepared for the Australian Census of 1881, that of 1891 and 1901, as well as that of 1911. The results for 1881 were deduced from quinquennial groups ; those for the latter Censuses from year-groups ; and they are shewn on Figs. 43 and 44. It will be seen that intervals of ten years cause considerable differences in the forms of the curves ; these differences are due of course to migration and to fluctuations in the birth and mortality rates. Commencing points of age-groups of one year at age indicated. Fig. 43. The curves in Figs. 43 and 44 are interpreted in the following way, viz. : — ^The ordinate or vertical distance to the curve at any point repre- sents in thousands the number of males (or females) in the age-group of one year commencing at the age in question. POPULATION AGGREGATES AND SEX DISTRIBUTION. 127 .n 1 r p* \ FEM. SiLES iV ^ o^^ \/ § \ A ■s \ \ \ \ 1 \ V v ® 20 \ \ ^ Pi DD \ k\ g \ \ , \ \ 10 ^ XV . \ N> ^ V. ~-^ ^=^ 5 iJo Commencing points of age-groups of one year at age indicate>1. Fig. 44. 13. Growth of population when rate is identical for all ages. A population P(, increasing at the instantaneous rate p per unit of time becomes, if that rate be constant, as we have seen, Pt=PQ e''* see II., §§ l-IO, formulae (1) to (14). Hence, it the numbers between the ages X and x-\-dx for the epoch i = 0, are represented by P^f(x)dx, in which case (326) J"/ {x) dx=l and the rate of increase be the same for all ages, then the numbers between the ages x and a; + da; at any later date t, must be (327) Ptdx = Po e"^ f (x) dx. the aggregate being Pg e"' ; that is to say /(a;) remains constant. Hence, if the age-groups be divided by the total population, the results will be identical, i.e., the relative numbers will be seen to remain the same and their graphs will be identical. If, however, the aggregate numbers, denoted by F (x), are graphed, the graphs will not be identical. For we have in the latter case (328) Ft{x)dx = eftPo (x) dx ; and by hypothesis p is not a function of x ; hence (329) ^) =e.^^ ; or tan 9, = e^" tan 0„ 128 APPENDIX A. that is, the slopes of the tangents to the graph of the population are increased in the proportion 1 : e''*. In the absence of all information of " migration" and " natural increase" (increase by excess of births over deaths) the rate of increase of the preceding period must be assumed to continue not only for the population as a whole, but also for each age ; which is expressed by (330) Pe=Poe'"/;/(.T)^.r. and (327) hereinbefore. 14. Growth of population where migration element is known. If the ages and numbers of migrants be known, as well as the ages and numbers of the dying, then it is possible to determine the numbers in each age-group by remembering that survivors after t years have increased their age by t years. Except for very small communities, this method of estimating populations according to age (and sex), is, however, perhaps impracticable. We shall, however, later consider it. Here it may be noted that the estimation may be most conveniently treated in single year age-groups, i.e., not by infinitesimal methods. The value of the method is that it would enable aU rates to be finally made up intercensally, whereas, after a Census has rendered the intercensal adjustments possible, they have always to be corrected. 15 . Growth of population when rate of increase varies from age to age. Changes in the birth-rate ( = rate of immigration at age 0), in the death- rate for various ages (= rate of emigration at age x), in the rapidity of migration and age of migrants (= rate of immigration or of emigration at age x) causes a change to take place in the form both of/ (a;) and F {x) referred to in. the previous section. The graphs of / (a;), i.e., of relative numbers, at different epochs all give an area of unity between the limits and CO (= end of the longest life) ; hence the curves for different epochs necessarily intersect ; those of F {x), i.e., of absolute numbers, give the areas Ft, and may or may not intersect. We consider the consequence of those variations which change the form oif{x) ; see Figs. 43 and 44. Where we have to interpolate to obtain intercensal populations, or to extrapolate to predict a population, we may assume that the tangents to the curve foix) change uniformly with time ; that is, they become those of fT(x) by a linear change with time, T denoting the intercensal period. Thus ,oqiN dft (x) dfo jx) , t _ dfT(x) dfo(x) that is, o- is the total change in the tangent in the intercensal period T. Hence, given the total population at the time t, we can effect its dis- tribution according to age by determining merely /j (x) on the supposition indicated. POPULATION AGGREGATES AND SEX DISTRIBUTION. 129 This supposition (i) is of a more general character than that of sup- posing that the number at any age changes Hnearly : supposition (ii.). Graphically, the difference between the two is that, according to supposi- tion (i.), the intercept on any ordinate between the graphs of /^{x) and /r (cb), divided in the ratio t/T, gives the position of /( (x), while according to supposition (ii.) it is the intercept between Fg {x) and -Fjt(.t) which is uniformly divided. The advantage of supposition (i.) is that only the form of ft{x) is fixed ; the graph of Ft {x) can then be made to agree with any intercensal estimate of population.^ 16. The prediction of future population and its distribution. — ^The graphs of population of various countries for the years 1790 to 1910, Fig. 3 hereinbefore, discloses no general law. All shew what may be called oscillatory development. The graph of the population of Aus- tralia from 1788 to 1914 (see Official Year Book No. 8 of the Common- wealth of Australia, p. 127) shews also this feature in a fairly well marked degree, and those of the individual States exhibit more striking oscilla- tions. Hence accurate predictions even of total population of any pre- cision are not possible. Figs. 43 and 44 shew that accurate predictions for age-groups are not only not possible, but may be even more misleading than the assumption of an unchanged distribution according to sex and age. It may be noted, however, that there is a general similarity, though there is by no means identity, in the forms of the graphs for males and females. The great fluctuation in the masculinity of the population according to age is also evident from a comparison of the results shewn on Figs. 43 and 44. This, however, wiU be discussed later. ^ See Census Report, Vol. I., Chap. IX., post-censal adjustment of population estimates for the intercensal period 1901-11. X.— THE MASCULINITY OF POPULATION. 1 . General. —The ratio between males and females in any population has been called its masculinity, and the fluctuations of such a ratio are obviously important. The following ratios of the aggregate number of males to the aggregate number of females in various populations will give an idea of how closely the number approximates to unity. Table XIX. — Mascnlinity of Various Populations (about Yeai 1900). Norway 1891 .932 Ireland 1901 .974 Australia . . 1901 1.101 Sweden 1895 .944 Italy 1901 .990 C. of G.Hope 1904 1.024 Scotland 1901 .946 United States 1900 1.044 India 1901 1.038 Eng. & Wales 1901 .954 Canada 1901 1.050 Ceylon 1901 1.140 Germiny . . 1900 .969 Newfoundl'd 1901 1.053 The results given hereinbefore, viz., in VIII., § 9, Table XI., shew that even when the total numbers for all ages for males is made equal to that for females, there are easily discerned differences between Eastern and Western populations. In the foUowing Table, viz., XX., the aggregate number of males in the different age-groups in the first eleven countries are divided by the aggregate number of females in the same age-groups, the results being shewn on line W ; for the last three countries the similar quotients are shewn on line E. Table XX. — Change of Masculinity with Age ; Aggregate of Various Populations, about 1900. Countries. 1-4 5-9 10-14 15-19 20-24 25-29 30-34 35-39 40-44 45-49 W E 1.024 1.003 1.016 .966 1.014 1.047 1.015 1.212 .992 1.073 .979 .919 .991 1.022 1.005 1.037 1.021 1.135 1.020 1.035 1.012 1.131 Countries. 50-54 55-59 60-64 65-69 70-74 75-79 1 80-84 85-89 90-34 95-100 All Ages. W E .988 1.005 .962 1.095 .934 .870 .927 .882 .906 .873 .895 .885 .847. .784 .873 .905 .674 .880 .588 .880 .9964 1.0390 MASCULINITY OP POPULATION. 131 The figures in the table shew the relatively large range of " mas- culinity" for different age-groups, and indicate the desirableness of the determination of a norm for purposes of comparison. We proceed to consider this aspect of the question. 2. Norms of masculinity and femininity. — ^The variations with the lapse of time, of the norm of distribution according to age for the male population of any community, and the same norm for the female popula- tion of the same community wUl not, in general, be identical. The pro- gressive changes, which may have both periodic and aperiodic elements, are best studied by observing the fluctuation of the masculinity or of the femininity of the population. These characters as ordinarily defined are the number of males to one female (or in practice usually to 100 females), and the number of females to one male, respectively. Thus if m = the number of males, / the number of females, and p = m -f- / the number of persons of any age, the masculinity fxi and femininity (f>i for that age wiU be expressed by the formulse : — (332) 1^1 = j; <^i =^ with suffixes to denote the age. When these quantities and their varia- tions are known, the changes taking place in the relative numbers of the sexes are determined as soon as the variations in the norm for the entire population (persons) are ascertained ; see VIII., §§ 8 to 10. The curve shewing the variations of the norms for both sexes at each age from epoch to epoch is not an essential, for their fluctuation is determinable from the fluctuation of the norm for persons, and the fluctuation of either the masculinity or the femininity. For this purpose a somewhat different definition of masculinity is desirable ; this we shall now consider. 3. Various definitions of masculinity and femininity. — ^For many purposes definitions other than that mentioned above have advantages. Both of the functions referred to for ordinary populations approximate to unity. But other functions may be adopted which hover either about I or about zero. For example, the ratio of males (or of females) to" the whole population, is a quantity which ordinarily approximates to I ; or yet again the ratio of the difference of the number of males and females to the total population is a number which ordinarily approximates to zero. Algebraically, the three methods and their interrelations are as follows : — Ist Method : — 771 1 1 (333).. Masculinity = /xi = 7 ; Femininity (^1= ^ = — Possible range to -|- oo ; ordinary value about 1. 132 APPENDIX A. 2nd Method : = (334). .Masculinity =11.2 = m m 7 Ml Femininity ^^2 __A_-^ m , , l+Mi m -\- f 1+01 1+Mi Possible range to + 1 ; ordinary value about \. 3rd Method : = (335) . . Masculinity = /is = m m-f _f -1 .Ml »»+ / w» , 1 Ml +1 ' /"^ , f — m m 01—1 Femininity = ^3 = j^:^ = ^-^ - ^^^f Possible range — 1 to + 1 ; ordinary value about zero. The mutual relations subsisting among these several quantities are set out in the following table : — Table XXI. — Relations subsisting between Masculinity and Femininity according to Various Definitions. Func- Expressed in terms of — tion. Ml M2 M3 01 02 03 Ml Ml Ml M2 1 - M2 M2 2n2 -1 1 + M3 1 — Ms Hl+Ms) MS 1 01 1 1+01 1-01 1 + 01 92 1-02 1-202 1-03 1 +03 i(l-0s) — 03 M2 M3 1 + Mi Ml — 1 Ml +1 01 02 03 1 Ml 1 1 1 M2 1 — M2 1-2(12 1 — Ms 1 +M3 HI -Ms) - Ms 01 01 1+01 01 -1 01 +1 02 1-02 02 202-1 1+03 1-03 i(l+03) 03 1 +Mi 1 — Ml 1 + Mi 4. Use of norms for persons and masculinity only.— Instead of having three norms, viz., one each for males, females and persons, it will often suffice to have one for persons, and one for masculinity. Thus in the norm of population the masculinity, by method 3, viz. (wi— /)/(m+/) is as follows for Europe (i.) and for India (ii.). MASCULINITY OF POPULATION. 133 Table XXn. — Change of Masculinity with Age. Age Group. 1 to 4 5 to 9 10 to 14 15 to 19 20 to 24 25 to 29 30 to 34 35 to 39 40 to 44 45 to 49 (i.) . . (ii.) . . + .013 .018 + .009 .037 + .008 + .004 + .009 + .078 .003 + .018 .009 .063 .003 .008 + .007 .002' + .013 + .045 + .012 .003 + .008 + .044 Age Grodp. 45 to 49 50 to 54 55 to 59 60 to 64 65 to 69 70 to 74 75 to 79 80 to 84 85 to 89 90 to 94 95 to 105 (i.) . . (ii.) . . + .008 + .044 .005 .017 .019 + .023 .032 .088 .037 .090 .049 .090 .054 .085 .089 .077 .130 .090 .000 .077 .000 .143 5. Relation between masculinity at biith and general masculinity of population. — It has been suggested that some tendency exists which, while not very strongly expressing itself, is nevertheless sufficiently evident to equate the numbers of the sexes in the population of any country, or at least that the masculinity at birth is in some way affected by the masculinity of the population. ^ Masculinity here denotes merely the ratio of males to females, that is, M/F. The population of Australia has enormously changed in its mas- culinity in a few decades, and consequently affords an opportunity of examining this supposition. The masculinity at birth is compared with that of the population for the years 1829-1913, the latter passing through a wide range of falling values. The results are shewn in the following table : — Table XXUI.- -Average Masculinity of Population and of Births, New South Wales, over Various Periods. Average for Masoulimty Period. Average for Masculinity Period. Years. of Popu- of Live Years. of Popu- of Live lation. Births. lation. Births. 1829-34 6 2.961 1.016 1840-49 10 1.625 1.034 1835-89 5 2.436 1.031 41-50 10 1.560 1.035 40-44 .. 5 1.752 1.026 42-51 10 1.510 1.036 45-49 5 1.498 1.038 43-52 10 1.412 1.036 50-54 . . 5 1.309 1.031 44^53 10 1.433 1.033 55-59 5 1.281 1.033 45-54 10 1.404 1.035 1830-39 10 2.680 1.026 46-55 10 1.375 1.032 31-40 . . 10 2.568 1.018 47-56 10 1.352 1.033 32-^1 .. 10 2.443 1.021 48-57 10 1.325 1.029 33-42 . . 10 2.314 1.020 49-58 10 1.308 1.032 34r-43 10 2.205 1.029 50-59 10 1.295 1.032 35-44 . . 10 2.094 1.028 60-69 10 1.233 1.058 36-45 10 1.979 1.028 70-79 10 1.196 1.045 37-46 10 1.877 1.026 80-89 10 1.209 1.050 38-47 . . 10 1.784 1.027 90-99 10 1.147 1.054 39-48 10 1.698 1.030 1900-13 13 1.186 1.058 1 Diising, Das Geschlechtverhaltniss inx Konigreich Preussen. 134 APPENDIX A. This table seems to shew that, on the whole, the masculinity of birth jLtj, can be expressed approximately by such an equation, for ex- ample, as (336). ...^„ = ^p = 1.06 - 0.0325 [fj.^ - 1) + 0.0333 (fj,^ - 1)^ ; /Xp denoting the total number of males divided by the total number of females in the population over the period considered. The tabulated mean values of the masculinity of the population, and the position of the curve which represents the formula, are shewn on Fig. 45. The result may, of course, not be directly due to the masculinity of the population : both may have varied through some condition itself varying with time. Fig. 46 shews such a variation. This, too, implies an opposite pro- gression ; that is, it indicates clearly that while the mascuUnity of the population was, on the whole, diminishing, that of the birth was, on the whole, increasing. The results for Victoria point less decisively in the same direction. They are as foUows : — Table XXTV. Period 1851-60 1861-70 1871-80 1881-90 1891-1900 Of Population 1.765 1.303 1.142 1.108 1.049 Of Births 1.046 1.047 1.044* 1.049 1.050 * In conflict with the general indication. These shew that as the masculinity of the population was diminishing, that of birth was increasing, with the exception of the decennium 1871- 1880. For the Commonwealth of Australia the results for the masculinity of the population at the beginning of a year compared with that of the births in the same year, set out in the order of the masculinities of the population, are : — Table XXV. — Masculinity in Australia. Masculinity . . 1909 1910 1911 1908 1907 1912 1913 Of Population 1.0764 1.0771 1.0787 1.0793 1.0824 1.0854 1.0885 Of Birth 1.0520 1.0638 1.0473 1.0493 1.0489 1.0454 1.0476 The trends are again in opposite directions, but not markedly. MASCXJLINITY OF POPULATION. 136 Mi ;. 45. 106 , -~~.^^ V--' 10 20 ^*"~ * ?0 The curve Is that given by formula (386) above. The dots are individual results. Masculinity of Population. V \ '^ ^~_ A _,^ -.— ' I ITBO S-f, The curve A denotes .S3 masculinity of population,' "a i B masculinity of live-births, g.j; The dots are individual ^ h^ results. 1000 1830 40 1850 60 70 80 90 1900 10 Year. Masculinity of Population and of Live-births, New South Wales, 1820-1913. Fig. 46. In the following table is set out the masculinity of the births, and in decreasing order of the population of a number of countries ; these give no definite indication : — Table XXVI.— Masculinity of Various Countries, Arranged in Order of Masculinity of Population. Year Masculin- Period for Mascvilin- Masculin- Country. of ity of which ity of all ity of Ex- Estimation Population M ^ F Determined. Births. nuptial Births only Greece 1889 1.1037 1881-85 1.118 1.059 Australia 1907 1.0793 1901-13 1.051 1.042 Servia 1890 1.0548 1885-89 1.047 1.035 Rumania 1889 1.0373 1886-90 1.077 1.034 Italy 1881 1.0050 1887-91 1.058 1.044 Belgium 1890 .9950 1887-91 1.045 1.022 France . . 1891 .9930 1887-91 1.046 1.029 Hungary 1890 .9852 1887-91 1.050 1.029 • Netherlands 1889 .9766 1887-91 1.055 1.047 Ireland 1891 .9713 1887-91 1.055 1.048 Finland . . 1890 .9690 1886-90 1'.050 1.052* German Empire 1890 .9615 1886-90 1.052 1.047 Spain 1887 .9615 1878-82 1.083 1.079 Austria . . 1890 .9578 1887-91 1.058 1.055 Denmark 1890 .9515 1885-89 1.048 1.050* Switzerland 1888 .9461 1887-91 1.045 1.016 England & Wales 1891 .9399 1887-91 1.036 1.044* Sweden . . 1890 .9389 1887-91 1.050 1.043 Scotland . . 1891 .9330 1887-91 1.055 1.059* Norway . . 1891 .9157 1887-91 1.058 1.059* Aver.(uaweighted) — .9838 — 1.0568 1.0446 * The masculinity of ex-nuptial births is greater in these instances than that of aU births ; in the other instances it is less. 136 APPENDIX A. 6. Masculinity of still and live nuptial and ex-nuptial births. — J. N. and C. J. Lewis^ studied the " variations of mascvdinity under different conditions" in 1906. Omitting seven of their quoted cases, in which the information is incomplete, they shew that stUl-births disclose a mas- culinity of 2 to 4 per centum greater than that for Uve-births. The un- weighted averages of their cases with the omission mentioned (see p. 162), viz., 17, give for the mascuUnity of live-births {M/F), 1.0504, and for that of stiU-births 1.3032 ; that is, a masculinity 1.2407 greater than that of Uve-birtlis. Results have been tabulated for Western Australia for the years 1897 to 1913 for live and still-births, and from 1908-1913 for ex-nuptial and nuptial stiU and live-births. These give the same general indication. The results are as follows : — Table XXVII. — Masculinity-ratios ior Nuptial, Ex-nuptial and StUl-biiths, Western Australia,* 1897 to 1913. Masculinity. 1897-1902. 11902-1907. 1908-1913. M. F. M-^F M. F. Mh-F M. F. Mh-F, Nuptial still- births . . Ex-nuptial still- births . . AU StiU-births . . Ex-nuptial live- births . . Nuptial live- births . . All live -births . . All birthst 507 759 15457 16216 16723 373 687 14658 15345 15718 1.359 1.1048 1.0545 1.0508 1.0639 672 982 21226 22208 22880 528 884 20108 20992 21520 1.273 1.1109 1.0556 1.0579 1-0632 804 49 853 1116 23941 25057 25910 641 37 678 1037 22882 23919 24597 1.254 1.325 1.258 1.0762 1.0463 1.0476 1.0534 * See Statistical Register, Western Australia, 1906 ; p. 12, 1914, Pt. I., p. 14. ■j- 1902 has been included twice in order to have 3 six -year periods. J Including, that is, stiU-births. The experience in Australia from 1901 to 1913 gave an unweighted average of the masculinities determined for each year, for all births, and for ex-nuptial births, the following results, viz. : — Australia ,» • ■ Various Countries (See Table XXV.) All live -births . . Ex-nuptial births All hve-births . . Ex-nuptial births Average Masculinity. 1.0508 1.0417 1.0568 1.0446 Kange of Masculinity. 1.0411 to 1.0638 1.0098 to 1.0621 1.036 to 1.118 1.016 to 1.079 The unweighted average ratio of the " ex-nuptial" to all live-births was 5.954 per centum for Australia. ^ See Jo\irn. Inst. Act., Vol. xl., pp. 154-188, April, 1906. total = m +/ = b nuptial = »*o + /o = 6„ ex-nuptial = »»i + A = 61 nuptial = '»2 +/2 = 62 ex-nuptial = '"H + fs = 63 total = m' +f' = 6' MASCULINITY OF POPULATION. 137 It was stated by R. Mayo -Smith in his " Statistics and Sociology, "^ that " among illegitimate" {i.e., ex-nuptial) " children the excess of boys is less than among legitimate" (i.e., nuptial). William Farr, however, pointed out in his " Vital Statistics,"^ that he beUeved that " it is assumed in the French returns that foundling children are illegitimate," but that such an assumption is probably invalid, and he considered the matter to be in doubt. The Australian results, however, tend to confirm those for Europe given in Table XXVI. 7. Coefficients of ex-nuptial and still-birth masculinity. — It is a somewhat remarkable fact that ex-nuptial and still-births shew increased masculinity, and that among stUl-biiths the ex-nuptial shew a somewhat different masculinity to the nuptial. For the analysis of this the follow- ing notation will be convenient : — - Live male and female births. StiU male If we call the ratio of the masculinity in the one case (say the ex- nuptial) to that in the other (say the nuptial) the masculinity intensifica- tion-coefficient k, its significance wiU vary according as we use ^1 , W2 > Ms J see Table XXI. It may easily be shewn that (337) For ^1; fc„=^.^A ; (338) For jt.2; ''n^'^^^: (339) For,x3; k„ = ''^^-^^; that is, in regard to any character in the first case it is the relative number of males born divided by the relative number of females born ; in the second case it is the relative number of males born divided by the relative number of births ; in the third case it is the ratio of the differences of the males and females, divided by the relative number of births. The coefficient intended can be indicated by suffixes and accents ; thus the intensification-coefficient of ex-nuptial stUl-births on total stiU-births would be yk'g ; of ex-nuptial on nuptial live-births, Aj^„ ; and so on ; see the preceding scheme of notation in the beginning of this section. 1 Maomillan, London, 1895, p. 77. ' E. Stanford, Loudon, 1885 p. 104. 138 APPENDIX A. The coefficients for Western Australia are as in the following table : Table XXVin. — Masculinity Intensification-Coefficients, Western Anstralia, 1897-1913. Ratio of Masculinity of To the Masculinity of 1897-1902 1902-1907 190&-1913 All stiU-births Ex-nuptial still-births Ex-nuptial live-births All live -births Ex-nuptial live-births Nuptial live -births . . 1.293 1.049 1.203 1.201 — 1.057* 1.052 1.029 * Depends upon limited numbers ; see Table XXVII. For Western Australia for 1897 to 1913 inclusive, the ratio of mas- culinity of all still-births, 1.287, on all live-births, 1.054, is 1.221. This agrees excellently with the result of a series of values for Europe shewn in Table XXVIII., the mean of which is 1.2397. Table XXIX.— Ratio of Masculinity of Still-Births to that of Live-Biiths, in various Countries. Years. Ratio. Years. llatio. Years Katio Paris Paris Livonia . . Montpellier Alsace- Ijorraine Netlierlands 8 10 10 10 5 1.157 1.179 1.205 1.208 1.208 1.210 ■ Germany W. AnstraUa Prussia Hungary Italy Amsterdam Mean 5 17 10 5 5 12 1.220 1.221 1.225 1.238 1.239 1.241 Austria Belgium . . Switzerland S\peden + Finland Sweden France Mean . . 5 5 5 9 5 1.249 1.264 1.292 1.299 1.300 1.360 Mean . . 1.195 1.231 1.294 8. Masculinity of First-bom. — ^It has been supposed that masculinity has some relation to primogeniture. For the six years 1908 to 1913 inclusive, there were in Australia 111,545 births, of which 25,708 were first births. The number of males and females gave the foUowing re- sults, viz. : — Masculinity of Australian Period. First-births. Other births. All births. 1908-1913 . . 1.05260 1.05001 1.05066 Tabulated according to ages between marriage and birth, the results were : = Period Masculinity of Australian First-births, the Interval after Marriage being — ■ Under 1 year 1 year 2-5 years &-25 years 1908-13 Difference from Mascu- linity of all live- births for same per- iod, viz., 1.0607 1.0534 + .0027 1.0514 + .0007 1.0578 + .0071 1.0091 -.0416 MASCULINITY OF POPULATION. 139 The numbers, however, are relatively small for the last group, in which there were only 3490 births. The difference between the different groups and the masculinity of aU live-births for the whole period is not more remarkable than the difference between the masculiaity of all live- births between one year and another. BertUlon's result from 1,140,860 births in Austria was 1.086 for first, and 1.054 for subsequent births ; while Gteissler's result for Saxony for 4,794,304 births was 1.054. Lewis for Scotland obtained from 85,964 births, for first births, 1.054 ; for subsequent births, 1.048 ; Streda for Alsace-Lorraine, from 47,198 births, for first births, 1.058 ; for subsequent births, 1.059.^ 9. Masculinity of populations according to age, and its secular fluctuation. — ^In any country where migration has a large influence, and especially where also the migration is of a somewhat specialised character, the masculinity is likely to shew considerable changes. In the following Table, viz., XXX., are given the mascuUnities (jus) in age-groups, for four Censuses, viz., 1881 to 1911, the masculinities in this case being {M — F)/(M-j-F). This character is strikingly different from that of England. The significance of the fluctuations of the masculinity are best seen in Fig. 47. a 3 .9 9 ® ^ "3 ■ g uS CO og — ^ «> a !» ».S "S "3 is go ■a z S o ES * (B CD a § A 20 i/ / "^ ^-^ / 1 /■ / #• "^ 10 / f" / "\ /'' ■\ \ / ' f K V ,-^-' V- \ *«^^ / , -^ y \ \ \ \ 1901 1 6- 56 3 40 60 6 Ages. Variation of Masculinity of Australian population according to age. Fig. 47. 1 See Joum. Inst. Act., vol. xl., 1906, p. 164. 140 APPENDIX A. Table XXX.— Masculinity* in Age-groups at Censuses 1881, 1891, 1901, 1911, Australian Commonwealth, and England, 1911. Computed from Smoothed Results. Age- Australian Commonwealth. England. Gbotjp. 1881. 1891. 1901. 1911. 1911. 0-4 .. 5-9 .. 10-14 . . 16-19 . . 20-24 . . 25-29 . . 30-34 . . 36-39 . . 40-44 . . 45-49 . . 60-64 . . 65-59 . . 60-64 . . 65-69 . . 70-74 . . 76-79 . . 80-84 . . 85-99 . . 90-94 . . 96-99 . . 100 .01018 .00898 .00943 .01332 .03493 .12482 .12489 .15176 .17886 .20734 .24498 .25646 .23988 .22504 .22228 .20038 .26350 .28965 .03175 -.05263 + .20000 .01374 .00975 .01195 .00389 .04192 .11802 .15534 .14833 .16100 .14761 .15267 .16233 .19446 .19310 .17717 .19886 .17799 .12313 .25424 .23967 .17647 .01227 .01105 .00981 .00223 .00157 .02183 .07807 .11272 .13292 .14744 .13833 .10217 .08809 .13194 .16770 .13247 .07707 .06902 .05306 .06215 .01588 .01064 .00869 .01485 .02472 .03155 .03485 .04356 .07038 .10160 .12294 .10885 .07725 .06274 .05417 .06685 .07253 -.02107 -.05164 -.04651 + .05263 + .00463 -.00060 -.00126 -.00804 -.05366 - .05440 -.04369 -.03459 -.03693 -.03811 -.04132 -.04883 -.06437 -.09299 -.14419 -.17745 -.21752 -.27160 -.36311 -.40237 -.43750 Mascxilinity of total Population .07983 .07362 .04824 .03840 -.03269 * (Males — Females) -4- Persons. An examination of these results shews that where there is a consider- able migration element, predictions as to the future movement of the masculinity, by extrapolation, are somewhat uncertain both for any age -group and for aU ages. Moreover, interpolations will lead to results which can be regarded only as fairly accurate. 10. Theories of Masculinity. — ^The results given shew that the masculinity of stiU-births is considerably higher than that of live -births, roughly in the proportion of about 1.15 to about 1.35 greater ; and that masculinity at birth generally is about 1.05 or 1.06. These facts are remarkable, and have given rise to various attempted explanations. J. A. Thomson in his " Heredity"^ says that, according to Blumenbach, Drelincourt in the 18th century brought together 262 groundless hypo- theses as to the determination of sex, and that Blumenbach regarded ' Murray, London, 1908, p. 477. MASCULINITY OF POPULATION. 141 Drelincourt's theory as being the 263rd. Blumenbach postulated a " Bildungstrieb" (formative impulse), but this was regarded as equally groundless. It has been suggested that war, cholera, epidemics, famine, etc., are followed by increase in the masculinity. These will have to form the subject of later investigations. At present it would seem that the first necessity is a sufficiently large accumulation of accurate statistic, as a basis for study. The one point which is clear is that death in utero (at least in the later stages) is marked by much greater masculinity than that which characterises live -births. This wUl be referred to later in dealing with infantile mortality. That the effect of war is not apparently discernible in existing statistics, is evident from the following table, viz., Table XXXI, shewing the experience of France from 1865 to 1876. It will be seen that the war- years, 1870 and 1871, and subsequent years reveal no change in the masculinity. Table XXXI.— Experience o£ France, 1865 to 1876. Deaths of Excess of Rates per 1000 of Mean Population. Children Males over Year. under 1 year of age per Females in each 1000 Marriage. Birth. Death. 1000 births. births. 1865 7.85 26.5 24.3 191 2.5 1866 8.00 26.4 23.2 162 2.6 1867 7.85 26.4 22.7 170 2.1 1868 7.85 25.7 24.1 192 2.3 1869 8.25 25.7 23.5 176 2.4 1870 6.05 25.5 28.4 191 2.3 1871 7.25 22.9 35.1 240 2.4 1872 9.75 26.7 22.0 152 2.3 1873 8.85 26.0 23.3 180 2.4 1874 8.30 26.2 21.4 158 2.6 1875 8.20 25.9 23.0 170 2.4 1876 7.90 26.2 22.6 165 2.3 XI.— NATALITY. 1. General. — The phenomena of human reproduction, as affecting population, and the whole system of relations involved therein, may- be subsumed under the term " natality." In one aspect they measure the reproductive effort of a population ; in another they disclose the rate at which losses by death are made good ; in a third they focus attention upon social phenomena of high importance (e.g., nuptial and ex-nuptial nataUty) ; in yet another they bring to light the mode of the reproductive effort (e.g., the varying of fecundity with age, the fluctuation of the frequency of multiple-birth, etc.) In this section we shall deal with the questions which relate more directly to birth-rate, and shall treat of those which relate more directly to nuptiality in section XII, and to fecundity in section XIII. Birth-rates are not immediately comparable. The physical and social development of two communities being identical, their birth-rates become roughly comparable only when the relative numbers of married and of single women at each age are identical. In regard to the initial qualification, it may be pointed out that any of the races of Western Europe, for example, may be immediately compared on the basis of identical numbers at the same ages ; but a population of the natives of India would not be comparable to one of Western Europe because of earlier physical development and earlier marriage. Comparisons of this special character, however, may sometimes be founded on principles indicated by the theory of " corresponding states" in physical investiga- tions. This matter will be referred to later. Populations similarly characterised in respect of features, material to any question at issue, may be called homogeneous in that respect. In order to compare the birth-rates of populations, otherwise homogene- ous, but differently constituted in regard to age, it is necessary to take account at least of three things, viz., (i.) the numbers at each age ; (ii.) the relative fecundity at each age ; and (iii.) the relative numbers of married and single women. In other words, a convenient and strict comparison can be made satisfactorily only on the basis of what may be called a " standard" or " normal" female population. This normal population should represent the mean of the whole series of populations proposed to be compared (i.e., the relative numbers of married and of single females at each age should be their ratio to the entire aggregate). Comparison is then effected by attributing to this population-norm the nuptial and ex- nuptial birth-rates actually existing in the populations to be compared with one another. Such a comparison is free from the effect of accidental differences in constitution as to age ; thus the relative magnitude NATALITY. 143 of the birth-rates and populations compared are revealed. The principles of developing norms of this type have already been considered ; see VIII., §§ 8 to 12. We consider first the ngiture of a birth-rate. 2. Crude birth-rates. — ^While the total number born in any population during any period, divided by the average number of the population during the period, i.e., the crude birth-rate, is one element of the rate at which the population is reconstituted, its nature and Hmitations are importa-nt from certain points of view. We propose to consider these. Since both births and population vary with time, we may regard their variations of rate as represented by the functions / {t) and F (t). Thus if Bjn denote the number of births occurring in a unit period (say 1 year), and P^ be the mean population during that period, the average period- rate (annual rate in the case supposed), which may appropriately be re- ferred to the middle of the period, is : — R R 1 (340) ^ Pw. P t^^F{t)dt the instantaneous value passing through the range of values which deter- mine the form of the functions / and F. P is the population as at the middle of the year, and B the rate per annum at which births are occurring at that moment. In general, no serious error wiU be introduced in the value of j8 if, instead of P^, the population at the middle of the year is used, though more accurate results wiU be to hand if population-determinations at the end of each half-year, or each quarter, or better stiU each month, are used to ascertain the mean. The necessary formulae would be respectively (341). .P„=i(P„+Pi) ; or = -f (P„ + 4Pi + Pi) ; or = ^ (Po+4Pi+2Pi+4Pi-FPi); or (P„ + 2P,., + 2PaH- . . . .2Px. + Pi ; or = i Kn + P.% + Pf. + -Pie + ^i) + 2Pa + 5 (P^-f Pa + Pa+ Ph) + 6 (P,3, + Pft)} ;i or any of these indicated in VI., § 2, Table VI. 1 The question of the formulae to be preferred was discussed for quarterly results in the Population and Vital Statistics Bulletin for Australia, No. 1, pp. 20, 21, and the coefficients adopted were 1, 4, 2, 4, 1, though previously 1, 2, 2, 2, 1 had been used. The use of formulae based upon integral functions supposes that the recorded population at the moment of record is substantially free from large deviations from the nvunber represented by the functional change. If the functional change is small, and the " accidental" deviation is large, the use of the functional formxilae does not yield the advantages expected, and has the disadvantage of multiplying the " acci- dental" deviation possibly by a very large or a very small factor (as the case may be); if the former, the result is not satisfactory. 24 144 APPENDIX A. Such formulae are, of course, more than abundantly accurate for all statistical purposes. Birth-rate is influenced by — (a) the sex and age constitution of the population ; (b) all forces restricting the fecundity of a population (e.g., frequency of, and the age of, marriage ; social tradition and habits ; etc.) ; (c) the frequency of multiple-births ; (d) infantile mortality (since mothers who lose their offspring are again exposed to the risk of maternity), etc. These influencing factors will be considered either in this section, viz., Xn., or in later sections. 3. Influence of the births upon the birth-rate itself. — ^Let it be sup- posed that the population of two communities be initially P and that in the same period B births occur in one and 2 B in the other, of which in each case the proportion s survive ; the numbers being thus sB and 2s B at the end of the period. If there were no migration, and no deaths, other than those arising from the births, the deduced birth-rates would be R o i> 9 7? a larger quantity. Hence the effect of an increase of a birth-rate, when a proportion of the births is incorporated in the population, is to somewhat diminish that ratio of births to population, which really represents the relative frequency of birth, unless at least the population is increasing in some manner which counteracts this. The preceding result is more obvious if put in the form — (342„)..2^a = p|:^{l+M-H4)Vetc.} ; ^, = ^ More generally we have — (3426).... iSi: ^2 ^' ^^ (r denoting the increase, supposed linear) ; shewing that the birth- rates and births are in the same ratio only if the mean populations are identical. Hence as measures of fecundity birth-rates need some sHght correction, owing to their influence on the magnitude of the population. They are strictly comparable in this respect only when two populations are homogeneous, and differences of birth-rate themselves disturb the homogeneity and thus involve the apphcation of some correction. ^ ^There is an analogous case in connection with the computation of interest earned on assurance and similar funds. Thus if I denote the interest earned in the course of a year, A and B the funds at the beginning and end of the year respectively, and i the effective rate of interest earned on the funds during the course of the year, then the value of i is approximately given by the following formula, now generally adopted in practice : — i = I/{^{A + B) - ^I } NATALITY. 145 4. Influence of infantile mortality on birth-rate. — Denoting the number of births by B, and of infantile deaths by M, and the number of women of child-bearing age by P, we shall have for the birth-rate /3, attributed not to the whole population but to the P women, and for |i the rate of infantile mortality — (343) j8 = B/P ; ,j, = M/B ; ^^ = M/P. Suppose [L to change to some other value fi' = M' /B' ; M' being the number of deaths and B' the number of births under the changed state of things, assumed to have become constant. Then, since mothers who lose their children are exposed to an increased risk of maternity, the ratio of which is only the proportion q (a proper fraction) of the full risk, we shall have for the number at risk as originally, viz., N, and also after a change in the prevailing rate of infantile mortahty, N' (344) N = P — B +qM ■ and N' = P — B' +qM'. If the reproductivity of these two groups is the same, then B/N = B' /N' ; from which it follows that — (345, ^L + J^_1.1 + i*_l and consequently, discarding the unit from each side and writing in the values of the quantities as by (343) above, we have — ■ (346) -J + qti =j^ + q^'; that is — (347) i8' = P {I +qp' ifi' - t.)} It will be found that this change is sensibly a Hnear one, or any increment in the rates of mortality will cause a sensibly constant but small proportional increase in the birth-rate. If we call the birth-rate, freed from the influence of infantile mortality, the normal birth-rate j8g, then — (348) j8o = ;8 (1 + k^i). in which k may be regarded as a constant for a particular community, and a particular epoch. The value of k was found on the average for Europe to be about + 0.033 fx, or Po = P + 0-033 fi, the birth-rate j8 being expressed per 1000 of the population, and the infantile mortality rate u, expressed per 1000 births. An examination of the data for differ- ent countries gave the following results : — 146 APPENDIX A. TABLE XXXn.— Influence of the Rate of MantUe MortaUty on the Crude Birth-rate for Various Countries, about Year 1900. Period. Value of /Sq and k COUNTBY. in;8 = ^0 + fcM.t Birth. MortaUty. New Zealand 1881-1905 1882-1906 13.2 k -^ 0.191 CoTTiTTionwealth . . 1887-1905 1888-1906 16.8 + 0.118 Sweden 1881-1904 1882-1905 17.1 -i 0.100 Norway 1881-1905 1882-1906 20.5 i 0.100 Prussia . . 1881-1905 1882-1906 19.1 + 0.085 Various Countries* 1901 1902 19.4 1- 0.083 Netherlands 1881-1905 1882-1906 22.6 + 0.063 France 1881-1905 1882-1906 12.7 + 0.061 Denmark 1881-1905 1882-1906 22.4 + 0.060 Japan . . 1 1881-1904 1882-1905 22.3 + 0.053 Ceylon 1881-1905 1882-1906 26.4 + 0.042 Jamaica . . 1881-1905 1882-1906 34.3 + 0.022 Switzerland 1881-1904 1882-1905 25.3 + 0.018 Ireland . . 1881-1905 1882-1906 25.8 — 0.026 England and Wales 1881-1905 1882-1906 38.6 — 0.058 Scotland 1881-1905 1882-1906 38.9 — 0.068 * For one year only. t The birth-rate being expressed per 1000 of the population, and the infantile mortality per 1000 births. The infantile mortaUty rate {n) in the table is expressed by the number of infants dying per 1000 of infants bom. The crude birth-rate (3) is the number of births per 1000 of the total population. It will be seen that the magnitudes of k, and therefore of q, have no general relation to the magnitude of the birth-rate ; that is, a particular value of the risk-factor is characteristic of a particular country. In an investigation made in 1908"^ it was shewn that the influence of infantile mortality was very irregular in its operation, and the following deductions were stated, viz. : — ^ (i.) When either all mothers of deceased infants, or any constant proportion thereof, may be regarded as subject to equal risk of fecundity (i.e., equally hkely to bear children) then equal increases in the rate of infantile mortality tend to be followed by equal though relatively small increases in the birth-rate. (ii.) The influence of infantile mortahty on the birth-rate must always be very small. (The contrary proposition is not, of course, necessarily true). This type of investigation aims rather at ascertaining the form of the function expressing the correction, so that the form being determined, the constants can then be ascertained from the data. It would appear that yearly irregularities of birth-rate are so great as compared with the influence of infantile mortahty that the latter is virtually masked by the former. Probably in any rigorous investigation of a measure of the fecundity of a population the birth-rate should be corrected in some such way as has been indicated. 1 By the writer. See Journ. Roy. Soc, N.S.W., Vol. xlii., pp. 238-250, par- ticularly Fig. 1 on p. 243 therein. 2 Loc. cit. pp. 241-2. NATALITY. 147 5. World-relation between infantile mortality and birth-rate. — ^In order to ascertain whether in a world-wide survey of infantile mortality and birth-rates any correlation manifested itself we may extend the pur- view of all countries where fairly accurate statistics are available, viz., the following : — Australia, Austria, Belgium, Chili, Ceylon, Demnark, England and Wales, France, Ireland, Italy, Jamaica, Japan, Netherlands, New South Wales, New Zealand, Norway, Queensland, Russia, Scotland, South Australia, Spain, Sweden, Switzerland, Tasmania, Victoria, West Australia. The populations are, of course, repeated with different rates, and are equivalent to 8776 millions,^ the results forming groups of available results ; according to the magnitude of the infantile mortality we get the results shewn in Table XXXIII. hereunder, the ranges of infantile mortality being shewn therein.^ In Pig. 48, graph A denotes the relative frequency of the given ranges of infantile mortality.* It will be observed that the graph is dimorphic, that is, that while the characteristic rate of infantile mortality is about .0150 (150 as usually expressed), there is also a second mode for the rate of about .0255. The corresponding crude birth-rates are about .029 and .048 respectively (or residual birth-rates, see hereinafter, about .025 and .035). It will be seen that there can be a very high rate of infantile mortality with low birth-rate, but it would appear, only for very limited populations.* TABLE XXXni. -Relations of Infantile Mortality and Birth-rate, various Countries, about Year 1900. Popula- Ranges of Infantile Crude tion Ee- Infantile Mean of Mean of Mortality Birth-rate 12 Months presented Mortality Infantile Crude of of Residual of (millions) for Individual Populations. .0688-.0959 Mortalities. .0821 Birth-rates. Aggregate. Aggregate. Birth-rate. 344 .0291 .0911 .02692 .02447 479 .1018-. 1232 .1120 .0291 .1119 .02889 .02566 2035 .1276-. 1474 .1371 .0288 .1387 .02865 .02468 2172 .1519-.1724 .1618 .0291 .1598 .02904 .02440 1116 .1762-. 1974 .1872 .0340 .1880 .03391 .02753 851 .2032-. 2 179 .2098 .0367 .2085 .03365 .02663 297 2213-.2372 .2286 .0380 .2279 .03808 .02940 696 .2406-.2559 .2490 .0480 .2491 .04757 .03572 668 .2601-.2771 .2688 .0479 .2710 .04763 .03472 189 .2800-. 2920 .2870 .0446 .2845 .04885 .03495 105 .3040-. 3290 .3133 .0385 .3075 .04549 .03150 147 .3325-. 3490 .3406 .0366 .3392 .03701 .02446 91 .3660-.4120 .3890 .0372 .3800 .03681 .02282 1 The method is, of course, not perfectly satisfactory ; for, as pointed out bv the writer (on p. 245), loc. cit. the populations are not homogeneous, and doubtless if more moderate-sized districts could be analysed the material would give a clearer indication of the true nature of the relation. '^ See also loc. cit., p. 246, and Fig. 2, p. 247, in the same paper. ' See page 150 hereinafter. * Similar indications are given by the analysis before referred to. See loc. cit. p. 248, Fig. 3. 148 APPENDIX A. This more general result shews that propositions (i.) and (ii.) in the preceding section can be regarded as true only for individual populations and probably for very Umited periods of time ; the effects are readily masked by more potent influences. In the table hereunder (XXXIV.), of results in the present century, the following countries have been included, viz., in column (i.) New Zea- land, 1913 ; Norway, 1912 ; Australia, 1913 ; Sweden, 1911 ; France, 1912 ; Netherlands and Denmark, 1913 ; Switzerland, 1913 ; Ireland, England and Wales, and United Kingdom, 1913 ; Finland, 1912 ; Scot- land and Ontario, 1913 ; Belgium, Italy and Prussia, 1912 ; Serbia, 1911 ; German Empire, 1912 ; Spain, 1907 ; Bulgaria and Japan, 1910 ; Jamaica, 1913 ; Austria and Hungary, 1912 ; Ceylon and Roumania, 1913 ; Russia (European), 1909 ; Chile, 1911 ; and in column (iv.) France and Belgium, 1912 ; Ireland, England and Wales, and Ontario, 1913 ; Sweden, 1911 ; United Kingdom, 1913 ; Switzerland, 1912 ; Scotland and Denmark, 1913 ; Norway, 1912 ; New Zealand, Nether- lands, and Australia, 1913 ; German Empire, Prussia, Finland, Austria and Italy, 1912 ; Spain, 1907 ; Japan, 1910 ; Jamaica, 1913 ; Serbia, 1911 ; Hungary, 1912 ; Chile, 1911 ; Ceylon, 1913 ; Bulgaria, 1910 ; Roumania, 1913 ; Russia (European), 1909. The results are the weighted means (or what is the same thing, the values are for the population- aggregates) of the populations, combined in successive groups of ten, arranged (in ascending order) according to infantile mortality in the one case, and according to birth-rate in the other. These results shew unequivocally that there is, in general, a relation between birth-rate and infantile mortality. The calculated results are as follows ; jS denoting birth-rate per unit of population, and /i denoting infantile mortaUty rate per birth : — Determined from groupings in the order of infantile mortality :— (349). .j8 = 0.00956 + 0.1405 fj. ; (which gives /x = 0.06804 -f-7.117 )3) ; and determined from grouping in the order of birth-rate : — (350). . . .^ =-0.03661 + 5.970/3 ; (which gives ^ =0.06132 +0.1675/x). The mean of these result.s is expressed with sufficient precision by — (351).. /3 =0.00785(1 +19.6^); fj, =0.0510(1—127^) jS being the rate per unit of population, and jj, per birth. NATALITY. 149 TABLE XXXIV.— General Relation between InJantUe Mortality and Birth-rate, Aggregates of various Countries, 1907 to 1913. InFANGILE MoRTAUTY and BtETH-BATi;. Birth -BATE AND INFANTILE MoBTALITY. Popula- Infan- Re- Popu- Tnffl.n - Re- tion in tile Birth- Calcul- duced tion in Birth- tile Calcul- duced Mil. Mor- rate, t ated, t Birth- MU- rate, t Mor- ated.§ Birth- lions. tality.* rate.! lions. taUty.* rate. * (i-) (ii.) (ii i.) (iv.) (V.) (vi.) 107.6 90 22.6 22.2 20.6 154.2 22.7 99 99 20.5 152.5 96 23.0 23.1 20.8 116.9 24.0 105 107 21.5 153.3 96 23.1 23.1 20.9 110.4 24.1 104 107 21.6 153.2 97 23.0 23.2 20.8 112.2 24.4 103 109 21.9 150.4 99 23.0 23.5 20.7 80.1 24.8 99 111 22.3 118.3 107 24.3 24.6 21.7 143.5 26.4 121 121 23.2 147.2 113 26.1 25.4 23.2 179.1 27.1 128 125 23.6 185.4 121 26.7 26.6 23.5 136.2 28.2 134 142 24.4 184.5 122 26.9 26.7 23.6 161.3 28.8 144 135 24.7 246.3 129 27.3 27.7 23.8 191.6 29.6 142 140 25.4 228.6 135 28.4 28.5 24.6 208.0 30.0 144 132 25.7 186.9 142 29.8 29.5 25.6 256.1 30.8 148 147 26.2 234.2 146 30.7 30.1 26.2 255.9 30.8 148 147 26.2 230.4 147 30.8 30.2 26.3 252.7 31.0 150 148 26.4 256.6 151 31.0 30.8 26.3 269.0 31.4 154 151 26.6 270.1 155 31.6 31.3 26.7 206.3 32.5 159 157 27.3 239.3 159 31.6 31.9 26.6 169.5 33.6 163 164 28.1 205.6 163 32.5 32.5 27.2 170.6 33.9 164 166 28.3 319.2 194 36.7 36.8 29.6 149.1 34.8 163 171 29.1 256.5 208 38.9 38.8 30.8 230.6 39.8 211 201 31.4 * Per 1000 births. t Per 1000 population. § By formula (350). t By formula (349). From these the lines B and C respectively are plotted and the cal- culated values in columns (iii.) and (vi.) are computed. The dotted Hues shew the positions of the other "graph for the purpose of comparison, and the line which represents formula (351) is between the two. That these results, though not identical, are very similar, is seen from the graphs B and C, shewing the two series of values. What they estab- lish is that, on the whole, the birth-rate and infantile mortality increase together. Moreover, when the birth-rate is reduced to its effective value twelve months later (that is, for one year of age), it is much more uniform on the whole. Since, as shewn, the. increase of risk of maternity is re- latively small (348), it follows that, on the whole, the social conditions which characterise a large birth-rate are those associated with a high rate of infantile mortahty. This, of course, is not necessarily so, but expresses the general fact. In short, a high birth-rate is usually associated with a high rate of infantile mortality, but high infantile mortality wiU, per se, not appreciably affect the birth-rate. The importance of this result is obvious. 160 APPENDIX A. GENERAL RELATION BETWEEN INFANTILE MORTALITY AND BIRTH-RATE. a s ^ 40 /I / // / / / f 30 f /y I / y / // i M / / / / / / / 10 J / / ; \ X 10 / / / ■20 \ _^^ ^ w 100 200 100- 200 For curve A.— Infantile mortality rate per birth. For curves B and C— Infantile mortality rate per 1000 births. Pig. 48. 6. Residual birth-iates. — Owing to the \evy high death-rate of infant.s, the crude birth-rate, taken alone, is not a satisfactorj' expression of the effective recuperative force of a population against the ravages of death. It is not practicable, however, to assign any particular age as specially appropriate for estimating the virtvxil efficiency of birth-rate, and as we have seen high birth-rates, however, are ordinarily associated with a high rate of infantile mortality. For example, New Zealand and Australia had birth-rates in 1912 of 26.5 and 28.7 per thousand population, and infantile death-rates {i.e., deaths under 12 months per 1000 bom) of 51 and 72, while Ceylon and Chile, in 1911, had birth-rates of 37.9 and 38.5, and infantile death-rates of 218 and 332. This question will be referred to later. Birth-rates corrected so as to represent the number hving after a given period may be called residual birth-rates, and the quantity multi- plied into a birth-rate to give its residual value may be called the survival coefficient, or survival factor. We shall consider these. Owing to the fact that of all the deaths which occur in 12 months, about 42 per cent, occur in the first month, the infantile mortahty may be referred to the same calendar year as the births without sensible error, or we may correct NATALITY. 151 it as explained hereinafter. Let ^ be the birth-rate and y the rate of infantile mortality, the first expressed per unit of the population, the latter per birth. Then the residual birth-rate ^j is^ — (352) ^, = p {I -y) The quantity in brackets is the '• survival-factor" and jS, is the " residual birth-rate." For a population in which the number of births was con- stant and the rate of mortality for the first twelve months was constant, the probability of persons of age living to age 1, viz., ^pi, would be the same as the survival factor, since under these conditions it would denote the ratio of those surviving one year, viz., li to the number born, viz., If,. Consequently, subject to this limitation — (353) (l-y)=^i=Zi/Z„. For a population in which the number of births is increasing, say, at the rate rt, and the rate of infantile mortahty diminishing,^ say, at the rate r't, these quantities become functions of time and are affected by the interval of time between the year for which the births are recorded and the somewhat later year for which the infantile deaths ought to be recorded, in order to properly refer to the birth-group. As, however, the error arising is of a small order as compared with the accidental deviations from year to year, it is questionable whether a correction is worth apply- ing. It may be mentioned that in Australia it was found by an investiga- tion for the years 1909 and 1910, that all children who die in the first year of life live on the average 99.3 days, and children are registered on the average 38.2 days after birth. ^ The difference, 61.1 days, or say two months, is regarded as the difference between the years. Thus the in- fantile mortality in the following table was calculated on the births occurring one-sixth of a year earlier. Similarly the birth-rate given for the equivalent year to n, say ^e, is — (354) iSe = i i8„_, + t i3n . It may also be noted that an investigation of the question shewed that of the deaths in Australia under 1 year of age occurring in any calendar year, 0.72 to 0.74 per cent. — average about 0.73 — arose from births which occurred within that calendar year, and 0.27 from those which occurred in the preceding year. This proportion is doubtless approximately true also for other countries. 1 These rates are commonly expressed per 1000 of the population, and per 1000 born respectively, in which case the formulae will be /S/ = /3' ( 1 — -JL- ) ; j8' and y being 1000 times greater than /3 and y. ^ Infantile mortality has for years past been steadily diminishing in many countries. ' This has ceased to be true because of the " maternity bonus." 162 APPENDIX A This would suggest that the coefficients in the above equation (354), should be ^ and f instead of ^ and |, but, only if the average late- ness of the registration of births and deaths were the same, which, how- ever, was not the case. The practical result of the difference is not great. It wiU appear from a rigorous investigation in the next two sections, that with the rate of infantile mortality as it stood during the years 1909 to 1913, the proper proportion is about 0.731, a proportion which wiU be modified only by the difference in the registration interval. This interval, owing to the payment of the maternity bonus, resulting in earlier registration of births, has now become smaller. TABLE XXXV.— Residual Birth-rates, AustraUa, 1904-14. Crude InfEintile Crude Year. Birth-rate, Death-rate t Birth-rate Survival Residual for Calendar Calendar for Equival- Factor. Birth-rate Year.* Year. ent Year. 1903 25.29 1904 26.41 81.77 26.073 .91823 23.94 1905 26.23 81.76 26.260 .91824 24.11 1906 26.57 83.26 26.497 .91674 24.29 1907 26.76 81.06 26.728 .91894 24j61 1908 26.59 77.78 26.618 .92222 24.54 1909 26.69 71.58 26.673 .92842 24.76 1910 26.73 74.81 26.723 .92519 24.72 1911 27.21 68.49 27.297 .93161 25.43 1912 28.66 71.74 28.410 .92826 26.37 1913 28.26 72.71 28.317 .92729 26.26 1914 28.05 71.47 28.083 .92853 26.08 • Per 1000 population. t Per 1000 births. The final column is the efficient birth-rate, the end of the first year of life being taken as an appropriate point of time for determining the efficiency, since the larger death toll from infantile troubles may be regarded as then past. 7. Determination of proportion of infantile deaths arising from births in the year of record, number of births constant.^Births, and infantile and other deaths, are recorded as occurring during successive equal periods of time, usually calendar years, half-years, quarters, months, etc. ; and the deaths during such periods are distributed accord- ing to a series of age-Umits, for adults usually whole years, 0-1, 1-2, etc. In the case of " infantile deaths" or deaths of children under one year of age, they are distributed according to age-hmits of weeks, months, quarters, etc. Consequently the infantile deaths occurring in any year are drawn from the births [and immigrants] both in the year of record < * NATALITY. 163 and in the previous year. More generally deaths of persons between the ages x^, and x^ recorded in any period of time, say — i^ to 0, are drawn [where there is no immigration] from those born [in the country] during the period — (a;2+*z ) to — (ajj+O).^ In the same way deaths recorded in any period — f2z to —t^ would be drawn from those born [either in the country or from migrants entering it] during the period — (a;2+*2z) to - (^1 + h )■ If the frequency of births be denoted by k' Fi (4), the number of survivors after any period of time, x, of persons born at the moment t, wUl, so long as the death rates at each age remain constant, also be this function multiphed by the. probability of surviving to the age x. Thus if this probability be denoted by Xx, or that of dying be denoted by Sa;, = I — \x, then the survivors of age x, say 8x, and those who have not attained that age, say Dx, will be — (355) 8x =XxFi(t); and Dx = 8xFi(t) for we may make A;' = 1 it ratios only are needed.^ With births increasing, the successive records of the dying of any given age wiU also shew a similar progressive increase, proportional to that of the births, the death-rates at each age being constant. Thus the aggregate of births between the times ' /' \ 1 ri i f \ / 0. n 4 kJ I ?' \ 1 ^..^ t , \ i'' ' i t\ 7 ' ' ,-» 1 ~" i > " / 1 / V / > 1 / -a — — y \ / \ ^ \ n 7 \ 11] y ' 10 - «. ^ ^ ^ ^'' 1 - \ \- y \ 0.07 ■ ,L \ r - s / _ 5« ^ r ~ Ja Fe Mr Ap My Jn Fig. Jy Ag 54. Se Go No De Ja birth and registration has, however, shortened since the introduction of a maternity bonus : see pp. 151 and 152. 168 APPENDIX A. The following procedure was adopted. The births registered were taken out in the several quarters ; these quarters were then equalised, the numbers being corrected to shew what would have been given by a constant population, since it was found that the increase of this last was sensibly at the rate 1 + 0.0247265 t. In this way the values shewn in Table XLII. hereinafter were obtained. These quarterly results may be subdivided into monthly values, as explained on the next section, so as to give the monthly values. These results are shewn by the curve B in Fig. 54. TABLE XUI.— Birtbs Registered. AustraUa, 1907-1914. Births aa Registered. Births as Corrected for Equal Quarters and » Ck)nstant Population. 236.462 243,191 254,141 242,860 241,457 .98891 244,914 1.00307 251,467 1.02987 238.830 .97816 The values for the individual months may be deduced as explained in the next section, and are as follows : — 123456 78 9 10 11 12 .9807 .9916 .9944 .9936 .9996 1.0160 I.Q333 1.0366 1.0197 .9924 .9922 .9699 and these monthly results are shewn by the small rectangles in curve B, Fig. 54. For the greater part of the year, at least, the results are substantially identical for the two sexes, as a compilation made for the four years, 1907- 1910, shews. The results were as follows : — TABLE XLm. — Seasonal Fluctuations* of Births, according to Sex. Australia, 1907-1910. Males, Females or Persons. Jan. Feb. Mar. April May. June. M F P .9874 .9903 .9889 .9169 .9229 .9198 .9949 .9950 .9949 1.0162 1.0079 1 1.0116 ' 1.0064 1.0069 1.0067 .9978 .9859 .9920 July. Aug. Sept. Oct. Nov. Deo. M F P 1.0321 1.0170 1.0249 1.0410 1.0583 1.0504 1.0299 1.0437 1.0367 1 1.0378 1.0465 1.0420 .9924 .9760 .9844 .9482 .9479 .9480 * The registration was on the avert^e 38.2 days after birth for the years 1907-1910. Reverting to curves A and B, Fig. 54, the curve drawn by lines may be taken as a probable representation of the fluctuation ; since there is no reason to suppose that the large oscillations are other than accidental. NATALITY. 169 As the theory of determining the Fourier curves to fit the group results presents certain special features, it is given hereunder. 16. The subdivision of results for equalised quarters into values corresponding to equalised months. — When quarterly results are available, they may (after equalising and also being freed from the annual pro- gression so as to give, as residuals, only the fluctuation elements) be readily resolved into monthly values, which have a high degree of probability. The most convenient form in which to give such results is the height of the monthly group. Let the mean of the heights of four quarterly groups be denoted by R, with suffixes corresponding to the quarter (viz., 1 to 4), and that of the monthly group by r, with corresponding suffices (viz., 1 to 12). Then the solution can proceed on one of two possible assump- tions, viz. (a) that the amplitudes of the component fluctuations are identical, and the epochs are different, or (6) that the epochs are identical and the amplitudes are diflEerent.^ That is, we may assume either (a) that — (375) y=a-\-b sin. (a; + j8 ) + 6 am. 2 (a; + y) ; or (6), that — (376) y =a +b sin. (x + ^ ) + c sin. 2 {x + j8). The data are, of course, inadequate in themselves to determine which assumption should be adopted, and the results are to that extent, uncertain. But this uncertainty, in general, is of small moment. In case (a) we have — (377). J -= bcos.^ = ^{Bi + Bi); m = ~bsin.^ =^{Ri+Ri), 3 3 3 3 (378). .2)= 6co«2y = -(E2+-B4) ; g=-hsin2y=-^y/ —^B.yB^-^Ei,B^) It will be seen that q is not independent of Z, m and p, since we must have — (379) g2 == (^2 _|_ ^2 _ 4 ^2) From this last, the value ^\/ —1 (RiRt -f RiRs) is deduced. Ob- serving that ^ ^/3— 1 = — 0.1339746 ; ^ (1 — \/3) = - 0.3660254 ; ^ y'3 = 0.8660254 ; we may put the values of ri to r^ in the following very convenient forms, viz. :— (380) n = — 0.1340 Z+ ^ m - ^ P -f 0.8660 g. (381) rg = — 0.3660 I + 0.3660 m - P ^ See Studies in Statistioal Representation (Statistiool Applioations of the Ppurier Series), by Gr. H. Knibbs, Joum. Roy. Soo. New South Wales, Vol. xlv., pp. 76-110, 1911. In partioular see pp. 88-89. 170 APPENDIX A. (382) rs =— 4" ^ + 01340m - ^ P— 0.8660 g. (383) U =- J ^— 01340 m + -i P - 0.8660^. (384) rs = ^ 0.3660 Z — 0.3660 m + P (385) re = 1.8660 Z — -^ m — — P + 0.8660 q. (386) ry =+ 0.1340 Z— ^ m — ^ P + OMGOq. (387) fa =+ 0.3660 Z — 0.3660 w — P (388) rs =+ 4 Z- 0.1340m —\ P— 0.8660 g. (389) rio = + -5^ ^ + 0.1340 m -\- ~ P — 0.8660 g. (390) rii = + 0.3660 Z + 0.3660 m + P (391) n% =— 1.8660 Z + ^ "* — 4 ^ +0.8660g. In case (fc) we have — (392).... Z= -- 8 6 cos = -^{Rz^R^); m== bsin^=-^ (Ri+ Ri). (393). . . . P = — I c co« 2j8 = I (2?2+i24) ; 3 = | V (^J (-^3 + -B4)* + (JJi + P4)^]-(iJi+ Rs)^\ Again, q is not independent of Z, m, and p, since we have — (394) *^^=i^;=-^^=^(^^+-^)-^^- which leads to the value of q above written. If c = 6, the last expression for q in (394), reduces to that first given, viz., in (379). It is obvious from this last value for q, that the ratio cjb is at our disposal, and provided it be so chosen that the whole expres- sion within the braces is not negative, there wiU be a real value for q. A unique solution will be that which makes the q term zero in the above series of equations for monthly values. This is given by making the expression within the braces in (393) zero. Hence for this we have (395) j= {Ri+Rs)/VHR3 + Ri)^+(Ri + R*)^'i If, therefore, the relation between j8 and y, and between b and c are both unknown, we may, with advantage, write g = in the series of equations 380) to (391). In short, if we assume that c = b then y is determinate. NATALITY. 171 If this relation be not assumed, but that y = ^ is assumed, we may, vnthin certain limits, still make the ratio of c to 6 whatsoever we choose, and, if we have no ground for believing that a particular ratio is to be preferred, the simplest solution of the whole problem is, making the epochal angles fi and y identical, to so take the ratio of c to 6 that the q term will be eliminated from the series of equations for monthly values, viz., formulae (380) to (391), etc. ; that is, we may determine this ratio by (SOS)-*^. It may be reiterated that the subdivision of the quarterly into monthly values by the preceding formulae assumes that the fluctua- tion involves only terms sin. x and sin. 2x. 17. Equalisation of periods o£ irregular length. — ^In order to apply the formulse of the preceding section, it has already been indicated that the crude data must be freed from any annual progression depending on a progression in population numbers and among the births themselves. It is preferable to operate, therefore, on rates, i.e., to divide the number of births (or marriages or deaths, etc.) each month, quarter, or year, as the case may be, by the mean population of the month, quarter, or year itself. Even then a correction is necessary, since for precise results it is still necessary to equalise the period, in fact, if the seasonal fluctuation (or armual period of oscillation) to be determined be small in amphtude, the equaUsation is an essential. Both months and quarters differ appreciably in length.^ For population-numbers and for birth-numbers, the equalising corrections will necessarily be made in a somewhat different manner. A table of corrections for the ends of the months or quarters is first formed. Numbers such as population-numbers and rate-numbers may be called continuant, B,TaA. those such as numbers of birth, marriages and deaths, etc . , accretional. For the purpose of corrections it may also be assumed that the daily values at the terminals of the unequal periods is the mean of the values for the adjoining periods.^ Then, except for the first and final period, there are two corrections. For a single leap-year there is no correction at the end of August, and none at the end of October. The equalised February is always in January, and excepting as above mentioned the terminal of the equalised month is always in the month follomng.* ' Suoh a solution has the further advantage of making the deviations from the averages for the respective quarters a minimum. ' The shortest month is no less than 8 per cent, short of the average, and shortest quarter 1.37 per cent. ' It is more rigorous, of course, to determine the function, the integral of which gives the result dealt with, but this process is tedious and ordinarily quite un- necessary. * There would have been some advantage if January had had 30 days, instead of 31, and February 30 days in ordinary and 31 in leap years, instead of 28 and 29 days. 172 APPENDIX A. Let 8 T and S T be the small periods to be added respectively to the beginning and the end of an unequal period to make it coincide with an equalised period, the length of this last being T^. Let also the periods preceding and following that to be corrected be denoted by T and T' ; and let the period to be corrected be denoted by T^. Then, the correct- ing periods Sy, etc., being small, we have very approximately, for continuant numbers, P, P„ and P', etc., denoting that corresponding to To, (397)....Po=P„+ ^ |{P-P„)8r+(P'-P„)8'T;. and for accretionai numbers, N, N„, N', etc., N^ denoting that corres- ponding to the period T^, = Nm+^J-{N+N^)ST+(N„ + N'}S'Tl The approximate identity of these expressions can readily be estab- lished.^ In regard to the sign of the corrections it may be observed that for continuant numbers the value is to be increased when the shift of either terminal of the unequaUsed period towards the terminal of the equalised period is in the direction of higher values. For accretionai numbers, the number is increased for an additive shift, diminished for a negative shift. 18. Determination of a purely physiological annual fluctuation of birth-rate. — ^The annual birth-rate fluctuation, as obtained in section 15, by means of the formulae of sections 16 and 17, cannot be regarded as furnishing the variations of the reproductive activity solely due to physiological causes, which variations may be presumed to repeat them- selves every year. The distribution of the frequency of marriage, and therefore of birth, throughout the year is afiected by the fetes observed, and particularly by the " movable feasts" (Easter, etc.). The number of years to be included to secure a true mean-determination must embrace the whole cycle of movement. The extent of this cycle has been referred to in a paper on the Statistical Application of the Fourier Series, by the writer.^ But even when this mean result is obtaiaed, what may be called the physiological fluctuation is not to hand, since the effect of the " movable feast" is distributed, not eliminated. By a systematic analysis, ^ The question of corrections of this kind has been dealt with at length by me in a paper read 5th July, 1911, at the Roy. Soc, N.S.W., see its Joum. xlv., pp. 79-85, wluch treats of the correction of an increasing population, and that for unequal months, quarters, half and whole years. " Vide Journal Royal Soo. N.S.W., Vol. xlv., pp. 76-110. NATALITY. 173 however, of the results for different years in which the place of the mov- able feast is as different as possible, the effect of this distribution can be ascertained and corrections applied to ehminate the effect. The diffi- culty of a perfectly satisfactory solution will be apparent from Fig. 55 hereunder. ' 19. Periodicities due to Easter. — As ecclesiastically defined, Easter Day is the first Sunday after the 14th day of the paschal " Calendar Moon," a fictitious ecclesiastical moon, which is from one to three days later than the real moon. The average position of Easter for the century 1800 to 1899 is April 8.55 days, and for the ceiitury 1900 to 1999 is April 8.89 days, or say for the whole period of 200 years April 8.72 days. In Fig. 55 the Easters in each decade are shewn on a single Une for the years 1800 to 1999 inclusive. An inspection of the figure shews that the points lie approximately on a series of 10 slanting lines, four days apart, these lines progressing at the rate of one half day per decade, and further that they are inversely symmetrical. For lines a, b, c, and e and a', b', c', and e' the symmetry is perfect ; for lines d and d' however the symmetry is not absolutely perfect. It is evident that no means derived from two decades nor from periods of 19 years, nor from centuries are exactly comparable. POSITION OF EASTER FOR 200 YEARS. March April 2zaitazizisaxii < z i i s 6 ? s a lO n iz u i4 is le i7 is lazozi zzbzazs -^Yr-^r--- p^-^-t^-^ — ^r-- '\ ' V S, ^, \ 1 S \ . V . , 1 , (1 . . 1 1 1 f 1 Y S, ^^ ^ , \, i\ \ , ^ i'' ■^t ^ -s zi i: :e> it^n^ s* ^^- V ^ . '. %. I' ■ ' i^ A V- '' , ■ , ^ S,^ 1 S, !,N \> t±^ =ldS± = ^t^^;STS^=^S":: U4--1^--A-.-^^_4^^^^^^ + ^ ^- ^ '' \ \ ■ ' v ' . ! ^ \ ^ \ ' '\i ';; ■^, \, 1 :.,, y, :;, \, t 1 V ' ^ . < \ \4 \: 1 X , , f V,, T ^ i N, ^ { \, '^v N ' ^ < i I ^ t',' , 1 ' , r '■ 1 -l; ,j_^i_^^ -^.-JJ 11 :i^. Jl . H-f-T-L-K - - -rMi- -\ -^-f - f 1 f - ^ H -hhW^^-^^^mHv^^J 1800' 9 1013 zoa 30-39 40-49 50-59 eo-G3 70-73 80-89 90-93 20-29 50-33 40-49 50-59 eo-es 70-79 80-89 30-99 Fig. 55. Since the tropical year = 365.2422 days and the synodic lunar month = 29.530588 days, the Metonic cycle, *19 tropical years is 6939.6018 days, and 235 complete lunations equal 6939.6882 days, differing only .0864 day from the nineteen years. 174 APPENDIX A. The following table exhibits the peculiarities for successive decades. TABLE XLIV.— MEAN POSITION OF EASTER FOB 200 YEARS.* - 1800 1900. Easters Mean of Mean of Easters Mean of Mean of DeoEkde. Mean. in March April Decade Mean. in March April March. Easters. Easters. March. Easters. Easters. April. April. 0-9 9.46 1 29 9.56 0-9 10.66 2 30.5 13.12 10-19 8.16 3 25.67 13.57 10-19 8.36 3 27 12.86 20-29 8.86 2 28 11.50 20-29 8.06 2 29 10.25 30-39 7.86 3 29 11.43 30-39 9.66 2 27.6 12.62 40-49 9.06 2 25 12.50 40-49 8.96 2 26 12 12 60-59 8.56 3 27 13.57 50-59 7.56 2 27 10.38 60-69 7.96 3 28.67 11.86 60-69 9.66 2 27.5 12.88 70-79 9.36 2 29.5 11.88 70-79 8.66 3 28.33 13.14 80-89 9.46 2 25.5 12.62 80-89 8.66 2 28 11.26 90-99 6.76 2 27 9.25 90-99 8.76 2 30.5 10.88 Means 8.55 2.3 27.48 11.70 Means 8.89 2.2 28.09 11.94 * The complete Eaater Cycle, restoring both the day of the week and of the month, is known as the " Dionysiaii" or " Great Paschal" period. Its length is 4.7.19 = 532 years. To obtain a normal periodic fluctuation it would be preferable, were it practicable, to combine the results, each for a series of years such as would give Easter an identical distribution. In the period such a series is, however, impracticably long. Hence in the case of marriage, birth-rate, migration, etc., it is necessary to consider the actual effect on the periodic fluctuation studied. In respect of marriages the effect of Easter is to reduce the number of marriages in the Lent period (6 weeks) preceding, and to augment them in the preceding and following periods. It may be noted that for the fluctuations of annual period in the marriage frequency, the great length of the Lent period, viz., 6 weeks, has the effect of throwing the increase of frequency as far back as Febru- ary. The migration frequency is often thrown back into March. Thus, as is evident from the preceding table and the diagram, decennial means will clearly be nearly but not exactly comparable. The data for a thor- ough study of periodic fluctuation would in these cases have to be weekly groups. Xn.— NUPTIALITY. 1. General. — -The phenomena of reproductioivhave a double aspect, viz., one a sociological and the other a physiological. Thus, from the standpoint of a- theory of population, both are important. The women of reproductive age in any community furnish the potential element of reproduction ; but the resolution into fact depends also upon social facts as well as upon physiological ; for example, the relative proportion of married and single, i.e., the nuptial-ratio, even more profoundly affect the result than physiological variations of fecundity. In Chapter XVIII. of the Census Report (Conjugal Condition), the numbers of married and un- married females have been given as at 3rd April, 1911, in Australia. These will be considered mainly in regard to the child-bearing age, in dealing later with fecundity. 2. The Nuptial-Ratio. — ^The " nuptial-ratio," j, may be defined as the ratio of the married, J, to the unmarried, U, which latter may be taken generally as including the never married, the widowed, and the divorced. This ratio, J/ U may apply to either sex and to any age, or age-group, or to the total for aU ages, etc. The nuptial-ratio in any community may be regarded as a measure of the social instinct, and also a measure of the reproductive instinct, modified by social traditions as well as facilitated or hindered by economic conditions. This ratio, for the case of females, is, of course, specially important in relation to fecundity. The significance of marriage in respect of reproductive activity depends upon the relative frequency of nuptial and ex-nuptial births, as well as upon the relative proportions of the married and unmarried, that is, it depends not merely upon the nuptial-ratio, but also upon nuptial and ex-nuptial fecundity, particularly during the reproductive period of life. The values of (399) j = J/U for various countries are given in the following table for women during the reproductive period, and for women of all ages, viz., from age to the end of life. TABLE XLV. — ^Ratios of Married Women in various Age-groups to Unmarried Women in the same Groups. Reproductive Ages. Female Nuptial Ratios. Ages of Aust. Census, 1911t C'wlth Aust. 1908. England and Wales. 1901. 1911. Scotland. 1901. 1911. Ireland. 1901. 1911. Bel- gium. 1910. Germany. Women. Metro porn. other. Tc(tal. 1900. 1910. 10 to 141ncl. 15 „ 19 „ 20 „ 24 „ 25 „ 29 „ 30 „ 34 „ 35 „ 39 „ 40 „ 44 „ 45 49 „ 50 „ 54 „ 55 „ 59 „ 60 „ 64 „ .0000 .0337 .03510 1.0945 1.8201 2.2491 2.5045 2.4617 2.0628 1.5747 1.0622 .0000 .0435 .4892 1.6325 2.8810 3.5996 3.9037 3.6935 3.1420 2.3651 1.5761 .0000 .0395 .4242 1.3613 2.3318 2.8938 3.1586 3.0324 2.6634 1.9470 1.3070 .0001 .0382 .4214 1.2997 2.4698 2.9805 3.1159 3.1068 2.6025 1.8482 1.5815 • .0000 .0000 .0157 .0121 .3731 .3184 [ 3.0124 ;|;0?«? } 2.3915 IIS } 1.3217 {i:«tit .0000 .0000 .0767 .0145 0.3049 .2758 [ 1-3759 -; Jill 1 2.2854 ||3«| I 1 aioQ 1 2-0750 r ^■'"■^ 1 1.6795 I 1 nnn? ^ 1-3061 r lOOO^ 1 0.9089 .0000 .0000 .0075 .0063 .1538 .1538 \ .8397 .8137] |- 1.6777 1.7040 1 }■ 1.4343 1.5443 1 \ .8686 1.0490 j -0000 .0271 .04482 1.6385 2.8324 3.4697 3.3632 2.8921 2.2601 1.5909 1.0929 .0000 .0000 .0161 .0139 .3977 .3959 1.8172 1.9359 3.63813.8471 4.2516 4.4905 3.80124.0635 3.00863.2488 2.16352.3415 1.48641.5995 .095901.0353 „ 106 „ .5231 .5198 .5218 .5159 .5528 .4293 .45X6 .3643 .3765 .5781 .5200 .5466 • Ages 60 to 61 only. f 3rd April, 1911. 176 APPENDIX A. The results in the table shew that there are considerable divergences between populations as regards their nuptial constitution, consequently even if the individual fecundity were constant, the birth-rates would differ. The results of the Australian Census of 1911 shew also that there are striking differences between metropolitan and extra-metropolitan communities, the marriage-rate being very much higher for the latter ; and they shew also that the nuptiality is very different as regards the sexes. See Vol. I., Chap., XVIII., Conjugal Condition, § 6, of the Census Report. 3. The Crude Marriage-Rate.— The lack of homogeneity in popula- tions, illustrated in the last section, renders the crude marriage -rate, viz., the ratio of the marriages, J, to the population, P, of uncertain signific- ance. The heterogeneity arises largely from divergences of social life and tradition, in respect of the relative frequency of marriage, and the frequency according to age. Inasmuch, however, as ordinarily the constitution of any population does not materially change, the marriage- rates for any particular country and for limited periods are comparable among one another, and their variations may generally be attributed to variations in the economic conditions of the population in question. Wars have, of course, a marked effect, see the points marked with asterisks, on Table XL VI., and also Fig. 56, giving the curve of the mean of the marriage -rates of a number of important countries. We shall denote the crude marriage-rate by n ; thus — (400) n = J/P. In some countries the marriage-rate is the ratio, not of the " marriages," but of the " persons married," to the population. In such cases the rates will be double those shewn in Table XL VI. hereunder, the which gives the marriage -rates for the countries for which in Table XXXIX. the crude birth-rates were given. This also gave the values of the marriage- rate. In Table XL VI., the mean in the final column is merely the un- weighted mean, and is therefore not the rate for the aggregate of the populations. The trend, thus determined, treats each population as equally important in regard to the revelation of the secular tendency, if any, of the marriage-frequency. For the constitution of a norm a weighted-mean would of course be needed. Fig. 56 illustrates the movement in the marriage -rate, and shews that movement in its relation to that of the western world generally (excluding America). Although the general trends shewn by broken lines of curves A and B, are by no means similar, there are often very similar fluctuations about this general trend, which appear readily enough if the general trend be regarded as a basic line about which the minor fluctuations may be regarded as moving. NtJPTIALITY. 177 1860 .004S .0040 .0036 .0030 .009 .008 .007 S .006 I .0075 .0070 .0035 .0030 .0025 .00015 .00010 .00005 70 Birth, Marriage and Divorce Rates. 80 90 1900 10 00000 ■--. "■••. '••. '••. * ••»-,.. ..r- • '■ I /^ J ^ z-^ ^ f- v.^ ?\ / r^'/ / V^ r" ■ V ^ h 1"* V 1 V ?v\-> r, /« i 'vs A ^ \=/ y^ «^ 1) Years. A a. Decennial mean of Australian birth-rates. A. Crude mairiage- rates, Australia .0080 .0075 B. Curve of marriage- .0070 rates ; mean o< various c'ntries Bb. Decennial aver- ages of birth- rates ; mean of various c'ntries 1.5000 1.0000 G. Begistration of marriages 1008 .6000 —1914, Aus- tralia. D. Belative frequency of divorce. 1860 70 80 90 1900 Fig. 56. 10 20 Yean. Curve Aa shews the successive decennial means of the birth-rates of Australia, the central year being changed one year at a time. Curve A shews the marriage-rates of Aastralia by the zig-zag line ; the fine dots shew the successive decennial means ; the broken line, closely following the decen- nial means, indicates the general trend. Curve B shews the mean of the marriage-rates of a series of countries ; the fine dots shew the successive decennial means of these ; the broken line indicates the general trend of the marriage-rates. Curve Bb shews the successive decennial averages of the means of the crude birth-rates of a niimber of countries. Curve C shews the mean annual fluctuation of the registration of marriages in Australia for the period 1908-1914. Curve D shews the relative frequency of divorce per unit of population for Australia, the portion a b being prior to acts facilitating divorce ; b c being the con- dition immediately following upon the passing of the facilitating Acts ; o d, and d e being the subsequent trends of the relative divorce-frequency. As regards birth-i'ates and marriage -rates, it will be observed that here there is some indication of a correlation between the phenomena. This correlation will not, of course, be well-marked, since the aggregate 178 APPENDIX A. of " first births" is not large compared with " all births," But the trend of the AustraUan birth-rate shewn by Curve Aa is strikingly similar to Curve A shewing the marriage -rate, and Curve Bb gives some indication of its connection with Curve B. TABLE XLVI. Mairiage-iates for Various Countries — 1860-1913 — ^per 10,000 of the Population. Teai > 1 1 1 to a 1 1 .3 1 < W ^1 So 1= 1860 84 86 70 79 84 78 82 73 85 801 1861 86 82 68 82 80 73 75 81 71 80 778 1862 88 81 67 81 85 71 74 79 71 88 785 1863 84 84 72 80 87 73 75 83 73 85 796 1864 86 86 72 48 79 87 80 70 57* 84 75 83 *756 1865 83 88 74 55 79 91 91 71 89 85 76 78 800 1866 76 88 74 54 80 78 57* 67 84 84 79 65* 82* *745 1867 75 83 70 54 79 93 68 61 77 84 78 97 104 781 1868 76 81 67 50 79 89 72 67 65 73 77 73 92 137 772 1869 73 80 67 50 83 90 80 72 57 74 77 74 104 110 772 1870 71 81 72 63 61* 74» 74 70 60 74 80 70 98 98 •740 1871 69 84 72 54 73* 80« 75 73 67 65 73 80 74 95 104 •759 1872 70 87 •76 50 98 103 75 79 70 70 75 83 78 93 108 810 1873 74 88 78 48 89 102 79 77 73 73 81 86 78 94 113 822 1874 72 85 76 46 83 97 76 83 77 73 82 84 76 91 107 805 1876 73 84 74 46 82 91 84 90 79 71 85 84 73 86 109 807 1876 71 83 75 50 79 86 82 82 77 71 86 83 72 83 102 788 1877 73 79 72 47 75 80 78 79 76 69 81 81 69 76 94 753 1878 74 76 67 48 75 78 72 74 73 65 74 78 67 76 96 728 1879 72 72 64 44 76 77 76 70 68 63 74 77 68 78 104 722 1880 72 75 66 39 75 77 70 69 67 63 76 75 71 76 92 709 1881 76 76 70 43 75 77 81 69 64 62 78 73 71 80 100 730 1882 81 78 71 43 75 79 78 69 67 64 77 72 70 83 103 740 1883 84 78 71 43 75 80 81 69 66 66 77 71 68 79 105 741 1884 83 76 68 46 76 81 83 70 69 66 78 72 68 80 103 746 1886 82 73 66 43 75 82 80 70 67 67 76 70 68 77 101 731 1886 79 71 63 42 74 82 79 70 66 64 71 70 67 79 97 715 1887 76 72 64 43 73 80 80 71 63 63 70 70 71 79 90 710 1888 80 72 64 42 72 80 79 71 61 59 71 69 71 80 94 710 1888 77 75 67 45 71 82 77 71 63 60 71 70 73 76 82 707 1890 76 78 69 45 70 82 73 70 65 60 69 71 73 76 82 706 1891 75 78 70 46 75 82 75 71 66 69 68 71 74 78 86 716 1892 67 77 71 47 76 81 75 72 64 57 68 72 77 78 92 716 1893 62 74 66 47 75 81 74 72 65 67 70 73 76 80 94 711 1894 61 75 67 47 75 80 75 72 64 58 70 72 75 80 93 696 1896 62 75 68 51 74 80 73 73 65 59 71 74 78 81 86 713 1896 66 79 71 51 76 83 71 76 67 60 73 75 81 80 81 727 1897 67 80 72 51 76 84 72 79 67 61 75 74 83 81 82 736 1898 67 81 74 50 74 85 69 78 70 62 76 73 83 79 84 737 1899 70 83 75 50 77 85 74 78 71 63 75 74 83 83 91 755 1900 72 80 73 48 78 86 72 78 69 62 76 76 86 83 89 752 1901 73 80 70 51 78 83 73 76 66 61 72 77 87 82 88 745 1902 73 80 71 52 76 80 73 74 64 60 71 76 81 78 87 731 1903 67 79 72 52 76 80 72 74 60 58 71 75 79 78 82 716 1904 70 77 71 62 76 81 75 74 60 59 72 74 80 78 92 727 1905 73 77 68 63 77 81 77 75 58 59 72 73 79 78 86 724 1906 75 79 72 52 78 83 79 77 59 62 75 75 81 79 88 743 1907 79 80 72 52 80 82 78 77 60 62 77 75 80 76 100 764 1908 78 76 68 52 80 80 84 76 61 61 75 72 78 77 92 740 1909 79 74 64 62 78 78 78 76 60 60 74 71 77 76 87 722 1910 84 75 65 51 78 78 79 73 62 61 73 72 79 76 87 729 1911 88 76 67 54 78 80 75 74 63 59 72 72 80 76 93 738 1912 91 78 69 53 79 80 76 73 62 59 73 76 80 74 86 739 1913 87 78 71 51 75 63 59 72 78 704 1914 88 M-I13 758 791 698 487 771 831 761 740 661 634 744 761 753 815 949 NUPTIALITY. 179 4. Secular Fluctuation of Marriage-rates. — ^Fig. 56, embodying the results on Table XLVI., reveals the fact that the relative frequency of marriage has been increasing in Australia since 1897, although it has tended to diminish recently in the old world. It is apparent from a comparison of the two curves, A and B, that there is no very marked correlation between the two progressions. The factors influencing the relative frequency of marriage probably have a very unequal incidence in different countries. In order to obtain an accurate measure of reaction of the larger economic influences on the rates, statistics covering long periods of time will be required. The characteristics of the longer or secular fluctuations will fully appear only when much more statistical material is available than exists at present. The period of the larger oscillations in the data shewn amounts to about 22 or 23 years in Australia, and about 30 or 31 years for the aggre- gate of the populations of the western world. The period of the minor fluctuations is very variable, and is somewhat ill-defined. In Table XLVII. are shewn the values of successive decennial means for the marriage-rates, and also for the birth-rates. These are shewn by dots on Mg. 56. TABLE XLVn.- -Decennial Unweighted Means of Marriage and Birth-rates, 1860 to 1909. Marriages per 100,000 of the Population. Decade Year. 1860. 1870. 1880. 1890 1900. Year.* A W A W A W A W A W 740 781 758 746 734 712 687 734 1 729 781 766 739 715 710 697 732 2 724 786 774 732 701 711 707 737 3 722 783 777 727 692 714 719 739 4 721 778 783 725 680 716 729 739 5 812 779 719 773 788 724 i 673 721 740 735 6 799 773 719 770 793 723 1 669 726 749 733 7 782 771 726 767 792 722 668 729 764 732 8 764 773 737 760 778 720 673 730 782 733 9 754 776 747 752 756 717 678 731 802 732 Decade Year. Births per 10 0,000 of the Popiilation. 3,894 3,396 3,634 3,382 3,435 3,179 2,743 3,009 1 3,832 3,397 3,532 3,365 3,382 3,161 2,702 2,984 2 3,793 3,397 3,526 3,343 3,313 3,142 2,683 2.960 3 3,739 3,400 3,532 3,326 3,239 3,122 2,669 2,935 4 3,688 3,396 3,533 3,311 3,155 3,102 2,663 2,915 6 4,141 3,478 3,659 3,397 3,522 3,284 3,082 3,092 2,657 2,890 6 4,102 3,436 3,625 3,381 3,519 3,272 3,005 3,082 2,651 2,862 7 4,059 3,423 3,598 3,401 3,512 3,240 2,932 3,063 2,652 2,827 8 3,997 3,420 3,572 3,393 3,503 3,218 2,863 3,052 2,671 2,776 9 3,955 3,407 3,546 3,382 3,483 3,206 2,788 3,026 2,700 2,732 A denotes the values for the Commonwealth of Australia. W denotes the values derived from the unweighted means" for the series of countries shewn on Tables XXXIX and XLVI. * The moment of time to which the values apply is the beginning of the years 0, 1, 2, etc 9. 180 APPENDIX A. 5. Fluctuation of annual period in the frequency of mairiage. — Social custom in regard to marriage expresses itself in a fluctuation of annual period, but the changes in the date of Easter make the results for any one year not comparable in general to those of any other. The movement of Easter has been already considered, see Part XI., Natality. The following results are for the period 1908-1914, and are corrected for inequality in the length of the month, and for an increasing population. The table gives the crude and the adjusted data. TABLE XLVm. Number of Marriages Registered in the Different Months. Australia, 1908-14. Period. Jan. Feb. Mar. April. May. June. July. Aug. Sept. Oct. Nov. Dec. 1908-14 . . Equalised . . Constant Population 21,462 21,060 21,325 21,106 22,691 22,924 22,732 22,420 22,599 28,358 28,663 28,817 19,714 19,205 19,271 22,059 23,232 23,268 20,752 20,357 20,434 20,733 20,369 20,299 22,824 23,154 23,022 22,138 21,760 21,579 21,140 21,455 21,343 25,534 25,106 24,790 Batio to Average . . .9490 1.0201 1.0057 1.2824 .8576 1.0350 .9093 .9033 1.0245 .9603 .9498 1.1032 These results are shewn, the rectangles and the probable fluctuation, by curve C, on Pig. 56, and represent the fluctuation of the registration of marriage. It is not certain that the returns made to the Registrars of Marriages by those who celebrate them have not also seasonal peculiarities, and consequently the fluctuation shewn is compounded of the two, and in reference to the time scale is in advance of the true position. The components of the curve can be found by applying formulae (90) to (101) of § 5, part III., Determination of Constants, etc.i 6. G«neral. — Conjugal Constitution of the Population. — The " general conjugal constitution" of a population is deflned by the number of persons therein who have never been married ; who are living in the state of marriage ; or of widowhood, etc. ; or who are living in the state of " divorced" persons. The actual unadjusted numbers of males and females in age-groups on the 3rd April, 1911, as indicated by the Census are shewn on the table of §4, Chapter XVin.,Vol. I., of the Census Report. These are represented on Pig. 57, which shews both the group-values and the curves, which give sensibly the same totals. The results as furnished by the Census are somewhat vitiated by misstatements as to age ; on the whole, however, they give a fair representation of the change in the 1 See also formulae (376) to (395), § 16, Part XI., Natality. NUPTIALITY. 181 Fig. 58. Mg. 57 — The rectangles shew the total numbers as at the Australian Census of 1911 in 5-year groups, and the ciirves give approximately the equivalent areas, the heavy curves denoting the results for males, and the light those for females. Curves A and B shew thenumbers of the "Never married" ; C and D the numbers of the "married" E and F the numbers of the " widowed" ; the former being for males ; and G and H (which can- not be distinguished) shew the numbers of the divorced." Fig.58 — ^The figures, which illustrate Table XLIX., shew the asym- metry of the distribution for the " never married, U ; the " married," M ; the " widowed," W ; and the " divorced." The scale of W is ten times that of U and M, and that of D is 100 times that of U and M. 10 29 30 40 60 60 70 80 90 Fig. 57. conjugal constitution with age. The general significance can be better grasped from the results shewn in the following table : — 182 APPENDIX A. TABLE XLIX. — Proportional Conjugal Constitution of the Australian Population, 3rd April, 1911, per 10,000,000 Total Population (Adjusted Numbers.) Proportion per 10,000,000 of Total Proportion per 1,000,000 of Population. same Sex and Age-groups. Age-groupB. Never Married. Wid- Di- Total. Never Married. Wid- Di- Married. dowed. vorced Married. owed. vorced Under 14 M 1,506,806 1,506,806 1,000,000 F 1,467,395 2 1,467,397 999,998 2 14 to 20 M 710,197 5,304 34 4 718,539 992,538 7,413 48 6 F 662,798 35,358 184 18 698,358 949,080 50,630 264 2S 21 to 39 M 875,496 699,580 14,646 1,731 1,591,453 550,123 439,586 9,203 1,088 r 602,222 862,948 24,658 2,265 1,492,093 403,609 578,347 16,526 1,518 40 to 59 M 231,079 746,217 55,057 2,941 1,035,294 223,201 720,778 53,180 2,841 F 116,157 621,059 107,535 2,229 846,980 137,142 733,263 126,963 2,632 60 to 79 M 58,438 194,935 61.309 595 315,277 185,354 618,297 194,461 1,888 F 18,608 124,159 134,718 285 277.770 66,991 446,985 484,998 1,026 80&above M 4,507 10,770 12,301 45 27,623 163,161 389,893 445,317 1,629 F 1,129 3,850 20,424 7 25,410 44.431 151,515 803,778 276 All Ages M 3,386,523 1,656,806 143,347 5,315 5,191,992 652,259 319,108 27,609 1,024 F 2,868,309 1,647,374 287,519 4,804 4,808,008 596,569 342,632 59,800! 999 1 The table is based upon 4,455,005 persons, of whicli 2,313,035 were males, and 2, 141 ,970 were females ; it shews the distribution of 10,000,000 persons on that basis. The ratios in the second part of the table shew the proportional distribution in each age group. This distribution is illus- trated in the small diagrams of Fig. 58, in which U denotes the males and females belonging to the class " never married " ; M denotes the " married" males and females ; W denotes the " widowed," 'of each sex ; and D the divorced of each sex. These small diagrams represent by the rectangular areas on the left of the median line the males, and on the right thereof, the females. The scale of U and M is identical ; that of W is 10 times, and that of D, 100 times as great. The age at which the married are equal numerically to the unmarried is about 29.49 for males when the proportion of the total at that age is 0.49557, and 25.27 years for females when the proportion at that age is 0.49699. The difference is 4.22 years, and the mean proportion 0.49629 is close to either. This is due to the fact that the number of widowed and divorced is very small at the ages in question. 7. Relative conjugal numbers at each age. — ^The progress of the conjugal constitution with age is completely defined by giving for each sex, the proportion living at each age, and the proportional division of each such number according to conjugal condition. In the following table, which represents the smoothed results for the population of Aus- tralia at the Census of 3rd April, 1911, the relative distribution of males and females is shewn in columns II. and III. These numbers multiplied by 0.2313035 in the case of males, and 0.2141970 in the case of females (see the preceding section) give the absolute numbers, smoothed. The distribution of 100,000 of these at each age is given for each conjugal condition, viz., in IV. and V., the unmarried ; in VI. and VII., the widowed ; and so on. Thus at each age a complete comparison is NUPTIALITY. 183 possible of the conjugal state. Assuming the constancy of the conjugal constitution of the population the results given in columns IV. to XII. are the probability of the number of males or females which will be found characterised as never married, married, widowed or divorced, in a total of 100,000 males or females of each year of age throughout life. Columns II. and III. shew, for the population of 10,000,000, a probable number of males or females living at each year of age throughout the lite -period on the assumption of an unchanging constitution according to sex and age. As a matter of fact the Australian population, however, has not reached a " steady" state as regards the constitution of its population. TABLE L. — ^Relative Conjugal Numbers at each Age. Australia, 3rd April, 1911. Proportion per 100,000 of any Age in each Conjugal Condition. Proportion per 10,000,000 of same Sex. Age Last Never Married. Married. Widowed. Divorced. Birth- day. Fe- Fe- Fe- Fe- Fe- Males. males. Males. males. Males. males. Males. males. Males. males. I, n. m. IV. V. VI. vn. VIII. IX. X. XII. 253,554 263,314 100,000 100,000 1 236,741 247,352 100,000 100,000 2 227,662 238,776 100,000 100,000 3 221,173 232,426 100,000 100,000 4 216,158 226,689 100,000 100,000 6 211,030 221,422 100,000 100,000 6 205,544 216,147 100,000 100,000 7 199,236 210,605 100,000 100,000 8 193,611 205,675 100,000 100,000 9 189,232 201,852 100,000 100,000 10 186,115 199,135 100,000 100,000 11 184,835 197,118 100,000 100,000 12 184,813 106,086 100,000 100,000 13 185,860 196,417 100,000 99,998 "2 14 188,588 198,425 99,993 99,958 "7 42 15 192,846 202,463 99,982 99,783 18 215 2 ... 16 196,742 206,660 99,945 99,207 55 789 4 17 200,105 209,910 99,842 97,445 156 2,547 "2 8 18 202,552 212,020 99,507 94,363 491 5,621 2 15 1 19 203,339 212,575 98,803 90,089 1,191 9,878 4 27 "2 6 20 202,932 211,646 96,862 84,638 3,111 15,290 23 59 4 13 21 201,908 209,144 93,784 77,311 ?>}^S 22,547 54 121 6 ?! 22 200,256 205,554 88,724 70,131 11,172 29,634 93 204 11 31 23 197,226 200,885 82,292 61,261 17,537 38,393 153 302 18 44 24 192,582 195,288 76,334 54,327 23,403 45,192 236 418 27 63 25 186,746 189,284 70,235 48,343 29,402 51,018 326 555 37 84 26 180,702 183,033 64,175 44,043 35,349 55,149 426 701 50 107 27 174,619 177,047 58,423 40,529 40,975 58,487 535 857 67 127 28 168,700 171,165 53,325 37,220 45,937 61,599 654 1,036 84 145 29 163,041 165,339 49,526 34,316 49,586 64,280 782 1,242 106 162 30 157,732 159,615 45,773 31,703 53,169 66,641 918 1,477 140 179 31 152,938 154,153 42,050 29,253 56,732 68,827 1,060 1,726 158 194 32 148,316 149,297 38,623 27,299 59,998 70,490 1,210 2,001 169 210 33 144,192 144,913 35,755 25,742 62,703 71,732 1,365 2,303 177 223 34 140,534 141,029 33,532 24,593 64,757 72,623 1,523 2,648 188 236 35 137,417 137,532 32,018 23,352 66,100 73,362 1,687 3,038 195 248 36 134,594 134,166 30,608 22,188 67,326 74,092 M®9 3,462 206 258 37 132,387 131,164 29,495 21,236 68,252 74,546 2,041 ^■??i 212 267 38 130,491 128,344 28,536 20,209 69,012 75,059 ^■5^i 4,456 219 276 39 128,870 125,725 27,727 19,392 69,612 75,368 2,436 4,957 225 283 40 127,499 123,036 27,035 18,423 70,089 75,785 2,645 ^?S^ 231 288 41 126,085 120,006 26,296 17,585 70,601 76,023 2,867 f'^°2 236 292 42 124,753 116,766 25,596 16,697 71,060 76,272 ?'S®? ^•m 246 294 43 123,297 113,820 24,815 16,111 71,584 76,108 3,345 7,486 256 295 44 121,810 111,075 24,004 15,481 72,121 75,913 3,606 8,311 269 295 184 APPENDIX A. Relative Conjugal Numbers at each Age. Australia, 3rd April, 1911. — Continued. Proportion per \ 10,000,000 of same Sex. j Proportion per 100,000 of any Age in each Conjugal Condition. Age Last Birth- Never , Marri<.,1 Married. | Manned. Widowed. Divorced. day. Males. Fe- males. Males. Fe- males. Males, Fe- males. Fe- Males. males. Males. Fe- males. I. 45 46 47 48 49 120,227 118,632 116,738 114,201 110,863 in. 108,316 105,580 102,742 99,455 95,281 IV. 23,206 22,540 22,062 21,696 21,673 V. 14,831 14,203 13,632 13,064 12,592 VL 72,620 72,959 73,124 73,169 72.924 vn. 75,875 76,693 76,231 74,888 74,455 VllL 3,887 4,195 4,503 4,831 6,191 IX. 9,000 9,911 10,846 11,760 12,668 X. 288 308 311 314 312 xn. 294 293 291 288 285 60 51 52 53 54 106,112 99,890 93,341 86,985 80, '(68 90,165 84,049 78,320 73,129 68,376 21,317 20,762 20,065 19,693 19,471 12,216 11,746 11,362 11,034 10,676 72,747 72,765 72,868 72,763 72,387 73,969 73,183 72,254 71,200 70,127 6,627 6,172 6,767 7,336 7,837 13,645 14,798 16,118 17,511 18,981 309 311 310 308 305 280 274 286 256 238 55 5« 57 58 59 74,798 68,840 62,865 57,474 53.237 63,726 59,053 54,585 50,631 47.302 19,317 19,106 18,994 18,896 18.694 10,316 9,966 9,653 9,316 9,031 72,094 71,915 71,433 70,806 70.294 69,009 67,531 65,772 63,932 62.064 8,300 8,704 9,291 10,012 10,728 20,484 22,342 24,427 26,611 28.770 289 276 282 286 284 192 181 148 141 136 80 61 62 63 64 49,602 46,433 43,873 44,724 39,870 44,622 42,424 40,678 39,258 37,904 18,757 18,920 18,940 18,879 18,781 8,764 8,434 8,166 7,741 7,556 69,406 68,371 67,491 66,677 66.863 60,052 58,108 56,959 53,939 51,669 11,558 12,439 13,313 14,206 15.138 31,063 33,332 36,763 38,202 40.761 279 270 266 238 228 131 126 122 118 114 65 66 67 68 69 38,149 36,378 34,574 32,771 30,912 36,588 35,276 33,782 32,078 30,164 18,382 17,709 17,457 17,620 17,790 7,256 6,870 6,690 6,409 6.068 65,254 64,647 63,624 62,309 61.063 49,263 47,147 44,855 42,859 40.966 16,160 17,542 18,832 20,000 20.979 43,371 45,868 48,363 60,634 52,872 204 202 187 171 188 110 106 102 98 94 70 71 72 73 74 29,096 27,341 25,460 23,562 21,669 28,194 26,359 24,608 22,890 21,121 18,276 18,817 19,358 19,817 20,152 6,861 5,668 5,422 5,185 6,131 59,406 57,938 66,563 55,468 53,990 38,856 36,880 34,906 33,008 31,012 22,140 23,087 23,943 24,687 26,738 66,194 57,367 59,692 61,730 63,784 178 168 186 128 120 89 85 81 77 73 , 75 78 77 78 79 19,861 18,123 16,459 14,639 12,568 19,281 17,400 15,453 13,441 11,545 20,026 19,323 18,126 16,539 16,480 4,868 4,789 4,649 4,558 4,466 52,024 50,334 48,595 47,549 46,784 29,141 27,327 25,442 23,500 21,890 27,841 30,248 33,202 36,824 37,668 65,932 67,849 69,848 71,885 73,591 109 96 78 88 68 69 65 80 81 82 83 84 10,817 9,023 7,263 5,824 4,630 9,762 8,189 6,830 5,640 4,650 14,788 14,960 15,833 16,704 17,647 4,371 4,291 4,236 4,160 4,080 43,965 42,166 41,072 39,347 37,348 20,214 18,566 16,882 15,183 13,495 41,167 42,789 42,976 43,876 44,912 76,366 77,109 78,842 80,667 82,425 80 95 119 74 93 85 86 87 88 89 3,662 2,832 2,166 1,634 1,258 3,842 3,142 2,516 1,975 1,636 18,654 18,473 17,964 17,196 15,808 4,012 3,960 3,891 3,830 3,770 35,419 33,588 31,936 31,746 31,615 12,063 10,288 8,809 7,764 7,264 45,809 47,786 49,900 50.794 52,234 118 153 200 284 343 90 91 92 93 94 968 722 532 363 233 1,186 878 616 416 280 14,732 16,669 16,260 17,867 20,370 3,720 3,681 3,640 3,601 3,563 30,367 28,743 26,016 23,810 22,222 7,204 7,621 8,581 9,882 11,637 54,464 56,688 57,724 58,333 57,408 447 95 S6 97 98 99 169 125 99 78 61 210 159 117 84 61 20,513 20,690 21,739 22,222 21,429 3,634 3,601 8,470 3,446 3,427 20,513 24,138 26,087 27,778 28,571 13,044 14,746 16,630 18,276 20,050 68,974 56,172 62,174 50,000 60,000 ■• ■• 100 43 42 20,000 3,411 30,000 21,811 60,000 Total 10,000,000 10,000,000 NUPTIALITY. 186 8. The carves of the conjugal ratios. — ^The smoothed results for each sex, representing the ratios which the " never married," the " married," the " widowed" and the " divorced" bear to each other (given in Table L ) are graphed in Fig. 59, and are represented respectively by the curves U^ and U/ , M^ and M/ , W^ and W/ , and D^ and D/ . Conjugal Ratios, Australia, 1911. 1.0 0.9 0.8 0.7 0.6 0.5 0.4 3 [0.3 0.2 0.1 0.0 0.003 0.002 0.001 0.000 V \ \ \ r ^, \ / \ \\ _ / \ \ / ^ h\ •> / r*' \ I / /■ \ \ / \ V /' «In ' \ \ \ j^ / HT ;^ r \, \ \ *\ / V, / \ 1 ,1 I \ ^ hi r m \ y \ y \ y 1 K \ / / ' V 1 1 ^ \ Ur 1 / i V, 1 1 \ \ s / \ \ / 1 Uf. \ S ^ / _/ \ ^ ■"y 1 N y ~" ^ "^ ^ ^ t- -/- / ^ / ' / 1 / ^ >^ -^ V. y J) Ktf — ' ir** -»— / 4 ^-, \ t N 1 / / ■£ ^ \ Pn T / / y \ V / / ^ / k' =*=^ s, {■■I 1 / %= !?^ Uf ^ ■^ Widows. 10 20 30 40 50 60 Years of Age. 70 80 90 100 Widowers. Married Males. Married Females. TTmnaTried Males. Unmarried Females. 0. Zero tor "unmarried," " married," and " widowed " curves. 0. Zero for " divorced ' curves. Fig. 59. These curves shew merely the proportion ol the unmarried, married, widowed, and divorced at each age, the number at each age being unity tor males, and also unity tor females. They thus shew the distribution for each age according to age, but not between one age and another. The results for males are shewn by small crosses in the figures ; those for females by small dots. The curves for the " never married" are somewhat of the type e"*^> where p is large. The critical features of these curves can be best shewn in a tabular form, and are as in the following table : — 186 APPENDIX A. TABLE LI.— Critical ieatuies in the freauencies of conjugal conditions. Aostialian Census, 3rd April, 1911. • Proportion Character of Critical Feature. Exact of Total Age. Age -group. Maximum proportion married, males 49.5 0.73100 ,, „ „ females 43.0 0.76160 Minimum proportion married, males 95.0 0.00217 ,, „ „ females 90.0 0.00063 Equal frequency married and unmarried males . . 29.49 0.49556 „ „ „ „ females 25.27 0.49699 Maximum proportion widowed, m.ales 90.5 0.89100 „ „ „ females . . 93.7 0.58400 Equal frequency unmarried and widowed, males 67.6 0.18600 „ „ „ females 49.5 0.12600 Maximum proportion of divorced, males 52.0 0.003115 females 44.0 0.00295 In general these results are for the smoothed curves represented in Fig. 59, as may be seen by a reference thereto. The ratios among one another of the various ratios given in Table L follow no simple law, and an examination of them was found to lead to no important results. 9. The norms of the conjugal ratios. — ^It is eminently desirable that a series of curves based upon the aggregate of all populations to be com- pared, should be tabulated and constructed on some such model as that indicated here for the population of Australia. Such a norm, representing the relative frequency of the never-married, the married, the widowed and divorced for the entire aggregate would constitute the best possible bases for comparisons of the position of individual nations and peoples. The international issue of graph paper on which such curves were already drawn, preferably in faint colour, would enable the statistician to see instantly the position of his own country in regard to the larger average in respect of the particular character compared. 10. Divorce and its secular increase. — ^The frequency of divorce is of sociological interest. The effect of the Divorce Act (55 Vict., No. 37) of New South Wales, and of Victoria (53 Vict., No. 1056), which came into force on 6th August, 1892, and 13th May, 1890, respectively, have had a conspicuous influence in increasing its frequency. In the former State the frequency was more than quadrupled for about three years ; in the latter it was tripled, as the result of the operation of these Acts. Table NUPTIALITY. 187 LII. shews the frequency of divorce per 10,000,000, for the several States of the Australian Commonwealth for which they were available up to 1886, and for the whole Commonwealth from 1887 onward. The populations up to 1886, used to compute the divorce-rate, correspond to the number of States for which the divorce results were available, and the number of divorces include the judicial separations. The results for the successive years are as follows : — TABLE LII.— Relative Frec[ueueies, per 10,000,000 population, of Divorces and Judicial Separations. Australia, 1874 to 1913. Year Rates* in Decades. Proportionf of Judicial Separations. of Decade. 1870. 1880. 1890. 1900. 1910. Period. Pro- portion. 237 377 981 1,066 1874-1879 .020 1 179 594 1,052 1,154 1880-1884 .052 2 113 684 1,024 1,464 1885-1889 .062 3 274 1,293 909 1,347 1890-1894 .043 4 140 176 1,261 1,014 1895-1899 .038 5 220 269 1,194 862 1900-1904 .042 6 350 229 1,039 860 1905-1909 .043 7 210 205 1,113 854 1910-1913 .023 8 140 297 1,024 997 1874-1913 .0381 9 120 361 1,043 1,163 •■ * Number per 10,000,000 of population, judicial separations and divorces together. f Ratio of judicial separations to The total number of divorces and judicial separations were 10,194 and 404 respectively, the total thus being 10,598. The relative fre- quencies, tabulated above, are shewn by the bottom curve in Pig. 56, viz., curve D. The proportions which judicial separations bear to the totals appear also in the table. Apparently divorce was increasing at first approximately at the rate 0.00000165 per unit of the population per annum, so that the number of divorces (V.) from 1781 to 1890 would be represented roughly by (401) V = 0.00000165 P (t - 1870), in which formula t denotes the year for which the number is required, and P the population at the middle of the year. The values according to this formula are denoted by the dotted line a b on Fig. 56. The relative frequency then rises in 3 years from, say 0.0000330 to the value 0.0001293 ; that is at the rate 0.0000321 per 188 APPENDIX A. person per annum — the line b c on the figure. The relief afforded through the change in the divorce acts, having apparently been secured in the short time mentioned, the relative frequency of divorce fell fairly regularly until about 1907, viz., at the rate of 0.00000333 per person per annum. Hence for this period the relative frequency is about (402). V = - 0.00000333 P {t - 1893). This is the line c d on the graph. The relative frequency of divorce then rapidly increases to about 0.0000100 per person per annum. This is denoted by line d e on the graph. 11. The abnonnality of the divorce curve. — Owing to the change in the divorce law being, as shewn, instantly followed by a large increase in the number of cases, the curve of frequency cannot be regarded as normal for the larger ages. For the purpose of estimating the rate of increase, previous to the legal change, the results for a few years before the change can be used. Similarly the results after the change can be carried backward to some common year in the changing period. This gives the following results : — TABLE Lin.— Shewing Influence of Divorce Acts on Number of Divorces. Australia. Average Increase per Annum (Number). Number as per Year at Change (1892). Factor State. Before Legal Change After Legal Change Before Change. After Change. of Increase N.S. Wales .. Victoria Commonwealth (1884-1891) 5.6 (1881-1889) 1.9 (1881-1888) 7.4 (1893-1895) 0.0 (1891-1893) 0.0 (1893-1907) 5.6 69.7* 32.5 116.6 306.7* 91.7 436.6 4.4 2.8 3.7 * Divorces and judicial separations together. In view of the fact that, as shewn, the change consequent upon the operation of the Divorce Acts is very marked in the frequency of divorce between 1890 and 1893, say 21 to 18 years before the (Census cf 1911, and that there is a remarkable decrease in the proportion of " divorced" for NUPTIALITY. 189 ages about 55, see the points marked a and a' in Pig. 59 (which would correspond to ages of about 35 in the year 1891), it seems more than probable that the left-hand branch of the divorce curves belongs to the later, and the right-hand branch belongs to the earUer divorce regime. To obtain the true tendency to divorce according to age of the parties, these irregular frequencies would, of course, have to be eliminated. Hence it is desirable to include in the statistics of divorce the age of petitioners and respondents. See later. 12. Desirable form of divorce statistics. — ^From what has preceded, it is evident that for divorce statistics to be of high value from the stand- point of sociology, they should fulfil the following requirements, viz., they should include the numbers both of petitions for judicial separation and for divorce, and should shew for each : — (1) The date and the ground of the petition ; (2) The action resulting therefrom (granting, refusal, or other action), together with the date of such action ; (3) The date of birth both of petitioner and respondent ; (4) and the date of their marriage. Statistics so kept would furnish results shewing frequency- according-to-age and age-differences and according to duration-of- marriage. The sociological value of such statistics is self-evident, for it would throw light upon the influence of age per se, of difference of age, and of duration of marriage, and thus would expose the conditions which are of danger from the standpoint of social stability. 13. Frequency of marriages according to pairs of ages. — ^The fre- quency of marriage according to pairs of ages can be well determined only for a considerable number of instances. For example, if assigned to groups, according to age last birthday, there are, between the ages 12 and 95 for brides, and 15 and 99 for bridegrooms, no less than 7140 groups. As for the last eight years the average number of marriages per annum was only 37,740, this gives a little over 5 per group on the average, a number insufficient to indicate the characteristics of the frequency. For this reason eight years marriages were taken, viz., 301,918, or the marriages of 603,836 persons, who were married during the years 1907 to 1914 inclusive. Of these marriages the ages of 57 brides were not stated, though the ages of the bridegrooms were given ; the ages of 19 bride- grooms were not furnished, though those of the brides were given ; and in 54 cases neither the age of bride or bridegroom was given. That is, there were 130 cases (or about 1 in 2322, or the 0.00043058th part) defective. These are disregarded. For single year groups the numbers of marriages are shewn in Table LIV. 190 APPENDIX A. TABLE LIV.— NUMBER OF MARRIAGES* ARRANGED ACCORDING TO * The figures denote the number Ages of Bride- grooms AGES OF BKIDES. 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 ■• 1 2 8 12 44 28 113 38 208 37 214 65 362 68 268 40 233 32 195 381 15' 25' 158 13 59 233 426 602 1158 870 742 603 460 405 311 247 182 133 94 78 54 50 26 37 27 22 18 13 6 1 3 16 17 8 51 309 778 1033 2076 1703 1494 1170 971 875 607 567 339 287 186 154 114 121 81 76 40 47 28 18 14 22 6 10 14 5 6 10 4 6 30 195 740 1261 2527 2342 2152 1458 1197 985 775 606 439 311 259 193 157 126 93 90 74 57 18 1 3 16 112 395 1075 2301 2384 2550 1885 2267 1824 1627 1238 1036 766 609 402 344 261 215 174 129 101 75 58 71 26 34 21 17 12 15 14 14 4 10 78 327 891 3764 3960 4114 3672 3170 2630 2171 1858 1364 1022 727 604 462 334 304 241 188 181 125 90 70 52 39 41 35 38 30 16 11 6 7 8 2 8 58 194 489 1845 3008 3468 3302 5 30 90 320 1185 1869 3128 3249 2981 3006 2555 2683 2050 2255 1783 1873 1353 1431 986 737 710 513 366 323 272 171 178 113 110 53 70 42 36 23 29 21 15 20 20 21 1088 769 668 553 422 313 273 259 205 128 112 87 64 72 49 37 32 21 37 15 15 5 7 9 12 47 160 738 1269 2032 2989 2821 2540 2141 1957 1484 1145 786 727 512 551 364 300 236 259 164 138 87 88 59 48 45 37 24 32 21 3; 6 24| 124 76 465^ 289 744 1247 1776 42 190; 526, 3071 835! 513 1311 769, 2664 16161 1089 1354 2382 2200 2073 1844 1736 1668 1457 1317 1172 1183 820 841 767 790 549 563 455 432 477 375 323 368 242 263 250 216 177 204 157 139 85 122 84 84 B4 68 57 62 72 45 43 52 26 29 30 27 27 26 1723 1532 1242 1121 982 752 621 536 449 337 327 215 190 161 109 112 27 2 11 28 118 226 309 545 832 948 1098 1320 944 754 673 541 449 387 324 254 354 197 172 116 114 91 65 72' 50 52 49 28 22 9 12 12 12 6 4 29 73 134 231 179 395 228 17! 58 34 28 450 584 730 879 1037 830 703 636 517 407 354 333 250 240 243 180 108 106 81 70 79 56 36 44 29 31 20 12 19 13 29 360 424 490 590 709 742 599 555 461 384 377 362 255 243 208 208 119 131 110 74 30 51 82 132 298 360 409 423 555 472 355 264 270 239 167 193 160 121 142 107 84 31 24 59 84 94 162 190 253 280 314 365 339 497 342 344 288 258 227 216 168 152 117 135 105 96 61 63 46 47 39 34 13 25 17 19 7 15 17 63 61 78 124 154 203 260 229 230 280 317 275 246 228 194 216 169 174 70 75 100 64 132 115 110 101 80 72 62 123 166 168 204 182 225 193 298 235 223 178 178 128 49| 60 49 48 33 34 86 114 132 153 157 173 180 197 233 190 153 187 154 173 106 132! 96 72 93 88 65 44 48 52 28 32 22 19 5 6! 4 22! 11 35 261 48 30' 55 76 82 66 88 66 35 100 100 126 130 157 164 188 153 172 147 118 101 124 86 76 95 87 64 55 58 40 38 24 21 24 36 76 76 94 90 95 103 126 165 160 142 124 73 86 88 37 21 25 26 35 53 60 61 46 67 64 95 94 115 111 189 139 126 76 111 85 80 73 66 69 81 58 54 36 40 26 20 15 15 19 24 31 38 46 48 23 42 50 66 87 75 103 120 151 124 86 97 81 84 78 54 47 60 55 57 34 34 27 20 1 473 489 2412 6907 13246 18140 2023132673 27950 26402 23903,20707 I I 17781 14440 12372 10010 8405 5848 5558 4341 3854 3521 2924 2438 200fi 1688 NUFTIALITY. 191 THE AGES OF THE CONTRACTEIG PARTIES, AUSTRALIA. 1907-1914. of abuples : not of persons. AGES OF BBIDES. Ages 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 69 60 61 62 63 64 65 66 i 2 1 1 1 2 2 67,68 69 70 71 72 73 i 74 75 76 :: 77 78 78 79 79 80 80 81 81 82 82 83 84 95 1 1 2; 1 of Bride- grooms "l 1 4 6 8 15 8 8 20 21 26 25 28 27 30 31 43 61 58 61 87 90 56 75 59 66 60 39 32 24 44 33 19 1 3 1 3 1 5 7 9 8 7 6 10 12 19 27 30 24 27 38 37 33 52 42 73 56 78 58 50 39 42 30 27 23 37 24 i i 2 1 3 2 3 4 6 15 10 6 14 10 20 21 15 18 27 35 30 28 42 46 53 61 40 49 41 36 30 31 40 25 30 "3 1 1 3 7 2 2 3 6 11 10 11 7 16 23 12 20 20 40 33 31 34 29 56 58 55 49 42 41 50 29 27 18 21 '3 1 2 1 3 8 4 4 3 2 6 9 12 7 6 8 20 21 15 23 15 19 32 33 51 42 38 37 35 18 34 18 26 i 2 2 3 1 3 6 3 7 3 6 8 7 10 13 14 19 18 22 26 12 27 29 34 3e 40 27 29 36 19 21 i 2 1 2 4 1 5 2 5 5 3 1 9 6 7 12 24 8 13 13 20 14 13 13 31 27 43 23 36 29 29 23 26 i 1 1 2 2 2 3 4 2 1 2 6 3 7 4 4 9 10 6 10 12 15 20 12 10 31 35 40 32 22 15 16 i '2 1 2 2 3 1 3 1 i 3 7 2 3 2 11 10 6 6 8 13 19 15 10 24 30 32 24 25 17 19 i 1 'i 2 1 5 5 2 2 2 3 6 7 4 2 4 10 7 15 18 12 16 20 13 17 'i i i 2 1 3 1 1 'i 4 4 5 3 9 3 3 5 13 12 8 18 15 16 24 19 19 3 2 3 1 2 6 5 5 5 4 7 10 14 8 15 16 1 1 1 i i 1 2 '3 2 3 1 3 '3 7 5 8 5 4 9 9 9 12 16 1 '3 "2 i 5 2 6 2 4 4 7 5 2 3 9 7 1 i i 1 i 2 2 5 3 "7 7 8 9 7 10 i i i i 1 1 2 2 '2 2 2 3 2 7 4 6 3 1 6 4 1 '2 1 i 1 2 1 i '2 3 4 2 4 6 3 2 8 1 i 'i 1 1 2 2 1 2 1 1 '2 3 3 7 2 3 4 1 4 1 1 i 2 1 3 1 3 1 i i i 1 2 '4 1 i 3 2 i "i 1 1 i 1 1 '2 1 'i 1 '2 3 2 1 i 1 i 2 '2 1 4 'i 2 1 'i .1 i i i 'i 1 'i i 83 1 1 2 2 2 2 4 2 2 4 2 2 4 1 2 2 1 2 2 2 1 2 2 1 1 1 i 1 1 4 51 239 1205 3353 6438 17374 19977 23655 24918 24650 23494 21012 19384 16113 13392 10349 9745 7712 6796 6066 5345 4411 4530 3737 3252 2336 2437 2058 1745 1847 1675 1350 1363 1146 1113 748 795 647 623 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 4? 48 49 50 51 52 53 54 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 'i 1 1 "% "\ 2 1 2 2 74 75 76 77 84 95 Total. Ages. 23 15 13 9 5 6 5 7 6 7 4 3 2 6 ■3 2 3 1 "1 1 1 22 16 13 15 9 6 4 7 8 4 3 6 5 5 1 "1 "1 "1 17 22 16 12 11 11 7 6 11 8 7 3 2 4 3 1 2 2 1 'i 1 4 20 15 19 22 6 14 13 4 9 6 12 8 6 2 1 2 4 2 2 3 'i 20 29 19 17 12 9 9 4 3 5 8 7 2 2 5 3 1 'i 26 IS IS. IS IS 11 12 5 IC 6 7 C 4 « 2 i 2 ] '4 : '] 21 29 21 22 9 16 9 10 9 6 6 2 6 7 4 6 2 3 1 3 2 1 .. i i 19 15 g 17 11 11 8 6 7 6 8 6 8 6 4 '4 1 3 'i 'i i 'i 15 19 12 14 15 17 11 12 10 7 3 6 3 6 6 5 2 S 16 12 9 6 7 10 11 7 7 3 7 4 9 4 1 3 5 '5 1 1 1 2 i 14 13 9 20 11 7 9 15 7 5 XI 4 5 5 4 4 6 2 2 1 11 8 8 15 9 11 4 5 6 9 7 5 3 5 2 6 1 1 1 2 '3 1 'i 10 6 12 10 8 12 3 9 6 6 - 7 4 4 4 4 4 2 'e 1 1 i i 1 12 10 11 6 6 9 3 9 10 2 6 9 5 3 2 2 1 2 '2 2 1 1 1 'i 12 12 8 6 9 14 6 3 6 4 5 4 4 7 5 6 2 1 2 3 1 1 i 1 1 3 12 14 10 9 7 6 8 5 2 4 6 11 3 '2 '2 3 4 1 1 1 7 8 10 11 14 8 3 6 1 8 5 3 4 3 7 3 1 1 2 1 2 2 1 1 3 1 5 1 6 6 10 10 5 3 1 6 7 6 1 2 2 1 2 2 1 2 4 1 4 8 3 5 8 6 4 6 5 6 9 5 6 5 5 3 1 3 %. 1 2 1 3 1 2 4 1 6 5 2 2 1 '2 3 i 1 1 '2 i i i 4 1 2 2 7 2 3 8 7 4 5 6 6 1 3 3 1 4 2 2 i 1 1 4 '2 4 3 2 1 6 5 7 4 7 5 2 6 4 '3 1 1 1 1 1 i 2 i 2 2 2 '5 3 4 4 6 2 6 6 5 1 '4 1 1 'i 5 2 2 2 1 3 5 i 6 8 7 2 3 4 3 2 4 7 3 4 i 1 1 '2 1 3 2 i 3 6 3 3 2 4 1 2 3 1 3 i 1 'i "i '3 2 1 2 3 1 2 3 4 4 5 6 4 1 1 '2 2 2 i '2 3 1 2 2 3 4 2 5 4 4 1 2 3 5 2 i 'i 1 i 1 '2 2 '2 2 1 3 4 4 2 2 2 3 4 3 i i i 2 i 2 2 3 1 2 1 1 4 5 3 1 1 'i 2 2 'i 1 2 2 1 1 I 2 1 1 'i 1 i i 1 2 1 '2 i i "i i 1 1 3 3 3 i 2 1 i i 'i 1 'i 2 i 1 '2 'i 'i 'i 2 3 1 i 1 1 'i 1 1 1 "% 1 i i i 'i 1 i 'i i 2 2 64 545 489 400 414 286 347 236 227 229 203 219 170 163 154 122 134 70 70 77 64 53 33 33 22 23 28 10 9 8 7 5 3 3 1 1 " 1 1 73 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 96 99 N.S. 1287 1065 948 946 712 663 649 493 481 308 361 238 229 179 189 165 159 98 136 58 93 " 68 95 58 54 51 43 36 21 21 14 11 10 4 6 6 4 2 1 3 1 1111 301918 Total 192 APPENDIX A, This table exhibits the various irregularities in the data. The num- bers are not quite trustworthy about the ages 21, for reasons which will appear later, as it is certain that in some cases misstatements are made by persons marrying under that age. This table is suitable for the analysis of the frequency at the lower groups of ages only. For the analysis of the frequency at the more advanced age groups, a second table of five-year groups has been prepared. (Table hereinafter). The frequencies exhibited by this large group of marriages can, without sensible error, be referred to the beginning of the year 1911 (i.e., to 1911.0), as the moment which they can be regarded as true, and from which any secular change may be reckoned, or they may be regarded as contemporaneous with the Census of 3rd April, 1911. 14. Numbers corresponding to given differences of age. — ^The mode of tabulation in Table LW., though satisfactory in respect of shewing the grouping according to age-groups for single years, is by no means perfectly satisfactory for the purpose of very accurately determining the frequency of conjugal-groups according to various differences of age. It is obvious that when all bridegrooms, whose age was say x last birthday, and brides whose age was say y last birthday (x and y being integers), are grouped, the group contains brides who are one-half year older than the difference X — y, as well as brides one-half year younger than this difference. This can be readily seen from the nature of the table itself. To obtain some rough idea of the defect of such a mode of grouping, we may first divide the numbers (having regard to second differences) into four parts, so as to get the probable numbers attributable to each half of the age-period analysed. These quarter (or half-year) groups, however, will evidently not agree with what would have been given by an original compilation into half-year groups, for the reason indicated above ; this will appear more clearly hereinafter. To properly determine the law of nuptial frequency according to specified differences of age the only perfectly satisfactory compilation would be one in which, for small age-groups of bridegrooms (say) the tabulation was according to a series of increasing age-differences (of the age of the bride), positive and negative, and (for complete analysis) a similar tabulation for small ranges of the age of the brides, with a series of increasing differences, positive and negative, of the age of the bride- groom. These two tabulations wovld not give identical results, but if the age-groups were small, they would be approximately identical. The data of the table are nevertheless of value, and give a result which is of high precision in regard to the characteristic features of the surface representing the relative frequency of marriages for given pairs of ages. The results given in Table LIV. are for 301,918 marriages occurring in Australia during eight years, and are drawn from populations (mean annual), which aggregated to nearly 36 miUions. The marriage rates were thus as shewn in Table LV., p. 193. NUPTIALITY. 193 TABLE LV.— Marriage Rates, Australia, Total Period, 1907-1914. Males Kates, Males 18,614,557 0.0162195 Females | 17,206,457 I 0.0175468 Persons 35,821,014 0.0168570 These rates may consequently be regarded as representing the pro- bability of a marriage occurring in a population of males, females, or persons, constituted as the average for the eight years, 1907 to 1914, both Inclusive, in Australia. The probability of a marriage occurring among the never-married, the widowed, and the divorced, cannot be so well ascertained. By excluding the unspecified, the probabiUty of marriage for any pair of ages can be ascertained roughly by dividing the numbers in Table LIV. by 301,864 ; the quotient is the chance of the marriage occurring in the group of the pair of ages in question, provided that the proportions to the whole population of the males and females in eacJi group is unchanged. Denoting this probability by pxy , the marriage- rate by r^, and the population-by P, the number of marriages, N.j.,/, to be expected of bridegrooms whose age last birthday was x, with brides whose age last birthday was y, is : — (403). . N^y = Pr^'p^y ; Na;y =P'r',nP=oy ; Nxy =P"r'inPxy ; P, P'and P" denoting the population of persons, males, and females, respectively ; and r„, r'^ and r^ similarly denoting the marriage rates based upon persons, males, or females, respectively. The numbers of the table would roughly give the chance according to " alleged age," not according to " actual age" unless the alleged is also the actual age. We shall proceed to examine this question. 15. Errors in the ages at marriage. — ^Before analysing the data giving the protogamic surface, it is desirable to determine the error of statement at ages earUer than 21. Here it may be mentioned^ that the curves of apparent frequency of birth at different ages from say 17 to 22 shew that the numbers are doubtless erroneous. The same fact is sug- gested by the pecuhar irregularities in the numbers graphed in Pig. 60, which shews the numbers of brides and bridegrooms at all ages ; see curve A in the figure shewing the result for brides and curve B shewing that for bridegrooms. The explanation is unquestionably that the group " 21 years last birthday" contains a number of persons whose real age was 18, 19 or 20, or possibly even younger than 18. From an investigation of birth-frequency during the seven years, 1908 to 1914, both inclusive, it was found that the numbers given at ages 18 to 21 needed to be multiplied by the factors 1.05701, 1.07918, 1.17022, and 0.82704 respectively. (This applies to females only. There is doubtless also an error for males). Correcting these factors so as to obtain the same totals, the figures in line (4) below are obtained ; these are the probable correcting factors to be appHed to the 1 The matter is dealt with fully hereinafter. 194 APPENDIX A. numbers furnished directly. That is let M' be the true number of marriages for brides of any given age, and let M be the alleged number : then m being the factor of correction, we shall have : — (404) M' = m M, hence, if the error occur solely through misstatements by persons of 18, 19, 20, and 21 years of age we should have, for each age of bridegroom, to form corrections of the type : — (405) . . (M'ls+M'ig+M'zo+M'zi) ={misMis+mi,QMig+r)HoM2o+m2iMzi) This would be the appropriate scheme of correction^ if corrections for only one sex were needed. The result would then be as follows in Table LVI. hereunder : — TABLE LVI.— Coirectiou of Nambers of Brides of Alleged Ages, 18 to 31. Australia, 1908-1914. (1) Age of Bride (2) Number of Brides . . (3) Ratio to Total for Ages 18-21 (4) Factor of Correction (5) Product of (3) & (4) 18. 13,246 0.1572 1.0572 19. 18,140 0.2152 1.0794 0.16619 0.23229 20. 20,231 0.2400 1.1704 0.28090 21. 32,673 0.3876 0.8272 18-21. Total, 84,290 Total, 1.0000 Mean, 1.03355 0.32062 Total, 1.00000 These figures imply that there are 5.72 per cent, more brides of 18, 7.94 per cent, more brides of 19, 17.04 per cent, brides of 20, and 17.28 per cent, less of brides of 21 than admit that they are the ages in question. Misstatement of Ages. Ages 15 Curves C and D. 30 35 i . X0,000 S ^v ■a / \ 7 ^ X I^ S ' ^~v i" t ^K^^ a -.rnn / /' \ \ '^ t\ ^ 7,500 h I t \ ^ _, ...^ L.-.\.. . °. T^ V S i- I ^ \ ' § Jl Z ^ -t t L J£ ^ ° i t \ ^ t - ^ \\ ° snnn % ^ V ^•^ ^ y-^ ^ 3 1 5 vi^ ■ ° - Zjl N. L I 5 ^Ip^^^lt « ^ i \ ^^ - ^^ SI -; « t t -K ^^ 1 ^ X ^ J5 r ^ ^ ^ 7 i s % J S i5.S0U If , \ ^J fc \ ''-I S 1 7 ^^ c^^ ^ S t ■^1-3 2"^^ ""— — W 7 oh.^.^_ : : ::::: ::: -t-^5:=^ = 3s==- = = Ages 15 50 go 25 SO 35 40 45 Fig. 60. ^^^^^ A *°* ^- Fig. 60a. The curves A and B denote the number out of a total of 100,000 marriages of brides and bridegrooms respectively ; married at given ages. The dots and circles represent the original data, the curves themselves being the smoothed result. Curves C. and D. — The areas of the rectangles shew the nimibers of brides and bridegrooms, respectively, married at the given alleged ages. The true nmubers are the areasjo the curves, which furnish the smoothed results. 1 If in any example the result needed a small correction to balance, it should be made proportional to these last m Jf -quantities. NUPTIALITY. 195 An attempt has also been made to ascertain, by smoothing, the probable misstatement on the part of bridegrooms as well as on that of brides. For the sake of comparison the factors for converting the crude data into the smoothed results are given for both bridegrooms and brides, and for males and females from the smoothing of the results of the 1911 Census. The actual smoothing and its effect is shewn on Fig. 60a, see curves C and D, the former being the curve for brides, the latter that for bride- grooms. The areas to the curves give the smoothed results, the areas of the rectangles themselves shew the crude data. In this way the results (1) and (2) are obtained. TABLE LVO . — Coirection-Factors for Males and Females of Alleged Ages, 18 to 21. Australia, 1911. Factor of 1 1 Correction for — How Obtained. 18. i 19. 20. : 21. ,1) Males .. Smoothing of Cm-ve shewing 1 Number of Bridegrooms 1.211 1.137 1.262 0.831 (2) Females Smoothing of Curve shewing Number of Brides 0.962 1.054 1.228 0.844 (3) Females Smoothing of Fecundity 1 Curves 1.0572 1.0794 1,1704} 0.8272 (4) Females Mean of (2) and (3) 1.010 1.067 1.199 ! 0.836 (5) Males .. Smoothing of Census of 1 Population, 1911 0.9843; 1.0273 0.9955| 1.0283 (6) Females Smoothing of Census of 1 Population 0.9924; 1.0217 0.9902 1.0504 The indications from the smoothing of the number of brides, with those from the smoothing of the fecundity curves (see later) are in sub- stantial agreement, so far as the ages of 19, 20, and 21 are concerned ; see Unes (2) and (3) in the table above. It will be observed, however, that they are not in agreement with the Census deduction. An agree- ment was not, however, to be expected- in the latter case, for the mis- statements occur in regard to the age at marriage, an occasion on which there is not infrequently a motive for the misstatement. * 16. Adjustment numbers for ages 18 to 21 inclusive. — ^The actual adjustment of a table of numbers according to pairs of ages, however, involves the deduction of a number of brides and bridegrooms; which shall be equal for each group. It is evident that, inasmuch as the factors for the two are disparate, different results are obtained if we first correct by the factors for one sex and then by those of the other, or correct in- dependently and take means, etc. For this reason the following method, though not ideally satisfactory, was adopted. Denoting the correction-factor for bridegrooms (males) of age x by rrix, and that for brides (females) of age yhjfy, the composite factor (/j.) 1 Chiefly, but not whoUy, owing to the attempt, by persoijs vmder 21 years of age, to avoid the legal requirements, 196 APPENDIX A. for the group of brides and bridegrooms of the respective ages, may be taken as : — (406). ■ fJ-xy = \/{'>nxfy\ that is, it is regarded as the geometric mean of the two. If we decide to make the totals of the groups 18 to 21 unchanged, we shall have to apply a small correction to these factors. Let gxy denote a group of marriages for the ages in question. If the sum of the products jugr be equal to the sum of the original groups, no correction will be required. If it be not equal, then the correction can be distributed in the ratio of the groups themselves. That is, | denoting the correction, the new values (g') of the groups will become : — (407). ■g' =g + ^ =g \^+{G - s i^g) / q\ G denoting the sum of the groups, that is to say, O = Eg. This method of correcting leaves the entire aggregate unaffected, though it adjusts its component groups. The results are shewn in the table hereunder. The ^ correction necessary was very small, amounting to only 18 in 17,862. See Table LVIII. TABLE LVm. — Coirection of Numbeis of Mairiages for Ages 18, 19, 30, 21. Australia, 1907 to 1914. Crude Results. Factors of Corebotion. 1 CORRECTED RESULTS. 18 19 20 21 Totals. 18 19 20 21 18 19 20 21 Totals 18 19 309 195 112 78 778 740 ; 395 327 694 2,240 Males Females Means Males Females Means 1.211 1.010 1.1059 1.137 1.010 1.0716 1.211 1.067 1.1367 1.137 1.067 1.1015 1.211 1.199 1.2049 1.137 1.199 1.1676 1.211 0.836 1.0062 1.137 0.836 0.9750 343 223 837 819 136 79 463 320 781 2,439 20 21 [ ! 1,0331,26111,075 891 4,260 2,076 2,527| 2,301 3,764' 10,668 Males Females Means Males Females Means 1.262 1.010 1.1290 0.831 1.010 0.9161 1.262 1.067 1.1604 0.831 1.067 0.9416 1.262 1.199 1.8301 0.831 1.199 0.9982 1.262 0.836 1.0271 0.831 0.836 0.8335 1,171 1,469 1,910 2,389 1,328 919 2,3063,150 4,887 9,755 rtia' 4,196 4,723 3,883 5,060 17,862 i ( j 4,2614,900 4,233 4,468 17,862 The effect at the dividing ages of this regrouping is to change the 2,022 S™"P^ 6:897^ 912 3,302 23,130 1,395 56,029 8,031 become into 2,222 1 998 6,939 17,703 3,502 23,172 1,481 ; hence the five-year groups The totals for brides require 55,701 that the original figures in Table LIV. should be corrected by + 65, -|- 177, + 350, and — 592, and the totals for bridegrooms corrected by + 87, + 199, + 627 and — 913. NUPTIALITY. 197 TABLE LIX. — Shewing the Number per 100,000 Bridegrooms, and per 100,000 Brides Married at Griven Ages. Australia, 1907-1914.t Crude Results. Adjusted Results . Age. Crude Results. Adjusted Results. Age. Bride- grooms. Brides. Bride- grooms. Brides. Bride- grooms. Brides. Bride- grooms. Brides. (1.) 12 13 14 (ii.) (Ui.) 1 1 24 (iv.) • 0.0 0.1 0.2 (V.) 0.5 1.5 24 (i.) 55 56 57 58 59 55-59 . . (u.) 181 162 133 137 95 (iU.) 59 63 55 53 32 (iv.) 184 167 151 136 122 (V.) 72 66 60 12-14 .. 26 0.3 26 48 15 16 17 18 19 1 17 79 428* 1,176* 162 799 2,288 4,409* 6,067* 0.8 M.9 79 428 1,176 162 799 2,288 4,409 6,600 70S 262 760 30 1 60 61 62 63 64 60-64 65 66 67 68 69 65-69 .. 70 71 72 73 74 70-74 .. 75 76 77 78 79 75-79 .. 80 81 82 83 84 80-84 85 86 87 88 89 85-89 90 91 92 93 94 90-94 . Unspeci- iied . 115 78 75 76 67 45 19 31 27 23 109 97 86 76 68 42 36 31 27 13-19 . . 1,701 13,725 1,700.7 14,253 2,340* 5,452* 6,615 7,834 8,253 6,817* 10,626* 9,257 8,745 7,917 2,542 4,997 6,868 7,834 8,253 8,020 8,920 9,200 8,745 7,917 21 411 178 436 160 23 24 73 56 54 51 41 32 19 18 17 14 60.0 53.0 47.0 42.0 37.5 21.8 19.5 17 1 20-24 .. 30,940 43,362 30,494 42,802 15.0 8,165 7,782 6,960 6,420 5,337 6,858 5,873 4,783 4,098 3,315 8,190 7,782 7,120 6,290 5,337 6,819 5,843 4,897 4,078 3,297 26 275 100 239.5 86.0 28 29 44 23 23 26 21 19 7 7 5 4 34.0 31.0 28.0 25.0 21.5 10.5 8.5 67 25-29 .. 34,664 24,927 34,719 24,934 5.1 3.6 4,436 3,428 3,228 2,554 2,251 2,784 1,937 1,841 1,438 1,277 4,383 3,603 3,003 2,603 2,278 2,670 2,155 1,760 1,470 1,260 31 137 35 139.5 34.6 32 33 34 17 11 11 7 7 3 1 2 2 1 17.2 13.6 10.9 8.7 7.0 2.8 2.1 1.6 30-34 .. 15,897 9,277 15,870 9,315 1.2 0.9 2,009 1,770 1,461 1,501 1,238 1,166 968 808 785 664 1,995 1,748 1,533 1,346 1,183 1,143 1,003 873 753 643 35 36 53 9 57.4 8.6 37 38 39 9 3 3 3 2 1 1 5.6 4.5 3.6 2.9 2.3 0.7 0.6 0.5 35-39 .. 7,979 4,391 7,805 4,415 0.4 0.3 1,077 774 807 682 580 560 373 426 353 314 1,040 912 800 713 649 547 465 397 343 303 40 41 20 2 18.9 2.5 43 44 2 1 i 1.8 1.4 1.0 0.6 0.3 0.2 0.1 40-44 .. 3,920 2,226 4,114 2,055 0.0 0.0 45 612 522 447 452 380 313 236 220 215 163 589 527 468 413 363 271 241 213 187 163 0.0 46 47 4 5.15 0.3 48 49 0.25 0.15 0.1 0.05 0.0 45-49 2,413 1,147 2,360 1,075 SO 369 248 263 214 . 206 159 102 116 79 76 319 282 251 225 203 138 118 102 90 80 51 52 0.55 53 54 24 37 Nil 50-54 .. 1,300 532 1,280 528 NU • These have been partially corrected for misstatement o£ age. for description of Table. t See Section 17, hereinafter. 198 APPENDIX A. 17. Probability of marriage of bride or bridegroom of a given age, to a bridegroom or bride of any (mispecified) age. — ^The correction of the data, as indicated in the preceding section, admits of the construction of a table shewing in say 100,000 marriages the number occurring for bride- grooms of any given ages, and for brides of any given ages, the age of the other partner to the union being unspecified. In columns (ii.) and (iii.) of Table LIX., hereinbefore, the data are given the corrections referred to having been applied : columns (iv.) and (v.) are the smoothed results. The original data are shewn by dots on Fig. 60, the smoothed results by the curve, the ordinates to which represent throughout the probability of a marriage occun-ing within one half-year either side of any given age : that is, they are the values of the integrals : — K \ xdx and V+i K' \ydy; Jy-i see section 19 hereinafter. 18. Tabulation in 5-year groups. — So small a number as 300,000 does not give sufficient data for the determination of the averages for single years, at the higher ages. Before 25 is reached over one-fourth of the marriages have been consummated, and before 30, over two-thirds (exactly 0.277921, and 0.691744 respectively). This leaves for groups of over 30 years of age only about 93,069 among 6500 groups or an average of about 14 per group. It is thus necessary to form 5-year groups. These are shewn in Table LX. hereunder. The corrections, referred to in last section, change these numbers as follows : — Oriqenal Data. Adjusted Data. 3,302 1,395 23,130 1 56,029 1 4,852 92,354 , 3,502 23,172 1,481 55,701 5,138 92.068 41,193 13,1151 Totals. 41,135 130,909 Totals. The numbers given in the table itself are the uncorrected data. It will be seen that they are still small for the higher ages. To determine the critical features of the surface representing the frequency of marriage both Tables LIV. and LX. are required. Were these two tables smoothed they would give the probabilities of a marriage occurring within the year groups of specified ages or specified quinquennia. None of the groups is perfectly regular, but the greater regularity of the larger groups exists only for a limited range of years. The matter will be dealt with more fully hereinafter, viz., in § 23. NUPTIALITY, 199 TABLE LX. — Number of Maiiiages Airanged According to Age at Marriage in Five Year Groups. Australia, 1907-14. Brides' Aees. Bride- Total,* Katlo 1 1 III [111 Bride grooms' Ages. 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 10 to 84. grooms to to to to to to to to to to to to to to to to 14. i9.t 24.t 29. 34. 39. 44. 49. 54. 59. 64. 69. 74. 79.184. 1 Total. 15-19t .. 9 1 3,302 1,395 124 17 3 2 4,852 1,608 20-24t .. 44 23,130 lS6,02g 11,302 1,437 325 6C 22 4 1 . .1 92,354 30,602 25-29 . . 18 10,637 1 50,597 34.896 6,739 1,368 282 7(i 20 1 1 i; •• 104,639 34,673 30-34 . . 1 2,795 15,513 117,366 9,130 2,476 525 146 26 4 1 47,983 15,900 35-39 . . Si 917 5,134 1 7,298 5,672 3,621 i.oas 1 313 65 15 2 2 24,080 7,979 40-44 . . 1 237 1,576 2,564 1 2,811 2,47 a 1,502 510 112 26 a 1 11,821 3,917 45-49 . . 2 115 598 1,077 1,313 1 1,653 1,27S 859 263 74 36 8 7,277 2,411 50-54 . . 41 183 384 538 1 76S 1 7b4 675 4011 117 37 '20 2 i 3,926 1,301 55-59 . . 11 73 129 197 3ia 36(1 1 445 2fiii 218 65 26 4 2 2,132 706 60-64 . . 6 28 71 79 152 162 1 207 1 208 144 lOR 60 16 2 1,242 412 65-69 . . 1 15 24 43 66 80 1 133 122 113 105 »7 19 7 826 274 70-74 . . 6 16 17 SO an 47 1 65 41 50 59 28 6 415 138 75-79 . . 1 2 3 8 6 n 13 17 i»1 14 21 25 fl 164 54 80-84 .. 2 2 2 2 K 1 10 7 4 1 9 4 1 8 4 62 21 85-90 . . 1 1 1 4 1 1 1 1 1 12 4 Total* 78 41,193 131,151 75,257 28,003 13,257 6,114 3,462 1,605 790 435 300 103 30 301,785 100,000 Batio ol Biides 26 13,650 43,459 24,937 9,279 4,393 2,026 1,147 532 262 144 99 34 10 2 100,000 .3313617J to Total The heavy faced type the mark of exclamation ( I ) denotes the maximum on the * Brides over 85 and bridegrooms over 95, and unspecified cases are omitted, denotes the maximum on the vertical lines ; " horizontal lines. t The values corrected for misstatement of ages, 18, 19, 20, and 21 give the following results :— For 3,302 and 1,395, 3,502 and 1,481 ; and for 23,130 and 56,029, 23,172 and 55,701. In the totals 41,193 and 131,151 become 41,435 and 130,909 : and 4,852, and 92,354 become 5,138 and 92,068. The ratios 13,650 and 43,459 become 13,730 and 43,378 ; and 1,608 and 30,602 become 1,703 and 30,508. J Factor of reduction to 100,000. 19. Frequency of marriage according to age representable by a system o£ curved lines. — ^Frequency according to pairs of ages (bride and bridegroom) can best be represented by a surface, the vertical height of which, above a reference plane, is the frequency for any pair of ages denoted by x, y co-ordinates. The numbers marrying in any given period, whose ages range between x-^ ^k and x -\- \k (for bridegrooms), and between y — \k and y -\- \k (for brides), as ordinarily furnished by the data, are denoted by Z, the height of the parallelepiped. This frequency may, of course, be expressed as for the exact age, or it may be for the age-groups. When k is not infinitesimally small, the difference between the two is sensible and important. We shall assume for the present that the frequency varies only with age (not with time). The exact (instantaneous) age-frequency denotes the frequency which would exist if the persons were all of the exact age (x) in question, instead of being of various ages between x ~ ^ k and x -\- ^ k. The age-group frequency denotes the frequency with the ages distributed between the limits referred to. For most practical purposes the latter is the more important. Suppose the exact frequency, z, for the population P, to be : — (408). ^=F{x. y) then we shall have for any group-value : — (409) Z = PJj F (x.y) dx dy 200 APPENDIX A. The group-values usually furnished are for single-year groups, hence the limits of the integral are « ± i> 2/ rb ^- It may sometimes be more convenient to use a series of functions of the form : — (410) y=^Fy{x); or F^(y) in which case the fixed value of 7 or of X will be the middle of the range 2/ ± i, or .T J: |. Then we shall have : — (411) Z =PJ Fr(x)dx; or = Pj Fx(y)dy These last expressions, with fixed values either of Y or of X, are thus appropriate for representing the vertical or horizontal columns of figures in Tables LIV. and LX. by means of equations. For the vertical columns the abscissa is x, the age of the bridegrooms ; for the horizontal columns the abscissa is y, the age of the brides ; and the constants of the equations relate only to a particular range of y in the first case, and of X in the second, as many equations being required as there are ranges taken. We consider the matter more fully in a later section. This scheme of representation is practically more convenient than a more generalised system, it shews for each age of bridegroom (or of bride) the frequency of marriage with a bride (or a bridegroom) of a given age. (See part v., § 10, formulae 211 to 216.) 20. The error of adopting a middle value of a range.— ^In dealing with group-ranges, in the manner referred to in the preceding section, the results are not strictly attributable to the middle age of the range, nor is the error of such an attribution by any means always wholly negligible. The function represents the value of a range of values of the argument, i.e., for example, all bridegrooms whose age last birthday was x, x being an integer, or the group of bridegrooms whose age last birthday was say, between 20 and 24, etc. Suppose, for example, that the progression of a series of numbers, representing numbers at successive ages is approxi- mately : — (412) y = a -\- inx ; so that xy = ax -\- mx^ ; then the true value of the product of the numbers into the ages is given by the integral : — X + 1 (413) \xy dx = a (x -\- i) + m, {x^ -\- x + ^) Consequently where we require the weighted mean-age, it is necessary to compare this value with that arising from the supposition that all may be regarded as of age x -\- \. If we make this last assuniption,then we should have for the product of the numbers into the age, supposed common to all (414) a (a; -f i) -f m (a;2 + X -f 1). The former expression is algebraically greater than this latter one by the difference of m/3 and m/4, that is m/12, which is sensibly equivalent to a NUPTIALITY. 201 shift (e) of the central position of the amount m/Yly. Thus, instead of the central value of the range of ages we should take the " weighted mean" xa, which is given by : — (41f5) x^ = x^\ +e =.f +1 + m 122/ In applying this we may take m as indicated by the mean of the differences of the groups adjoining on either side. Thus if the groups for the ages 20 (and less than 21), 21, and 22 were respectively 76,132, and 224, then, instead of taking 21.5 as the mean age-value, i.e., the middle age of the range 21 (which include everyone whose age last birthday was 21), we could take m as the mean of 132-76 and 224-132, that is, m —\ (56 -f 92) ; or, as is obvious, \ (224-76), i.e., 74. Consec[uently by the rule above, viz. (415),' we have «„= 21.5+74 / (12 x 132) = 21 .54671. A curve which would give the group-results indicated is 60 + 20^ + 18p, the origin of abscissae being x = 20, so that ^ = 1 for a; = 21, and so on. The integral of the curve is 60 ^ + 10 f^ -f- 6^ If we put ^ = a; - 20 we obtain the curve y = 6860 — 700 a; + \%x^ with the origin at a; = 0, hence the integral between the limits a; = 21 and a; = 22 is 3430 a;2 — 233| x^ -f 4^ a;*, which gives the result 2844J as the sum of the xy products. Dividing this by 132, the number in the group, the average age is found to be 21.54671 as before. Let three successive groups for equal ranges of the variable be denoted by .4, M, and B ; and let x^ be the middle point on the range of abscissae of the middle group, M ; "then the mean value required (i.e., in the case under review, the average age of the persons in the group) is : — (416) Xfl = x^ + ITT A; ~ 24 M. in which h is the range of the variable comm.on to the three groups. If the curve of instantaneous values be of the second degree, this last formula is rigorously accurate. By means of it, the average values can, as a rule, be written in by inspection, and it can be ascertained where the correction e = jV^^ ^ (jg — A) / M is. sufficiently large to be taken into account. 21. General theory of protogamic and gamic surfaces. — ^The ages of husbands being adopted as abscissae, and those of wives as ordinates, the infinitesimal number dM in an infinitesimal group of married couples, consisting of husbands, whose ages lie between x and x -{- dx, and their wives, whose ages lie between y and y + dy, will be : — (417) dM = Z dx dy = kF (x, y) dx dy. Thus Z = k F {x, y) is representable by a co-ordinate vertical to the xy plane. Since Z denotes an actual number of persons in a double age-group, between say the earliest age of marriage and the end of life, viz., {xi to Xz) and (yi to ys), it is necessary, if we desire to institute comparisons between different populations, that Z should be expressed as a rate, z say : that is, z = either Z/P ; or Z/M ; that is to say, the 202 APPENDIX A. vertical height wUl represent the relative frequency of married couples whose ages are, in the order of husband and wife, x and y, in either the whole popuation P, or the married portion of it M. Thus we shaU have (418) P,OT M =kff F (x,y)dx dy. If the value of the double integral be taken for the limits denoting the range of ages of the married, say about 11 to 105, we shall have either M/P, or unity, as the result ; according as we denote by the frequency in reference to the total population or to the total married. Thus the marital or gamic condition of a community is completely specified by the gamic surface F {x, y, z), the unique mode of which is the summit of the conoidal solid represented by (418) above. Its first principal meridian is the Une joining the modes of the curves x = a, con- stant, or 2/ = a constant, passing therefore through the unique mode. The curves, z = any constant less than its maximum value, are necessarily closed curves, and may be called isogamic contours. The orthogonal trajectory passing through the unique mode is the second principal meridian of the surface. The values of x, y, and z for the unique mode of the surface may be called the gamic mode of the " population," or of the " married population," according as the constant k, in (418) above, gives M/P, or unity for the value of the double integral between the widest age limits. The gamic characteristics of a population are more briefly, and of course less completely, defined by the two principal meridians which we may call its gamic meridians, and the position (and magnitude) of the gamic mode. Reducing these tn their simplest numerical expression we have, for the briefest possible statement of the gamic characteristics of any community the values of x„, y^, and z„ ; and of the skewness of the profiles of the first and second principal meridians. The sign of the skewness may be determined by always making the right hand branch of the curve that for increasing age for the first principal meridian, and increasing age of the husband for the second principal meridian. A surface representing the frequency of marriage at particular pairs of ages we shall call a protogamic surface, and one representing the number of persons of particular pairs of ages living together in the state of marriage we shall call simply a gamic surface. Curves of equal frequency on these two surfaces, we shaU call isoprotogamic and isogamic contours, respectively, or more briefly, isoprotogams and isogams, and curves cutting such contours orthogonally will be called protogamic and gamic meridians.^ Let s denote a distance measured along a slope, so that ds is an element thereof. Then when — (419) dz/ds = sin ^ 1 The word " isogamy " has aheady been appropriated in a different sense in biology, viz., to denote the union of two equal and similar " gametes" in repro- duction. This, however, will obviously lead to no confusion. The isogamy of a people might be regarded as of two kinds, initial or nuptial isogamy (isoprotogamy), and characteristic or marital isogamy (or simply isogamy). NTJPTIALITY. 203 = a maximum or a minimum, the element ds is an element of a meridian ; such meridians are the principal meridians above referred to ; i.e., the principal meridians are the lines of greatest and least slope. 22. Orthogonal Trajectories. — ^The general theory of orthogonal trajectories may be stated as foUows : — ^Let the co-ordinates of a system of curves (isogams or equal marriage frequency in the case considered) be denoted by x and y, and those of the trajectory, cutting the system orthogonally, by ^ and tj ; then, although for any point of intersection of the two X = ^ and y = yj, dy/dx is not the same as dr)/d^, Since the tangents to the two curves are at right angles, we have the geometric relation dy/dx = - d^/dyj or <**> '+S-'S='' For any system of curves we have then (421)..../ {X, y, a) = Op where a is a constant ; then, employing S/Sa; and 8/Sy to denote partial differentiation with respect to x and y, we have also U22) _^ , _V ^ _ '*^^^ 8x + Sy -dx ^"' an equation by means of which a may be eliminated, so that a relation may be obtained between x, y and dy/dx. Let this relation be denoted by:- (423) i,(x,y, %)=0 This last expression is the differential equation of the system of curves we require. For orthogonal trajectories we have i = .i\ rj = y and dy/dx = — 1/ {dr]/d^), hence the differential equation of the system of orthogonal trajectories is : — (424) ^(^,^, - J-) =0 In the system we are considering, the curves (isogams) do not con- form to any simple specification, hence the present imperfect data do not indicate any unique system of curves of a simple character. If they did, it would be preferable to deduce the principal meridians of the surface by means of the general equation thereto. An examination of the surface, however, shews that there is no practical advantage in attempting to express it analytically. 23. Critical characters on the protogamic surface. — ^A review of the figures in Tables LIV. and LX. reveals the fact that, in general, if we regard the numbers of marriages corresponding to any given age for brides (the columns), there is a clearly-defined maximum value ; but that if we regard those corresponding to any given ages for bridegrooms (the rows), there are in many cases two or even three maximum values. 204 APPENDIX A. In this latter case, too, the maximum is often less clearly defined. The positions of these maximum points and the numbers (frequency) cor- responding thereto, are important, as they disclose the characteristics of the surface. There are two ways of estimating the position and fre- quency at the maximum (or any other point). One is to ascertain the position and frequency for the maximum of the frequency integral taken over the range x — ^ to z -\- ^, or over the range y — ^ to 2/ + J ; the other is to determine those elements for the maximum instantaneous frequency ; that is to ascertain the point when the frequency for an indefinitely small range is a maximum (expressed, however, per unit of age-difference, say one year). The latter only will be ascertained. By applying formulae (292) to (294), see Part VII., § 11, p. 92, the position and value of these maximum points (viz., those on the surface for ages of brides constant that of bridegrooms being variable, or for ages of husbands constant and that of brides variable), may be obtained. In this way the results given in the two following tables are deduced, viz., Tables LXI., and LXII., and in connection therewith it is to be remarked (a) that for results of high precision, the quinquermial grouping can be used only for the small groups at higher ages ; and (6) that the grouping in fives, not only tends to obliterate characteristics readily discernible in year -groupings, but gives a frequency of the order of about 25 times the magnitude of those groupings. Thus for very young ages and for the older age-pairs, the large grouping gives the best indication.'^ 1 The values are obtained in the following way : — The position of the maximvim of one group (say of bridegrooms) corresponding to the range of another group (say of brides) is found from the succession of the group-totals of the first, for any one range of the second, and is attributed to a mean age of the second, computed from the progression of numbers in the series of group totals of the second. By way of illustration consider the group of 59, for the age-group 65-69 of brides, and 70 to 74 of bridegrooms ; viz., the following figures : — Instances in Group. Adjoining Group Totals. The surrounding group-totals are as shewn. rf the arrac nf U^Aa^ V^ tol,.>« o„ „* <-V.„ 4^ JI„ „I 3.4.1.1.2 2.1.1.2.2 4.2.0.3.2 7.3.2.5.3 3.0.0.2.4 60 105.97.19 41.50.59.28.6 14.21.25 4 the years, i.e., as 65J, 66J, etc., and of the bridegrooms as 70J, 714, etc., the actual weighted-mean ages (deduced from the iudivl- vidual numbers) are as shewn hereunder. Slightly different results are obtained if the ages are deduced from the vertical and horizontal columns, viz., 97, 59, 21 ; 50, 59, 28 ; and from the diagonal totals, viz., 105, 59, 25 ; and 19, 59, 14. These different results are for bride and bridegroom respectively : — Middle Values of Groups. Actual Weighted Group-means. Computed from Vertical Groups, etc. Computed from Diagonal Groups. Years 67.5 72.5 67.35 72.64 67.48 72.45 67.46 72.40 This series of results shews that the error of assuming that the entire gi-oup is repre- sentable by the middle ages is not ordinarily considerable. XUPTIALITY. 205 TABLE LXI. — Critical Positions on the Piotogamic Surface for Teai-gioups. Marriages in Australia, 1907-1914. (Greatest frequency for various combina- tions of Age at Marriage). Mean Age of Age of Bride- ! ' Proportion of Brides in groom for i Difference of Maximum AU Brides Maximum Maximum Age. Frequency. of same Group. Frequency. Age -Group. 13.5 21.2 1 7.7 1 0.250 14.7 22.4 7.7 17 0.233 15.7 21.6 5.9 69 0.141 16.6 21.6 5.0 372 .1504 17.6 21.7 4.1 1203 .1742 18.5 21.7 3.2 2164 1986t .1621 .U92t 19.5 21.9 1 2.4 2600 i .1434 21.8 2.5 2500t .1364 20.5 23.4 2.9 2573 .1272 .1256 21.5 23.3 1.8 4156 .1266 .1295 22.5 23.7 1.2 3511 .1256 23.5 24.3 1.2 3269 .1239 24.5 24.6 0.1 3040 .1272 25.5 25.7 0.2 2744 .1325 26.5 26.6 : 0.1 2247 .1276 27.5 27.7 0.2 1753 .1214 28.5 28.5 0.0 1328 .1073 29.5 29.5 0.0 1045 .1046 30.5 30.7 0.2 768 .0913 31.5 31.6 0.1 565 .0966 32.5 32.5 0.0 510 .0916 33.5 33.5 0.0 320 .0737 34.5 34.6 0.1 305 .0791 35.6 35.5 0.0 236 .0670 36.5 36.5 0.0 190 .0650 37.5 1 37.9 0.4 167 .0685 38.5 38.6 0.1 194 .0801 39.5 39.5 0.0 153 .0765 40.5 40.3 —0.2 121 .0717 41.5 41.2 —0.3 74 .0657 42.5 43.1 + 0.6 94 .0730 43.5 45.2 I +1.7 80 .07512 44.5 45.3 + 0.8 63 .0664 In determining any critical point, however, the ages deduced as shewn above are not what is required. "^Vhat is definitely sought is the position and value of the maximum frequency, referred to a mean-age of bridegrooms (a;), (or of brides {y) ) ; that is the value of y (or of x, respectively) at which the ma xi mum value occurs. The data from which these are deduced are the series of parallelepipeds the heights of which may be taken as the group-totals. Thus, the horizontal series of group- numbers 50, 59 and 28, treated as ordinate-values bounded by a curve, gives 66.13 years as the' age of brides, corresponding to a maximum frequency of 62.18. If the 41 group be included, the maximum wUl be changed to age 67.50 years, and the frequency to 60.29. The mean age of the bridegrooms should be ascertained on the vertical line 67.50 for brides, but without incurring sensible error it may be taken as 72.50 — 5 (97 — 21)+ (24 x 59) =-- 72.23, see this part, section 20, formulae (412) to (415) ; the factor 5, however, appearing because the unit is 5-years. Bespecting (with sufficient approximation) X»i = J and i or in years five times these amounts, or 2i and 12^. This gives 39 Jf and 60^^ as the frequencies at t^e maximum and miniTmiTTir 206 APPENDIX A. Mean Age of Bridegrooms in Maximum Group. Age of Bride for Maximum Frequency. DiSerence of Age. Maximum Frequency. Proportion of all Bridegrooms of same Age-Group. 15.5 16.6 1.0 1 0.250 16.5 17.5 1.0 14 .274 17.5 17.7 0.2 60 .250 18.5 18.4 —0.1 318 .264 18.5t 0.0 352t .272t 19.5 18.9 —0.6 820 897t .2416 .2654t 20.5 19.5 — 1.0 1279 1496t .1986 .2117t 21.5 19.7 —1.8 2558 .1472 19.7t -1.8t 2410t .1465t 21.5 21.4t — o.it 32501 .1968t 22.5 21.6 —0.9 4110 .2057 21.8t —0.7 3424 .1714 23.5 21.7 — 1.8 4250 .1839 22.lt -1.4t 3508t 24.5 21.8 —2.7 3766 .1511 22.8t — 1.7t 3333t 25.5 21.5§ — 4.0§ 3276 § .1329§ 23. 3t — 2.2t 3026t .1225t 21.9 —3.6 3342 26.5 21.6§ — 4.9§ 2710§ .1158 23.4t — 3.0t 2694t .H47t 21.9 —4.6 2774 27.5 22.6 —4.9 2271 .1080 20.6 —6.9 2230 .1061 21.8 —5.7 2293 28.5 24.3 —4.2 1977 .1199 21.9 —6.6 1973 29.5 24.7 —4.8 1492 .0932 22.0 —7.5 1458 30.5 26.0 —4.5 1195 .0892 21.9 —8.6 1080 31.5 26.2 —6.3 849 .0820 32.5 26.1 —6.4 809 .0830 22.7 —9.8 719 33.5 26.3 —7.2 565 .0733 23.4 — 10.1 557 .0722 34.5 24.5 — 10.0 560 .0823 34.2? — 0.3? 309? .0455? 35.5 25.5 — 10.0 486 .0800 36.5 26.5 — 10.0 371 .0694 37.5 27.4 —10.1 332 .0753 37.2 — 0.3 171 .0388 38.5 28.5 — 10.0 364 .0804 38.3 — 0.2 195 .0430 39.5 29.5 — 10.0 246 .0658 39.2 — 0.3 157 .0420 40.5 30.3 — 10.2 217 .0667 41.5 31.5 — 10.0 144 .0617 41.2 — 0.3 76 .0325 42.5 32.5 — 10.0 137 .0562 42.3 — 0.2 90 .0361 43.5 32 9 — 10.6 108 .0625 42.9 — 0.6 94 .0457 44.5 32.7 — 11.8 99 .0567 43.6 — 1.0 57 .0326 J The restilts include corrections for misstatements of age. § These maxima disappear altogether when corrections are applied for misstatements of age. NUPTIALITY. 207 TABLE LXn. — Critical Positions on the Fiotogamic Surface, for 5- Year Groups. Marriages in Australia, 1907-1914. Maximum age-gronp of brides Mean age of brides in maximum group . . 10-14 •14.3 ? 15-19 •18.3 20-24 •21.6 22.5 25-29 30-34 •26.6 ' »32.2 27.3 32.1 35-39 ? 37.2 40-44 ? 42.2 Age of bridegroom for maximum freauency Difference of age Maximum frequency . . Proporiiion of all brides of same age- group 22.9 8.6 46.7 0.600 23.1 t23.8 4.8 2.2 1 1.3 24685 I t72500 124727 ' J72170 0.599 I 0.553 t0.601 I t0.551 27.5 0.9 0.3 36722 0.488 32.1 -0.1 -0.0 9397.6 0.336 37.5 0.3 3716.5 0.280 43.4 1.2 1541.1 0.251 Maximum age- group of bridegrooms Mean age bridegrooms in maximum group 15-19 •18.4 1 20-24 i 25-29 22.3 [ 27.3 1 1 30-34 35-39 32.2 , 37.1 40-44 42.2 45-49 47.3 Age of bride for maxi- mum frequency Difference of age Maximum frequency . . Proportion of all bride- grooms of same age- group 17.8 0.6 3800 t4000 0.783 0.779 1 22.1 23.6 0.2 3.7 59496 51865 {59166 0.644 i 0.496 0.643 1 25.9 27.9 6.3 I 9.2 18290 7465.5 0.381 0.310 32.1 10.1 2837.0 0.240 37.4 9.9 1683.8 0.231 Maximum age-group of brides Mean age of brides in maximum group . . 45-49 ? 47.3 50-54 ? 52.2 55-59 ? 57.3 60-64 1 62.3 65-69 •67.3 67.3 70-74 •71.5 72.1 75-79 •76.50 76.8 80-84 •81.8 82.2 Age of bridegroom for maximum frequency Difference of age Maximum frequency . . Proportion of all brides of some age-group 48.3 1.05 887.5 0.254 52.7 ■ 0.5 417.1 0.260 57.9 0.6 225.8 0.286 64.9 2.6 111.0 0.255 67.5 0.2 0.2 100.1 0.334 73.7 2.2 1.6 28.5 0.277 78.0 1.5 1.2 8.3 0.280 77.5 -4.3 -4.7 4.3 0.610 Maximum age-group of bridegrooms Mean age bridegrooms in maximum group 50-54 52.1 55-58 57.3 60-64 62.3 65-58 67.3 70-74 72.2 75-79 77.3 80-84 82.4 8!>-89 87.4 Age of bride for maxi- mum frequency Difference of age Maximum frequency . . Proportion of all bride- grooms of same age- group 39.6 12.5 785.5 0.200 46.8 10.5 457.9 0.215 48.8 13.5 213.9 0.172 45.2 22.1 139.0 0.168 62.8 66.1 9.4 6.1 66.6 62.2 0.160 0.150 57.5 72.5 ? 32.3 25.9 0.197 0.158 47.4 62.5 72.5 ? 10.2 9.6 8.7 0.165 0.155 0.140 47.5 7 4.3 0.360 * Calculated from yearly group results. t It is impossible from the data to determine these valuet with precision. { With partial corrections for misatatements of age. Fig. 61 shews the graphs of the maximum values. It is evident from these graphs that the greatest frequency of marriage is not well- defined according .to alleged ages. The surface shews ridges on the lines Aa, Ab, Acde, Afg and Ah. The highest point is for the group bridegrooms about 23.4, and brides 21.6 years of age, the frequency attaining to about 4,200, or about one seventy-second part (0.013911) of all the marriages. 208 APPENDIX A. The maximum group is 4114, or 0.13626 of the marriages. These figures are, however, somewhat uncertain, for reasons which will be pointed out in the next section. 24. Apparent peculiarities of the protogamic frequency. — Fig. 61 shews, by dots, the positions of maxima on the (vertical) columns, that is according to the ages of brides ; and, by dots with circles, the positions of the maxima on the (horizontal) rows, that is accord- ing to the ages of bridegrooms. If the ages have been correctly given there is no unique mode on the horizontal lines ; and this is a matter which demands special consideration. In Part X., § 6, Fig. 42, p. 115, it is shewn that the number of under-statements by women amounting to 10 years, is quite abnormal ; it does not follow the progressive diminution which characterises understatements amounting from 1 to 11 years. In the figure the line bAde would be the characteristic summit if the greatest frequency of marriage was in the case of parties of the same age. The Une f g would be the characteristic if a large number of men married wives 10 years younger than themselves ; while for the line Af to hold good, very large numbers of men of ages 22 to 31 must marry women of 21 years of age, irrespective of the disparity of age. To give the line of maxima Ah, a considerable number of men must marry women whose difference of age is one-half their age above 22. Such characters in a protogamic surface, are, a 'priori, extremely improbable. They would also characterise the apparent protogamic surface, if a considerable number of women, really of ages 22 to 32, all gave their ages as 22, when marrying men of from 22 to 32 years of age, and if a considerable number of women of 32 and upwards understated their ages by 10 years. This explanation probably does not differ very materially from the fact. Hence Tables LIV and LX must be regarded as of inferior value. It is, of course, much to be regretted that social organisation does not admit of the social- psychological fact of conjugal frequency at equal and disparate ages being accurately ascertained. 25. The contours of the protogamic surface. — ^The tedium of a rigorous analysis of a surface, when the measure of uncertainty is so large as is the case with the protogamic surface for Australia, is not warranted. A rough smoothing of the 5-year groups was, therefore, effected, and attributing the smoothed values to the centre points of the groups, and a series of contours for the proportions of 5, 10, 20, 40, etc., in a million of total marriages of all ages, were inserted by graphic methods. These gave fairly smooth contours. Regular curves being drawn, so as to ignore the minute undulations of the contours the results shewn on Fig. 61 are obtained. These represent with considerable precision the actual data from which they were derived, and will enable such data to be reproduced. They disclose the frequency distribution, for all combina- tions of ages, NXrPTIALITY. 209 Curves of Equal Marriage Frequency. — The Frotogamic Surface. Ages of Brides. 10 SO 30 40 50 60 70 80 90 100 100 Fig. 61. Note. — The pairs of ages which give equal frequency of marriage are found by following the course of any isoprotogam. The frequency indicated is per million marriages of all ages. The co-ordinates of any two points, whatsoever, on any isoprotogam are equivalent age-pairs, that is pairs of ages which are characterised by the same frejjuency of marriage. The protogamio surface, indicated by the family of curves or isogamic contours, is not the surface of frequency for indefinitely small ranges of age, but the surface for 5-year ranges of age ; see hereunder. These contours or " isogams" are numbered 5, 10, 20, etc., denoting the doubling of the frequency. The point denoted by an asterisk near A, is the summit of this surface, i.e., its ordinates are the centre of the 5-year ranges of age for which the frequency of marriage is greatest. From the sum- mit it falls most rapidly in the directions A, B and A, C, and least rapidly in the directions A, B and A E, the directions being shewn by broken lines. The values on the protogamio surface can be thus interpreted : — Assuming that the frequency of majriage for given pairs of ages, is as in AustraUa during the eight years, 1907-1914, in every 1,000,000 marriages of brides and bridegrooms of all ages, the number to be expected in any 5-year group over the range of 2J years earlier to 2J years later than the ordinates of the point taken, in the case of both bride and bridegroom, will be that shewn by the corresponding isogam, along which there will be equal frequency of marriage. Thus, for example, following the varia- tion with age contour corresponding to 10,240 marriages out of a total of 1,000,000, the frequency indicated will be very approximately that for the 5-year ranges, the middle values of which are brides 20 with bridegrooms 37 ; brides 24 J with bride- grooms 40 ; brides 30 with bridegrooms 42 ; brides 35 with bridegrooms either 41 J or 29 ; brides 37 with bridegrooms either 40 or 33 ; and so on. The contours thus shew the centre values of a 5-year range of age, at which there is equal frequency of marriage within the range. That is, if the co-ordinates of any point on a contour be X and y, the frequency of marriage is for the ages bridegrooms x — 2J to a; + 2^, with brides y — 2^ to y + 2^. Hence if M be the total number of marriages, the actual nimiber will be the number on the contour divided by 1,000,000 and multi- plied by M, 210 APPENDIX A. Characteristics of the Frotogamic Surface. Age of Brldesi Age of Bridegrooms. 10 20 30 40 10 20 30 40 SO S I h s ■f' .•• o» c A nnn A A ^ A 3 000 \ 1 j^ \a \ T) 2,000 / \ J> \ \ 1,000 ■ \ 1 i \, 1 \, } v. -^ y S, s.*. 10 20 30 40 10 Age of Brides. Pig. 62. 20 30 40 Age of Bridegrooms. 50 Curves ABC and D shew the various vertical features of the protogamio surface. Of these : — Curve A shews the projection of the profile on the y or age-of-brides axis, the dots indicating the values according to the data, and the continuous line shewing the probable true position of the surface profile. The outer Curve B shews the projection of the profile on the x or age-pf-husbands axis, the dots and circles indicating the positions according to the data. The inner curve indicates the position of a series of second and fairly well-defined maxima. All the points shewn are maxima of some kind. Curve C shews by dots, and a zig-.zag line joining them, the proportion which the frequency at the various maxima bears to the totals for the same age-groups of brides. The general trend of this frequency as a function of age is shewn by a broken line. Ciu-ve D shews by dots with circles and by a zig-zag Une, the proportion which the frequencies at the various mSixima bear to the total for the same age-groups of bridegrooms. The broken line shews their general trend. Each contour is twice the height of the contour immediately outside it ; thus the surface rises with great rapidity, and is very steep on the top, and also the left hand side in the figure. The proportion per million marriages for a 5-year group, ranging between a; ± 2^ and y ± 2^ is defined by the numbers written along the contours. The projection on the y—axis of the ridge running from the top left-hand corner to the NUPTIALITY. 211 bottom right-hand corner is shewn by curve A, Fig*. 62 ; and its pro- jection on the x—axis is shewn by Fig. B. The proportion which the frequency at the maximum bears to the total for the same age-group of brides is shewn by curve C, and for the same age-group of bridegrooms by curve D. In these two last curves the zig-zag lines shew the successive principal maxima, and the dotted Unes the general trend. It is probable that in a large population, when the ages at marriage are correctly given, the results would yield regular curves of the types drawn. The contours do not indicate curves of great regularity, but that is doubtless due (at least in part) to the inexact statement of age and the paucity of the numbers for higher ages. 27. Relative marriage frequency in various age-groups. — ^For socio- logic purposes, a table shewing the relative marriage frequency in various age-groups is of obvious- importance. Given an Australian population, constituted as to numbers of married and unmarried in age-groups as was its population during 1907 to 1914, 1,000,000 marriages are found to be distributed as follows : — TABLE LXin.— Relative FieoLuency of Marriage in Various Age-Groups. Australia, 1907-1914. Age- group of AOE-OKOOT OP BEIDES. All Bride- grooms 10-14 15-19 20-24 25-29 30-34 35-39 40-44 45-49 50-54 55-59 60-64 65-69 70-74 75-79 80-84 85-89 Ages.* 15-19 20-24 25-29 30-34 35-39 40-45 45-49 50^54 55-59 60-64 65-69 70-74 75-79 80-84 85-89 30 146 60 10 7 5 4 3 2 1 1 1 1 11,605 76,788 36,249 9,262 3,039 785 381 136 43 20 7 5 3 1 4,920 184,576 167,668 51,407 17,013 6,222 1,982 607 182 93 43 23 7 6 1 411 37,452 115,639 67,647 24,184 8,496 3,669 1,273 414 209 83 40 13 9 1 56 4,762 22,331 30,265 18,795 9,315 4,351 1,783 686 331 143 66 20 10 2 12 1,077 4,537 8,206 11,999 8,196 5,477 2,499 978 457 219 99 28 14 3 7 199 935 1,740 3,440 4,978 4,239 2,545 1,293 547 265 146 38 22 5 3 73 259 484 1,037 1,690 2,827 2,237 1,425 686 366 186 48 28 10 2 13 66 it 371 872 1,346 1,027 689 431 215 64 33 8 1 3 6 13 50 86 246 388 697 624 431 215 85 29 5 "l 3 7 13 30 80 166 215 351 315 166 92 23 3 "2 3 5 10 27 53 99 199 182 113 73 13 2 "1 3 7 17 50 63 73 47 8 1 "3 6 9 13 21 27 4 1 "2 3 5 7 11 2 "1 1 1 17,048 305,080 346,765 159,019 79,797 39,183 24,058 13,046 7,086 4,169 2,571 1,377 558 202 42 All Ages* 271 137,324 433,760 249,345 92,906 43,799 20,398 11,358 5,438 2,778 1,465 781 270 84 30 3 1,000,000 • Theae totals are about ten times those in the final oolumns of Table LX., p. 199. Though in substantial agreement they are not absolutely identical because these results have been slightly smoothed. The above table is founded upon the results given by a slight smooth- ing of the actual numbers, and gives the roughly adjusted relative- frequency of marriage' according to age-groups, based upon the marriages of the 8-year period, 1907 to 1914 inclusive, the 1911 Census being re- garded as giving a sufficient indication of the relative numbers of married and unmarried for the computation of any derivative relations. The middle point of time would be Jan. 0, 1911, while the Census is April 3rd, 1911. The total marriages were 301,922, or about 37,740 annually; half of them had occurred by about April 28, 1911, that is 25 days after the Census, hence a correction is not required, 212 APPE^TDIX A. 28. The numbers of the unmarried and their masculinity.— The smoothed results of the Census give the following numbers of unmarried at each age, viz., those shewn in Table LXIV. From these the ratios of the males to the females {M/F) have been computed ; they are shewn opposite the letters " Mas." in the Table. From the numbers given the mascuUnities can be computed of the various age-groups, which are required hereinafter for the computation of the probabihty of marriage according to pairs of ages. TABLE LXIV. — Number of Unmamed Males and Females and the Masculinity (,M/F) at each Year oJ Age. Australia, 3rd April, 1911. Year of AaBS. Decen- nium in Age. 10 20 30 40 50 60 70 80 90 M F Mas. 58,648 56,401 1.03984 43,049 42,654 1.00926 45,466 38,370 1.18493 16,700 10,839 1.54073 7,973 4,987 1.598T 5,232 2,340 2.2359 2,152 830 2.593 1,230 360 3.417 370 92 4.0 33 8 4.1 1 M F Mas. 54,759 52,982 1.03354 42,753 42,222 1.01258 43,799 34,634 1.26462 14,875 9,659 1.54001 7,669 4,623 1.6588 4,797 2,127 2.2553 2,032 760 2.674 1,190 320 3.719 312 80 3.9 26 6 4.3 2 M F Mas. 52,659 51,145 1 02960 42,748 42,001 1.01779 41,097 30,878 1.33094 13,250 • 8,730 1.51775 7,386 4,226 1,7477 4,332 1,938 2,2353 1,922 716 2,688 1,140 280 4,071 266 69 3.85 20 4 5.0 3 M F Mas. 51,158 49,785 1.02758 42,990 42,072 1.02182 37,541 26,360 1.42418 11,925 7,835 1.52201 7,077 3,940 1.7962 3,942 1,780 2.2146 1,822 690 2.641 1,080 240 4.50 225 58 4.05 15 3 5.0 4 M F Mas. 49,998 48,556 1.02970 43,618 42,484 1.02669 34,003 22,725 1.49628 10,900 7,278 1.49766 6,763 3,707 1.8244 3,642 1,549 2.3512 1,732 650 2.665 1,010 205 4.927 189 48 3.94 11 2 5.5 5 M F Mas. 48,812 47,428 1.02918 44,598 42,273 1.03062 30,338 19,600 1.54785 10,177 6,791 1.49860 6,453 3,441 1.8753 3,342 1,363 2.4519 1,622 600 2.703' 920 180 5.11 158 38 4.16 8 2 4.0 6 M F Mas. 47,543 46,298 1.02689 45,482 43,915 1.03568 26,823 17,267 1.55342 9,529 6,319 1.50799 6,185 3,212 1.9256 3,042 1,248 2.4375 1,490 550 2.709 810 155 5.23 121 28 4.32 6 1 6.0 7 M F Mas. 46,084 45,111 1.02157 46,212 43,813 1.05475 23,597 15,370 1.53526 9,032 5,910 1.52826 5,957 3,000 1.9856 2,762 1,145 2.4122 1,396 500 2.792 690 135 5.11 90 20 4.5 5 1 5.0 8 M F Mas. 44,783 44,055 1.01652 46,620 42,854 1.08788 20,808 13,646 1.52484 8,613 5,630 1.52984 5,731 2,783 2.0593 2,512 1,032 2.4341 1,328 450 2.951 560 120 4.67 65 14 4.6 4 1 4.0 9 M F Mas. 43,770 43,236 1.01235 46,470 41,020 1.13286 18,677 12,153 1.53682 8,265 5,303 1.55855 5,522 2,570 2.1486 2,302 910 2.5297 1,272 405 3.141 450 105 4.28 46 10 4.6 3 1 3.0 100 and over — Males, 2 ; Females, 1. Totals under 13, 662,764, 611,873 =1.08317. Note. — ^The masculinity is for the year-groups, and may be assumed to be the masculinity at age a; + J, where x is the tabular age, viz., the " age last birthday." NUPTIALITY. 213 The change of masculinity with age follows no simple law, as will be seen from curve A on Fig. 63. The irregularities after 80 are due to the relatively small numbers on which the curve is based, and must be re- garded as accidental. The masculinity diminishes in the earlier years, because of the greater mortality among males. Its constancy between the ages 25 and 37 is remarkable, as also is the sudden increase commencing at 66 years of age, and continuing to 76. CA S mOQ I ^^^ a. «.°a ■a w ■« r.ili E-l H Number of Males and Females Marrying and Living together in the State of Marriage, and the Masculinity of the Unmarried. 9 180 8 160 7 140 6 120 5 100 4 80 3 60 2 40 1 20 AlteB 10 !L — — /' \ / \\ \V"^ B M-^ >s- 1 iv \; i- i \ \ i s \ \ ; 1\ \ S ,/ 1 D -\ \. \ 1*1 1 ' M \ , /-^ r y \ ^ \ ' A k / \ i A v p^ / 4I L V A^" -\ \ / " \ \. y t^*^ X V "^ -"Hill r- \ .> -^ k- Ml s \ s b._^ b^ _J2_ V -♦^^S ^^ ^s ^ ^ ^ i 3 2 1.1 1 1.0 30 40 50 Mg. 63. 70 80 90 100 Curve A denotes the variation with age of masculinity ('M/F) of the unmarried. The small lozenge-shaped dots are the values according to the data ; the continuous line shews the general trend. The scale for the masculinity up to nearly 20 years has also been plotted on ten times the scale. See Table LXIV., p. 212. Curve B denotes the number of married females of marriages living with their husbands in a total of 1,000,000 couples. See Table LXVIII., p. 224. Curve C denotes the number of married males of various ages living with their wives, in a total of 1,000,000 couples. See Table LXVIII., p. 224. Curve D shews the adjusted number of females of various ages, per 100,000 marriages, occupying between 1907 and 1914. See Table LIX., p. 197. Curve E shews the adjusted number of males of various ages, per 100,000 marriages, occurring between 1907 and 1914. See Table LIX., p. 197. 214 APPENDIX A. 29. The theory of the probability of marriages in age-groups. — ^The data do not exist for a definite and rigorous determination of the pro- bability of marriage in age-groups ; nevertheless a fairly accurate esti- mate is possible by means of a somewhat empirical theory, which will now be indicated. The deduced results are shewn in Tables LXVI. and LXVn., see pp. 219 to 222.i For convenience the adjusted numbers from the Census are given in Tables LXIV. and LXV. hereinafter ; the corresponding numbers of marriages occurring in each age-group are also given. The values of q given in the tables enable the number of marriages likely to occur in each age-group to be computed when the numbers of unmarried males and females in the group are known. Thus, q being the tabular number, the number of marriages, N, may be computed by means of formula (431) or formula (434) hereinafter. (See next section.) Suppose that in any age-group there are M unmarried males and F unmarried females ; and that in a unit of time N pairs of these marry. The probabiUty wiife F females in the group, of a particular marriage occur- ring among the M males is obviously N/M ; and with M males in the group, the probability of a particular marriage occurring among the F females is similarly N/F. Such a statement of probability, however, lacks general- ity. To obtain a more general one, an expression is needed which, given a definitive tendency towards the conjugal state in males and in females, though not necessarily of the same strength (or potential) in each sex, and not necessarily independent of the relative numbers of the sexes, nor even independent of the lapse of time, will give the number of marriages occurring in a group, constituted in any manner whatever in regard to the numbers of either sex. We shall call the tendency to marry the conjugal potential under a given condition. In the case of males let the conjugal potential be denoted by y, and in the case of females by y'; y and y' vary with age, doubtless also with time, and (we may assume) with the relative frequency of M and F. Without d.sserting it to be exactly the law of variation, we may suppose that the conjugal potential varies somewhat as some constant, multipUed into some power of the ratio of the numbers of the unmarried of each sex. Put p for the constant in the case of males, p' for the constant in the case of females, then the conjugal potentials are of the type p. f {-^), which function can, for all practical purposes, probably take the form (425) y=2,(^)';andy'=p'(-|.)' formulae in which r and s are indices to be ascertained by experiment. ^ These results are on the basis of 10 million males, and the same number of females. Hence if they are multiplied by one ten-miUionth of 1,508,623, and 1,277,259 respectively, they will give the absolute numbers, since these were the number of unmarried males and of unmarried females respectively, on 3rd April 1911. NUPTIALITY. 215 Thus y = p and y' = p' when the numbers of unmarried of either sex are equal ; ordinarily they do not differ sensibly therefrom. Again, if the number of females be large, the y potential is doubtless smaller ; and if the number of males be large the y' potential is smaller. This appears to be confirmed by experience. The expressions (425) can be made to fit the facts by appropriately determining r and s. From (425) we have at once for the ratio of the conjugal potentials <-' f'-H'Y- where w = r + s, from which it is evident that it is not necessary to ascer- tain r and s individually, but only their sum, w. And if the conjugal potential vary with age, it could be ascertained only by comparing a series of results for the one age-group when the numbers of males and females were very divergent ; all other circumstances promoting marriage remain- ing constant. For this reason, with the limitations of existing data, We must assume (which doubtless, as already indicated, is not exactly true), that, when the numbers of the unmarried of each sex are equal, the conjugal potential and probability of marriage vary in the same way . That is (427) y / y' a: p/p' ; or the probabiUty of marriage is the effective measure of the conjugal potential ; or in other words (subject to what has been said above) we may suppose that, with equal numbers of unmarried males and females, the frequency of marriage is a normal measure of the conjugal potential. If we make still another assumption, viz., that indicated hereunder (in the passages in italics), a crude type of solution becomes possible, and the problem may then be envisaged as foUows : — If there be M males in any age-group and F females in any other age- group, it is obvious that there can hei MF marriages of particular pairs among these groups : and if a group out of these of N males and N females be taken, it is similarly seK-evident that they can form N N marriages of particular pairs. GonsequerUly assuming that the marriage of particular pairs is equally probable, avd that the relative magnitude of M and F does not influence the probability, p, then the chance of N marriages occurring is (428) pxy =- N^Ny/ (M^Fy) X and y denoting the age-groups referred to. The value of p cannot possibly become unity unless M = F = N. This probability does not, however, enable us to compute the likelihood of N marriages occurring with particular values for M nd F, since obviously N is not y/p.-\/{M.F), although that is a solution of eqi ation (428) ?■ Subject to the assumptions 1 For example, given M constant, N would depend upon ^JF, which is certainly not correct if M be large and F small. In this ease iV would evidently vary as F, not B& ■^F. 216 APPENDIX A. made, the function representing the chance of N marriages occurring must clearly vary approximately as -^{MF), when they are sensibly equal, and must vary sensibly as F (or M) when M (or F) is relatively very large. In order to obtain an expression that will readily fulfil the necessary conditions, we may observe that if we put (429) N^y=q^y.4> (Mx) . ^ (Fy) and for ready computation assume that the functions ^+P . F^+' Consequently we may write instead of (429) : — F M 1 1 (431). . N^y = q^y . M^+^ . F^+P = q^y . M,l,^+ =q^y . J-jni+z* and to find q from the results furnished in Tables LXIV. and LXV. we have, N-Xy Nxy (432) qxy= JH = XI ; or (432a) \ogqxy = log Nxy - fT—'^^^ ^ ~ YTTj, ^°S ^' X and y denoting the central values of the age-groups, i.e., a; ± | Jfc, «/ J; \k where k is the range of the group. The apphcation of this formula can be greatly facilitated in the following way : — ^Let Sxy = Mx + Fy, that is, let ' 4 is the quantity denoted by . q^y : 1 in which R'^ is the tabular value ^^ + '' and R'^ is the tabular value ^^ + *, the q quantities being a.s before. Values of iJ' = 2/x i+M TABLE LXVa. {or computing the effect of unequal numbers of unmarried males and females on the frequency of marriage. M F M F M F M F — or — R' — or — R' — or — R' — or — R' F M F M F M F M 10 1.2328 60 1.0694 200 1.0267 700 1.0094 20 1.1533 70 1.0616 300 1.0192 800 1.0084 30 1.1159 80 1.0556 400 1.0150 900 1.0076 40 1.0958 90 1.0507 500 1.0124 1,000 1.0069 50 1.0797 100 1.0467 600 1.0107 2,000 1.0038 The table shews very clearly that as the unmarried females (or males) become relatively fewer the number of marriages varies more nearly in the proportion of the number of females (or males). 30 . Masculinity of the unmarried in various age-groups. — ^The results embodied in Table LXIV., make it possible to compute the mascuUnity of the unmarried for any combined age-groups, since this affects the number that may be expected to marry. The masculinities are shewn in two tables, viz., Table LXVT. and Table LXVII., the former giving the results for 2-year age-groups for ages 15 to 44 for bridegrooms, and ages 13 to 44 for brides ; and the latter the results for 5-year age-groups for ages 15 to the end of life for bridegrooms, and 10 to the end of life for brides. From the values of M/F, = fi, (or F/M, =ff>,) the values of F / {M+F) and oi M / (M+F) may be readily computed if required. Thus^ (436). F 1 _^ M + F l+[i l+ = 2 M M + F !+^- = Ffif' then the quahfication as to the masculinity being approximately identical disappears.'^ It is not unimportant, however, to remember that the fundamental assumption would have to be very erroneous (and that would seem to be impossible) in order to seriously prejudice the precision of the result obtained by the application of the formula (434). The error in any real appHcation of the formula can be a differential one only, and if the constitution as regards numbers of the population be approxi- mately therefore that from which it was derived, any defect in the theory of variation with relative numbers of the sexes, formula (430), has no sensible effect. 32. The relative numbers of married persons in age-groups. — ^The Census of 1911 disclosed the fact that the number of married persons living together on the night of the 3rd April, 1911, was 623,720. The number of wives absent from their husbands was 112,129, and husbands absent from their wives 110,053. There were 616,738^ (out of a total of about 734,000 married couples) whose ages were fully specified, and who -were living together. This may not be a perfect sample of the entire population, for although the date of the Census, viz., 3rd April, is well chosen, the number of spouses of each age apart at a given moment is probably not sensibly proportional to the total number. As the totals, however, are only about one-fifth greater than the number for wiiich the information is complete, the 616,738 may be taken as fairly representing the popula- tion. The results are shewn upon Table LXVIII. 1 fi^ and 01 are the same as /i and above ; f,..^ and ^ are defined in Table XXI. p. 132 hereinbefore. ' This number is made up as follows : — Husbands and wives com- pletely specified as to age, and living together . . 616,738 Living to- Living to- gether but gether but Wives Total Wife's Age Husband's Absent. Husbands. Both ages unspecified . . 506 not Age not stated. stated. 617,244 + 4,108 + 2,368 -|- 112,129 = 735,849 Living together but wife's age not stated . . . . 4,108 || Living together but hus- age not stated . . . . 2,368 1.19313 Husbands absent . . . . 110,053 X Total wives .. = 733,773 = 1.18976 x 616,738 224 APPENDIX A. TABLE LXVm. — Number of Married Persons per 1,000,000 Married Couples, Living Together on the Night of the Census, 3rd April, 1911. In 5-year Age-groups. Wives' Ages. Total, Hus- I 1 1 10 bands' X Ages, t 0! 15 20 25 30 35 40 45 50 56 60 65 70 75 80 8590 95 to o to to to to to to to to to to to to to to to to to 99 1 4 19 24 29 34 39 44 49 54 59 64 69 74 69 84 89 94 99 15-19 . . 577 347 39 8 3 974 20-24 8 6,771 24,015 7,168 1,090 217 "63 "28 " 6 38,366 25-29 2 3,574 40,354 54,338 11,871 2,015 383 112 44 "11 2 1 112,707 30-34 . . 1,090 17,907 54,009 54,757 12,145 2,264 516 123 29 11 5 2 142,858 35-39 . 376 5,845 24,489 61,157 47,891 10,786 1,966 379 89 16 11 3 ' '2 ■] 143,009 40-44 . 130 2,048 9,082 25,695 47,680 44,462 9,936 1,934 452 92 36 10 3 141,660 45-49 . 44 760 3,287 9,610 28,654 43,595 40,083 8,644 1,450 340 96 16 10 131,489 50-54 . 24 258 1,090 3,124 7,694 19,245 35,589 29,716 5,800 1,138 311 50 13 "3 104,056 55-59 . 11 94 334 921 2,380 5,567 13,677 22,851 16,769 3,478 666 154 41 11 66,954 60-64 . 5 45 135 357 798 1,899 4,506 9,790 13,578 10,622 2,330 478 81 18 i'i'.'. 44,645 65-69 . 23 62 156 413 830 1,840 4,081 6,684 9,571 7,639 1,629 292 42 8.. .. 33,270 70-74 . 8 26 58 180 319 718 1,505 2,616 4,405 6,040 4,533 1,004 118 16 6 2 21,552 75-79 . '. "2 5 23 29 57 131 268 517 820 1,600 2,996 3,322 2,238 399 37 8 .. 12,452 80-84 . 2 2 3 16 24 42 79 152 227 472 751 1,166 1,111 655 84 26 .. 4,801 85-89 . 2 2 6 10 16 28 53 34 148 198 267 183 91 6 .. 1,094 90-94 . 2 2 6 1 3 B 34 37 31 36 18 15 . . 183 95-99 . 2 2 5 5 5 6 3.. .. 28 100-104 . •• 2 1 3 T.otals 1 15-104 1 11,606 91,713 154,087 158,750 145,157 129,598 109,339 79,771 48,584 31,841 21,070 11,593 5,098 1,471 258 52 2 1,000,000 33. Conjugal age-relationships. — For certain estimations it is important to know, for given ages of husbands, the average difference of the age of the wives ; and also for given ages of wives the average differ- ences of the ages of the husbands. These relationships as at marriage, i.e., initially, may be ascertained from marriage records. They may be called the protogamic age-relationships. The instantaneous relationships at any moment, however, are disclosed only by a Census, and may be called the gamic age-relMionships. The age-groups, wj^h the age of the husband as argument, and those with age of wife as argument, lead, it will be found, to different results, which have no obvious direct mutual relation. Hence this, in common with other analogous groupings of a non-homogeneous character, must be independently made, for a reason which we shall now more definitively indicate. In cases of the kind under consideration two formulae are needed ; in one the argument is the age of the husband (or bridegroom), in the other the age of the wife (or bride). 34. Non-homogeneous groupings of data. —If , associated with any group-range, viz., x^j. to x^ + i say, of any class of elements (ages of hus- bands in the case under review), there is a class of related elements (ages of wives), viz., «/fc_a to 2/4+6 say, where a and b, in general, have large values ; and if, reciprocally, a group-range, 2/4 to yi^^i say, is associated NUPTIALITY. 225 with the group x^_^ to x^.^^ say, A and B also having large values, the result obtained from the former will have no simple relation with that based on the latter. For a result based on the argument x, has not the same constitution as one based on the argument y. If the distribution about the mode in such cases be not symmetrical in each, in fact if it be not similar in all respects, no direct functional relationship subsists between results for groupings arranged according to the values of x, and those for groupings arranged according to the values of y. Groupings subject to this limitation may be called non-homogeneous groupings, and require special consideration. 3d. Average differences in age of husbands and wives, according to Census. — In Chapter XIX., Vol. I., § 2, of the Report on the Australian Census of 1911, results are given for a series of age-groups of husbands and of wives. The results are also given in greater detail in Vol. III., Table I., pp. 1106-7. The difference for the central-age of the group, which is sensibly, though not exactly, the mean-age, of those included therein, is as shewn on Fig. 64,^ the curve marked A, representing the excess of the age of husband over the average age of their wives, as determined from groupings according to the age of the husbands, and the curve marked B, representing the excess of the age of the -wiie over the average age of their husbands, as determined from groupings accord- ing to the age of the wives. The differences are given in Table LXIX. hereunder. The tangent line to curve A is coincident with the curve for the ages 40 to 60 inclusive (beginning point of year) ; hence for this interval the relation is — (438) D„ = -I- 0.098 a;^, for ages 40 to 60, D„ denoting the average excess in years of the age of the husband over the average age of the wives, and X)^ being the age of the husband. The tangent is coincident with curve B for the ages 30 to 67 inclusive, and the age of the wife is greater than the average age of the husbands by the amount Df^, where (439) !>/,== - 6.275 + 0.058 x„, for ages 30 to 67, in which x^ denotes the age of the wife. It is obvious from the table that the assumption ordinarily made is invalid. The characteristics of a table of values of the differences will be evident from the table itself. See pa^ 227 226 APPENDIX A. TABLE LXIX. — ^Differences of the average Age of Wives for Husbands of various Ages, and of the Average Ages of Husbands for Wives of various Australia, 1911. Age of Calculated Result, Curve A. Calculated Result, Curve B. Hus- band Position Ordin- Smooth- Crude Position Ordin- Smooth- Crude A; of ate to ed value value of ate to ed value value Wife Tangent Curve. of Dw from Tangent Curve. of Dh. from B. Data. Data. 14i + 1.42 —5.43 —8.4 15J 1.52 —6.52 — 5.00 —5.0 5..38 —.5.02 —lOAO 10.4 m 1.62 5.27 3.65 0.9 5..32 3.07 8.39 9..1 I'i 1.72 4.52 2.80 2.8 .5.26 2.25 7.51 7.5 m 1.81 3.78 1.97 1.2 5.20 1.76 6.96 7.2 m 1.91 3.48 1.57 1.1 .5.14 1.44 6.58 6.6 20J 2.01 3.12 1.11 0.6 5.09 1.16 6.25 6.2 23 2.25 2.38 —0.13 —0.4 4.84 .80 5.70 5.7 27i 2.70 1.33 + 1.37 + 1.2 4.68 —.17 4.85 4.7 30 t 2.95 .90 2.05 4.54 .0 4.54 32J 3.19 .56 2.63 2.5 4.39 .0 4.39 '4.4 37i 3.67 .10 3.57 3.6 4.10 .0 4.10 4.1 40 * 3.92 .0 3.92 •• 3.96 .0 3.96 m 4.16 .0 4.16 4.2 3.81 .0 3.81 3.8 474 4.66 .0 4.66 4.7 3.52 .0 3.52 3.4 52i 5.15 .0 5.15 5.2 3.23 .0 3.23 3.1 57i 5.64 .0 5.64 5.8 2.94 .0 2.94 3.0 60 * .5.88 .0 5.88 2.80 .0 2.80 62^ 6.13 + .08 6.21 6.5 2.65 .0 2.65 2.9 67 t 6.55 .19 6.74 2.50 .0 2.50 67J 6.61 .20 6.81 7.3 2.36 + .08 2.28 2.3 72i 7.11 0.66 7.73 8.1 2.07 .70 —1.37 —1.3 77i 7.60 1.58 9.18 9.2 1.V8 1.96 +0.18 + 0.4 82^ 8.09 3.14 11.23 11.3 1.49 3.76 2.27 2.2 87J 8.58 5.70 14.28 14.4 1.20 6.70 5.50 4.2 92J 9.07 9.10 18.17 18.6 0.91 12.01 11.10 11.1 97i 9.56 14.90 24.46 22.3 0.62 + 25.62 + 25.00 + 25.0 102i 10.05 29.95 40.00 40.0 —0.33 *f The asterisks and daggers denote the ages between which curves A and B, respectively, are straight lines. In the figure the curves A and B are very approximately the smoothed values. The tangents are shewn by dotted lines ; the data by the dots ; it is instantly evident that the difference is not constant, but is a definite function of age. A and B are the curves of the gamic age-relationship. 36. Average differences of age at marriage. — A similar table to the preceding can be constructed for the ages at marriage . In order to eUmin- ate the uncertainties due to paucity of data the results for the eight years 1907 to 1914 were combined. The combinations shewed the same tend- ency as was revealed by the Census, vIts., for the numbers to be unduly large for the ages ehdii^ with the digits and 6. The niimbers for the purpose of the following table have, however, not been smoothed ; the smoothing in the table itself making that .uivaeeegsary. NUPTIALITY. 227 Differences between Ages of Husbands of any Age and the Average Ages of their Wives, and between the Ages of Wives and the Average Ages of their Husbands. Curve C + 20 Curve C + 10 Zero of Curve C Curve A + 10 Zero of Curve A o t.^ Curve A - 10 Curve A -20 Ages / / /' li ^ / _-■ -- ■<'- ^ ii- -^ i ■y^ ^'-' ^ ; '^■r' — ' = = ^^ — — + 10 Curve U ZeroofCurveD TM T'h -10 Curves s|° Zero of Curves T'l. -10 Curve B. 10 20 30 40 50 60 70 80 9D 100 Age6 Fig. 64. Curve A. — ^Excess of the husband's age over the average age of their wives, at the 1911 Census. See Table LXIX., p. 226. Oa is the zero for the curve. Curve B. — ^Excess of the vpife's age over the average age of their husbands, at the 1911 Census. See Table LXIX., p. 226. Ob is the zero for the curve. Curve C. — ^Excess of the bridegroom's age over the average age of their brides, 1907-1914. See Table LXX., p. 228. Curve D. — ^Excess of the bride's age over the average age of their bridegroomst See Table LXX., p. 228. The results are shewn by curves C and D in Fig 64. The tangent to curve C, which is analogous to curve A, is identical with the results for ages 42^ to 67| years ; thus : — (440). .D\ 1.745 + 0.266 x^ ; for ages 42^ to 67|. For curve D, the difference of ages is analogous to curve B. The tangent is parallel to the age -axis at the distance (441). .D' = - 1.76 ; for ages 32^ to 60. The table shews the differences outside these hmits. Towards, the ends of the curves the results for all four ciirves are of course somewhat uncertain. C and D are the curves of the protoganiic Skge-relationship. 228 APPENDIX A. TABLE LXX. — ^Difference of the Average Age of Brides for Bridegrooms of various Ages, and of the Average Age of Bridegrooms for Brides of various Ages. Age of Calculated Result, Curve C. Calculated Result, C'lu^e D. Bride- groom Position Ordin- Smooth- Crude ! Position Ordin- Smooth- Crude C; of ate to ed value value 1 of ate to ed value value Bride Tangent Curve. of X)'„. from Tangent Curve. of D\. from D. Du,. Data D), 13i __ -1.76 -11.04 . 12'.80 12.80 144 1.76 8.45 10.21 1 10.21 15J + 2.38 — 5.35 —2.97 —5.50 1.76 7.10 8.86 •i 9.18 m 2.64 5.08 2.44 2.36 1.76 6.10 7.86 7.86 17J 2.91 4.85 1.94 1.08 1.76 .5.24 7.00 6.95 18i 3.18 4.56 1.38 0.81 1.76 4.50 6.26 6.25 19J 3.44 4.35 0.91 0.37 1.76 3.92 5.68 5.66 20i 3.71 4.08 — fj.37 —0.18 1.76 3.42 5.18 .5.26 23 4.37 3.49 -0.88 -0.49 ).7li 2.24 4.00 3.94 27i o.ol 2..52 3.05 2.72 1.7(j .70 2.46 2.46 32it 6.90 1.48 5.42 5.35 1.76 .00 1.76 1.76 37i 8.23 .56 7.67 7.67 1.76 .00 1.76 1.72 42^* 9.56 .00 9.56 9.45 1.76 .00 1.76 1.91 47i 10.89 .00 10.89 10.95 1.76 .00 1.76 ' 1.66 .52i* 12.22 .00 12.22 12.30 1.76 .00 1.76 1.75 57i 13.55 .00 13.55 13.42 1.76 . .00 1.76 1.31 60 t 1.76 .00 1.76 1 62i 14.88 .00 14.88 15.03 1.76 .06 1.82 ..30 67i 16.21 .00 16.21 16.16 ;, 1.76 .28 2.04 2.08 72J 17.54 .90 18.44 19.52 : 1.76 .73 2.49 1.31 77i 18.87 2.30 21.17 19.93 1.76 1.54 3.30 5.83 82^ 20.20 4.50 24.70 37.05 1.76 3.00 4.76 7.14 871 21.53 8.09 29.62 29.62 1.76 .5.. 30 7.06 97i 22.86 1.76 10.00 *f The asterisks and daggers denote the ^es between which the curves C and D , respectively, are straight lines. 37. The gamic surface. — ^The data furnished in Table LXVni. may be used to construct the gamic surface, on the same principle as was followed in the construction of the protogamic surface, dealt with in § 25 hereinbefore. The results are shewn on Fig. 65, from which it will be seen that the isogams are more elliptical in form than isoprotogams, and are more regular; see Fig. 61. The principal meridians AB, AC and AD, AE are in much the same positions as on the protogamic surface, but the point of maximum frequency A, and the line of greatest slope are for higher ages than on that surface. The interpretation of the curves is, mutatis mutandis, the same as that for the isoprotogams ; in the case of Fig. 65, however, everything applies to persons " living in the state of marriage," instead of to " persons at the moment of marrying. "' NUPTIALITY. 229 Curves oJ Equal Conjugal Frequency.— The Gamic Surface, 1911. 10 Ages of Wives. 30 40 50 60 70 80 90 100 Fig. 65. Note. — The pairs of ages for which an equal frequency of married couples existed at the Census of 1911 are found by following the course of any isogam. The remarks in the footnote to Fig. 01, p. 209, apply, mutatis mutmidis, to the contours of the Gamic Surface. 38. Smoothing of surfaces. — ^Let it be supposed that the nature of statistical data is such that the most suitable representation is by means of the heights of series of parallelepipeds, as for example, in the case just considered, of the numbers of marriages of bridegrooms between given age limits and of brides between the same or other given age limits. For simplicity we may assume that the combination is according to age last birthday, and thus is in single year groups. Since the general equation of a surface of a second degree will involve nine constants,we can deduce the constants of a surface representing its integral between the limits a; = 0, 1, 2, and 3, and y = 0, 1, 2, and 3, the deduced expression will give totals corresponding to those of the nine contiguous groups. By means of the corresponding surface equations, deduced from these, for lines parallel to the a;-axis, or parallel to the 2/-axis, we can find the hei^t to 230 APPENDIX A. this surface, along the four edges of the central parallelepiped. If this operation be then repeated, making each of the four adjoining parallele- pipeds the central ones in a group, we shall obtain a second series of values for the distances along the four edges to the surface ; if these do not differ very greatly then the means of each pair of values may be taken, in general, as the smoothed result. In this way the greater part of the entire surface can be dealt with, and the series of verticals to the surface thus found will have reduced the original irregularities, and may be regarded as a first smoothing of the surface, conforming, however, a.s nearly as pqpsible to the general series of group-heights. The results so obtained, however, are " instantaneous values," that is, they are the heights corresponding to the ranges .»; to r + dx, and «/ to y + di/. U the numbers be very irregular the process above indicated is extremely tedious, and of little value. It may then be preferable to regard the group results as vertical ordinates with the central values of the group-ranges as the horizontal co-ordinates. The procedure then in- volves the independent smoothing of a double system of curves, and the taking throughout of the means of the pairs of verticals so found. The whole procedure is then repeated, with the means thus obtained, until the smoothing is satisfactory. The criterion of good smoothing is that the " accumulated deviations" in either of the two directions (at right angles to one another) do not attain to appreciable values, and that they alternate in size. It should be noted that smoothing in this way does jiot give " instantaneous values," that is where k is the extent of the range, the heights now denote values true for the ranges | fc on either side of the values x and y, these being the ordinat€s of the centre of the ranges . There is another possible scheme of solution, viz., to ascertain the constants of an equation, which will give at once the group values for groups of the same double-range, the arguments being the ordinates of the centres of the groups. The method is analogous to that treated for a surface in Part V., § 10, formula (211) to (216), pp. 72-73, and the solution by a process analogous to that indicated in the section immediately following, will give the group-height for any value of x and y, the range being a; ± i ^, y ± ^k. 39. Solution for the constants of a surface representing nine contiguous groups. — ^The most general expression for a surface, every section of which parallel to the .r-axis and parallel to the y-axis is a curve of the second degree is (442). .z=A+Bx+Cy+Dxy +Ex^+FxY + Gy^ +Hx^y + Ixy* NUPTIALITY. 231 Let the values of the groups be denoted by the letters I, m. according to the following scheme : — y = 3 .0 = 2 * = 3 The integral of the above, divided by xy, the area of the base, is ; (443). . . . ^fJF (X, y, k) dxdy =4+ ^Bx +Y^y +X^^2/ + jEx^+jFy^ + j Gx^y + 1 Hxy^ + j Ix^y^ from which we deduce, by putting x {or y) successively 1, 2, 3, and making y (or x) equal 1, 2, or 3, the following values of the constants J to Z in terms of Z, m, t. The results are : — (444). ... A = - (q+ 0-2^3)+ {n+l - 2m)+ (I - p) - 3 (s - p) + (< - ?) + 3 (r - 0) + (p - o) - -2- (m - Z) (445)....5 = 3(g+o - 2p)- 3(»i + I - 2m) + 8(s -p)-3(t - q) - 5 (r — o) - 2 (jj - o) + (m - Z) (446). . . . C = 2 (g+o-2^)+2;)-2(p-o)+9(s-p)-9(r-o)-3(«-9) (447). . . .D = 4 (2j-o)-6(^+o— 2p)-24 (s-p)+15(r-o)+9(t-g) (448) . . . . .0 = - (w + i - 2m) {q^o-2p)-3[s -p) + I (« - 9) + I ('• - o) ■232 APPENDIX A. (449). . ■ . -f = I (r - O) _ |- {« _ p) + A (< _ 9) (450).... G!= - ^(t-q)- ^ (r-o)+9(s-p)+^{q+o-2p) (451). ...H=U(8 -p) - ^-^(r -o)- ~(l -q) (452). . . . I = I (t - g) + -^ {r - 0) --^ (s - p) It will be .seen that the arithmetical labour of deducing the constants of a surface which will exactly reproduce any square system of 9 contiguous group-values, is very great, and ordinarilj- prohibitively so. In general, therefore, less rigorous methods have to be adopted, and are ordinarily quite satisfactory, particularly in view of the fact that in practical calculations values according to a given double-range are required. 40. Naptiality and conjngality norms. — It would appear desirable to establish decennially, what may perhaps be called a nuptktUty or pro- togamic norm, and also a conjugality or gamic norm, on the basis of an aggre- gation of the marriages of a large number of populations for the former ; and of the Census results for the latter. The norms should preferablj' shew single-year re.sults up to 24 years for brides, and 29 years for bride- grooms ; and up to 34 years for wives, and 39 years for husbands, respectively. The protogamic norm will reflect the trend in regard to the early institution of marriage, and the gamic norm the modification of this by change in longevity, the frequency of divorce, etc. These norms could include the curves of the totals according to the- age of the males (bride- grooms and husbands), and according to the age of the females (brides and wives), and could include also the frequency of the group-pairs. The norms of the conjugal state, '■ never married,'" " divorced,'' and " widowed," might, with advantage — as well as those of the " married" — also give the frequencies according to group-pairs. 41 . The marriage-ratios of the unmarried. — ^It has already been shewn that the probability of marriage depends, among other things, upon the relative numbers among the unmarried of the sexes. So long, however, as a population does not greatly change its constitution according to sex and age, the crude probability of marriage according to sex and age may be regarded as varying approximately as the aimual rate. This probability maybe called the peithogamic coefficient^ for the sex and age in question. It will be further discussed in Part XIII. in connection with fecundity. 1 From vcWu to prevail upon, (flfiffii the Goddess of Persuasion) and yafUKoi, of or for marriage. Xra.— FERTILITY AND FECUNDITY AND REPRODUCTIVE EFFICIENCY. 1 . General. — ^The phenomena which directly concern the measure of the reproductive power of the human race will be dealt with in this part. These phenomena are in general complex, the variation of the repro- ductive power being in part of physiological origin, and in part of the result of the reaction of social traditions upon human conduct. This will appear in any attempt to determine the laws of what has been called bigenous^ (better, digenous) natality, or natality as affected by the ages of both parents, as distinguished from those affecting merely monogenous natality, or natality as related to the producing sex. In deducing the most probable value for certain of the phenomena it will be necessary to minimise the effect of misstatement of age. This can probably be done more effectually than would at first sight appear probable. The final results, however, must be subject to some small degree of uncertainty. The question of the reproductive efficiency of a population has in part been dealt with in Parts XI. and XII., dealing with Natality and Nuptiality ; this, however, is derivative and depends in its turn upon the age -distribution and conjugal condition of the producing sex. Many questions concerning the measurement of fertility and fecundity can be settled with sufficient precision without recourse to a differentia- tion depending on the age of the father, the better in Australia, perhaps, inasmuch as the decay of virility with age is not well marked, and in this aspect the digenous fertility stands in marked contrast with that of Hungary. 2. Definitions. — ^It is desirable , initially, to define the sense in which several terms will be used hereinafter. Monogenous fertility and monogenous fecundity will denote the fertility and fecundity of the female considered without regard to the age of the associated male. Digenous fertility and digenous fecundity will denote the fertility and fecundity of the female, as modified by the age of the associated male, and therefore is considered in relation to the ages of both males and females . Consequently computations of monogenous fertility or fecundity will be based upon the age of the female. It foUows from this, that two popula- tions will be (i.) exactly, or (ii.) approximately, comparable, only when the conjugal age-relationships are (i.) sensibly identical, or (ii.) are similar. 1 By Joseph Korasi, see Phil. Trans. Lond. B., 1895, p. 781. -34 APPENDIX A. Isogeny will deuote either equal fertility or equal fecundity, the former to be called initial isogeny or isoprotogeny ; the latter general isogeny, or characteristic isogeny, or simply isogeny. A curve, passing through a series of pairs of ages plotted as co- ordinates, in such a manner that it will pass through all ages which give either equal initial or equal general fertility or fecundity, will be called an isogen as appUed to either. The curves may therefore, Ln the cases considered, be called isoprotogens , and isogens. The terms '' fertiUty" and " fecundity"'^ though ordinarily sensibly identical in meaning, have sometimes been assigned different meanings by statisticians, one being employed to signify the qualitative, and the other the quantitative, aspect of reproductivity. Owing to their phonic resemblance the words " sterility" and " fertility" are the more appropriate to employ in order to denote the difference between producing or non-producing; while "fecundity," which biologically is used without quaUficative to imply producing in great numbers (a meaning which requires the qualification "great" when fertility is used), is obviously the more appropriate word to denote " multiple fertihty."! 1 In Latin, although " fertiUtas" and " feovmditas" have no marked difference of meaning, the latter word seems to be the preferable one for denoting frequency of bearing offspring. The root of fecundus is " feo" (obsolete), or PE = Greek (pu ; e.f. Sanskrit bhu ; Zend bu ; see 0i)u Liddell and Scott's Greek-English Lexicon, 8 Edit., p. 1703. The root of "fertUis" is "fero"=Greek root 0e/) : e.f., Sanskrit "bhar"; Zend "bar"; A.S., "bear-n"; the radical meaning being to bear or carry. See LiddeU and Scott op. cit., p. 1662. In regard to "sterilitas, " o.f., Sanskrit "stari" (vacca sterilis). In other languages the following correspondence might be suggested : — Enghsh. French. Italian. German. Danish. Swedish. Fertility ; FertiUty ; Fertility ; Fruehtbarkeit or Frugtborhed Frukteamhet; Gebarfahigkeit Fecundity. F^eondite. FeconditiL Ergiebigkeit or Avledygtighed Afvelsamhet. Fruehtbarkeit Inasmuch ' Fruehtbarkeit," " Frugtbarhed" and " fruktsamhet" ought, if possible, to be appropriated to the one meaning, the first suggestion as regards the German is to be preferred. That is, it is better to adopt " Fruehtbarkeit" for fertility and •' Ergiebigkeit " for fecimdity. KOrOsi suggests "Ergiebigkeit der Ehen." " iluttersohaftsfrequenz " and " Maternitatsfrequenz " refer only to cases of maternity. J. Matthews Duncan, in his " Fecundity, fertility, sterility and aUied topics,"' 1866, 2nd Edit., 1871, has used " fecundity" to imply the qu&lity of producing •' without any superadded notion of quantity," and " fertility or productiveness" " the amount of births as distinguished from the capability to bear." For the reasons indicated in the text, it is better to adopt the terms " sterile " and " fertile" as contrasted, that is, as meaning " non-productive" and " productive" without reference to quantity, and the term ' fecund" as conveying the idea of quantity. The matter seems of sufficient importance to abandon Duncan's usage. FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 2:ir Physiological or potential fecundity is, at. present, not ascertainable : what is discoverable is only actual fecundity. Both rise to a maximum and fall away, the latter .very early in life, while it is improbable that this is true of the former. The difference is theoretically (and of course practically) important. , The following definitions make the matter clear : — (i.) Physiological fecundity at a given age is the probability that a female of that age, subject to a definite degree of physiological risk, uniform for all ages, will reproduce. (ii.) Actual fecundity at a given age is the probability that a female of that age, subject to average actual risk (as modified by social traditions, etc., and also by reproduction itself, and not necessarily uniform for all ages), will reproduce. Inasmuch as physiological fecundity is probably not identical in populations of different races or nations, or even in populations of differ- ent localities and times, and is, moreover, dependent upon general health and mode and standard of living, the obtaining of its measure is in a high degree important, though at present impracticable. Actual fecundity is, naturally enough, different for married and un- married females. While it does not, even with married females, measure without correction the urgency of the reproductive impulse, or in un- married females measure the force which this impulse opposes to restric- tions created by social environment, it throws, as we shall later see, important light on this question. 3. The measurement of reproductive efficiency. — ^The determination of an unequivocal method of measuring the reproductive efficiency of a population is not without difficulty for the following reasons, viz., that — (a) The life of women varies in duration ; (6) The reproductive period is only a limited portion of it ; (c) FertiUty and fecundity are neither uniform for all ages, nor for all women ; (d) It appears to be qualified by the age of the associated males ; (e) Marriage and child-bearing initiate at different ages ; (/) Reproductive efficiency must take account of the duration of life of the children ; and that (g) The exercise of the reproductive function is subject to ad- ventitious influences. By way of enforcing the penultimate point, it may be noticed that gener- ally a high birth-rate is associated with a high rate of infantile mortality, and the rate measured by taking account only of survivors at the end of one year or other prescribed period may give quite a different indication to that derived from births only. The following outline of various schemes of measurement, some of which have already been dealt with, will indicate the nature and limitations of each. 236 APPENDIX A. Rate JtEASCRF.n by- Numerator. Denominator. Deduced Result known as- Keniarks. Total births, B \ Total popula- ation. P ; Crude btcth- I rate, B/P Is dependent on age, sex, and conjugal constitution of total population, and there- fore not strictly comparable as between different populations ; it measures merely one element determining increase. Total births, B Total female population, F Birth-rate re- ferred to total number of women, B/ F la dependent on female population onlj' and is affected of course by the age and conjugal condition of that population. Tot-al liirths, B \ Female popula- j Birth-rate re- Indicates reproductive efficiency of all tion of repro- ferred to "women within the reproductive period, ductive age women of re- Owing, however, to the Umits of this period i (viz., from productive being iU-deflned at the initial and terminal about 10 to age only ages, to the largeness of the number of 60), F', say BJ F' women at those ages, and to the fact that itis dependent on the age-constitution with- ! in the group chosen to represent the repro- ductive age, the rate is not as definite as is I ' desirable. The denominator, however. Is a good crude measure of the potential of I I reproductiveefficiencyof the population. Births in each I age-group, B^ The women in same groups, F, Birth-rate re- ferred to women of each age- group In question, Is uncertain for comparison because the ratio of married to umarried women may vary, and the relative frequency of mater- nity in each is not identical. Nuptial births iu each age- group, B\ JIarried women in same age- group, M^ Nuptial mater- nity rate for each age- group, B'^/M^ Shews only the average frequency of maternity (average probability of mater- nity) for married women in each age-group. E>:-nuptial births in each age- group of un- married women, B",. Unmarried women in age-group. Ex-nuptial maternity rate for each age -group. Shews only average frequency of mater- nity (average probability of maternity) for unmarried women in each age-group. Appropriately weighted sum of birth-rates of the married and un- married Unity llodifled *' Nuptial Index of Natality" This attributes tlie reproductive facts of an existing population to a supposititious " standard" population, in which the re- lative number of married and unmarried females is the general average (norm) for the groups of populations to be compared. The comparison so attained may be re- garded a suitable comparative measure of reproductive efficiency (natality). 4. Natality tables. — The preceding methods of measuring productive efficiency are all more or less defective. A more satisfactory scheme is to construct a monogenous age-group " natality table" for married, and one for unmarried, females . Such tables shew for each age the probability of the occurrence of a birth and the average number of children per con- finement : see hereinafter. This, without doubt, is a more definite method, and stands in much the same relation to statistics of births, as a mortahty table does in relation to statistics of deaths. It is, however, not perfectly satisfactory, because, as already indicated, it would appear that the age of the father as well as that of the mother affects the probabil- ity of maternity. This will be dealt with hereinafter. Tables of digenous natality, i.e., double-entry tables, shewing the natality for every com- bination of age, are more complete and exact, and would be perfectly so, if the fertility at any age were unaffected by the number of previous con- finements. This, however, is probably not the case. These matters will be dealt with in the various sections and tables hereinafter in this part. FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 237 5. Norm of population for estimating reproductive efficiency and the genetic index. — ^In order to eliminate the effect of variations in the con- stitution of populations, it is desirable to establish on as wide a basis as possible the norm of its female conjugal constitution, preferably for every 5 years of the reproductive period. This norm would shew for a total of 1,000, 10,000 or 100,000, etc., women of all reproductive ages, the number aged 10-14, 15-19, 55-59 ; that is from the 10th to the 59th year of age inclusive.* For each age -group there would be (at least) two classes, viz., the "unmarried" which might include widows and divorcees not remarried, and the " married." If, then, to these numbers in the age-groups of the " married" we attribute the nuptial birth-rates* and compute the births, and to the " unmarried" we similarly attribute the ex-nuptial birth-ratesf, which are actually experienced by any popula- tion considered, we shall have comparable measures ; and the aggregate (divided if desired by 1,000; 10,000 or 100,000, etc.) will be the " Index of Natality" based on the women of reproductive age. In short, the birth-rates actually experienced in the various age-groups of females of reproductive ages, for a series of populations to be compared as regards reproductive efficiency, are attributed to a common standard population (the norm). The sums in the various cases are the comparable measures of reproductive efficiency. Symbolically this may be described as follows : — Let^i andp'i, p^ and p'^, etc., denote the ratio of the married and of the unmarried respectively in age-groups 1, 2, etc., to the total number of women married and unmarried of reproductive ages in the norm or standard population ; that is, to the total of all the reproductive groups of that population. Then the sum pi+p2+- ■ ■ ■p'i+p'2-\-- . . • =1- Hence the index of natality, v, which measures reproductive efficiency, is simply — • (453) v=I:1:{pP)+2TAp'P') where |8 denotes the nuptial, and j3 ' the ex-nuptial, birth-rate based upon the numbers of the married and unmarried respectively, and not upon the total population of each group. In practice these results may of course for convenience be actually multiplied by 1,000, or any higher number. This index of reproductive efficiency we shall call the genetic index. It is formed in a manner identical with that adopted to determine the index of mortality. 6. The NataUty Index. — ^Following a procedure similar to that dealt with in last section, let gji and g''i, 9^2 3'Udg''2, etc., denote the ratio in the standard population of the married and unmarriedj respectively to the * By dividing the nuptial births in each age-group by the mean number of married women in that group, b„,/M. f By dividing the ex -nuptial births in each age-group by the mean number of unmarried women in that group ; 6„ /U. When desirable to distinguish them " never married " may be used instead of " unmarried," the latter would include " widowed " and " divorced." I See preceding , note. 238 APPENDIX A. total of the standard population. Then these quantities will be smaller than^i, p'l, etc., in the ratio of the sum of all females of reproductive age in the standard population to the total standard population, male and female. Hence if we attribute to each age-group-ratio the birth-rate experienced in the population to be compared, we get a total also smaller in the same ratio. This then would give the nataUty -index v ' That is — (4M)....v' =S(qP)+S{q'^') = P' where P' denotes the females of reproductive age in the norm, and P denotes the total population, male and female, in the norm. 7. Age of beginning and o! end of fertility. — ^The determination of the age at which fertility begins and ends is of importance, and also the range of the reproductive period, which, of course, may not extend in individual cases from the initial age to the terminal age for a large population. What will be discussed here is the latter. The limits may best be deter- mined from the usual statistical data by considering the nature of the frequency as the limits are approached. Keeping in view the fact that the numbers from which the experience is drawn do not vary appreciably, the absolute numbers may preferably be used for judging the age-terminals We get, therefore, for the old-age limit the following results for the period from 1st January, 1907, to 31st December, 1914, for Australia, the popula- tion being nearly o milUons. TABLE LXXT — ProbabiUty o£ Birth in Old-age, Australia, 1907 to 1914. Age of Mothers ' i 1 1 , < 1 Line Nuptial and 48 49 50 51 ' 52 53 54 55 ; 56 , 57 58 1 59 60 Totals. No. Ex-nuptial. 1 1 j 1 No. of births ill i 1 ' ; 1 1 ' '! 8 years 322 113 39 13 6 5| 3, 2| li 11 0' 506 2 Decrease at the 1 ■ rate of e" 319 117.S| 43.2. 15.9 5.8i 2.1 0.8, 0.3 0.1 0.04 0.014 .0053 .0020 504.613 3 1 Decrease at 1 1 j varying rate . . 322 113.4 42.0 16.4 6.8 3.0 1.4 0.7 0.4| 0.2 0.1 0.069 0.053 606.522 4 JElatio of decrease 2.84 2.70 2.66 2.42 2.28 2.14 2.00 1.86 1.72 1.58 1.44 1.30 5 " Equivalent 1 ' , nnmhei" of , 1 1 married women 16938 16105|15113;13898|12759 11716 10819 ,9940 8989 8071; 7269 6608 6033 6 Probability per I i 1 100,000* 2,377 877' 323 117i 59 53' 35i 26 14 15 7 Harried women ! i of same age 1 ; ' ' per annutnt . . 2,377 877 320! 117 71; 1 49 351 25 171 10 4 ? 2' ? 1 • Crude result. t Smoothed result, see formula (464). The above results indicate that towards the end of the child-bearing period the numbers decrease (above 48 years of age) roughly at about the rate e*, where x is the number of years ; see line 2. This at least holds from 48 to 52, when it would appear that the decrease is much more slow. A closer correspondence can be had by forming the numbers according to a formula varying the rate of decrease such as — (455) »^^i = {2.84-0.14 (a;- 48) i n, where n^ denotes the number of mothers of age x, last birthdaj"-. FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 239 The figures in line 1 in Table LXXI. are 8 years' experience of nuptial and ex-nuptial births with women of from 48 to 60 years of age in Australia. During this time there were 476 of the former to 26 of the latter, the number of married and unmarried females of the ages men- tioned being respectively 136,781 and 21,615, giving one case of maternity in 287.3 and 831.3 women, respectively. The frequency of maternity with unmarried women between the age-Hmits in question is thus 0.346 that of married women (or that of married women is 2.89 times that of unmarried women). If, therefore, we add to the number of married women 0.346 times the number of unmarried that will be the total " equivalent number" of married women to whom the cases of maternity can be ascribed. These, divided into one-eighth^ of the numbers on line 1, give the crude probabilities of maternity for married women of the ages in question. The values, as calculated from the data, are given in line 6 ; the smoothed values obtained from these are given in Une 7. Although a probability is given for age 58, the actual fact is that in over 7,000 possible cases (see line 5) no birth occurred ; 57 is the greatest age at which a birth actually occurred. The values are shewn as curve A and on a larger scale, as curve B on Pig. 66. It will be noted that the continuation of the curve for ages 49 to 51 (see a b) on the figure, suggests that 53 is the age at which the value approaches z^ro, point c, and the curve for ages, 51 to 60, b d in figure, seems to be quite a different curve. No simple exponential relation, however, will bring these two curves under a single formula. 2 See page 244 for Pig. 66. Por .the lower limit we have the following data, viz. : — TABLE LXXn.— Probability of a Birth in Early Age, Australia, 1907-1914. Line Age. 11 12 13 14 15 16 17 18 1 Nuptial births, 8 years • 4 30 170 1,138 4,062 11,761 2 Ex-nuptial births, 8 years 5 21 126 537 1,500 2,980 4,504 3 4 Total births, 8 years . . Ratio of ex-nuptial to nuptial births . . 5 00 25 5.2 156 4.2 707 3.16 2,638 1.32 6,942 0.73 14,265 0.38 5 Married women . . 1 18 93 349 1,145 2,651 6 7 8 f Never married " women Probability of nuptial maternity per annum per 1,000 Probability of ex-nuptial maternity per annum per 1,000,000 unmarried women 42,222 42,001 1.6 42,071 ? 500 6.6 42,484 ? 208 37.1 43,273 228 155.1 43,915 408 427 43,813 443 850 42,854 576 1,313 1 Approximately, see § 8, p. 240. ' Results deduced from the initial value 2377 by means of the formula — "x+l ={2-75-0.15 (a;-48)} ,.^. would bo in substantial agreement with (455), and are as follows : — 2377 864 332 136 59 27 14 7 4 3 2 1 They are less probable, however, than Jbpse. given on line. 7 in the table. 240 APPENDIX A. The results on line 8 do not need smoothing. Those on line 7 for the ages 13 and 14 are, of course, very uncertain, the normal values would probably be much smaller than 200. It is evident from the above, that the cases of ex-nuptial maternity throw most light upon the question of the com- mencing age of fertility. These are shewn on line 2, and will be given very nearly by the equation. ^ (456) nx_i = ; 1.50 + 0.50 (18— x); n^. The results are shewn as curve D, and on a larger scale as curve E, on Fig. 66, on page 244. The general result of the investigation as to the terminal condition.s is that the null-points can be taken as say 11 and 60, the values being very small from ages 53 onward, and from 1 1 to 12.* The initial null -point is consistent with the curve of frequency of the first menstrual appearance, which would give a null-point of about 9 years' and a maximum just after 16 years of age are attained. The curve as shewn in Fig. 66, curve C, gives, according to AVhitehead, the group-numbers of single year age- groups for a tots^l of 4,000 cases under observation. These group-num- bers are shewn by small circles, see p. 244. 8. The maternity-frequency, nuptial and ex-nuptial, according to age, and the female and male nuptial-ratios. — Let g, m, and u, denote respectively the number per annum (i.) of brides, (ii.) cases of nuptial maternity ; and (iii.) cases of ex-nuptial maternity, and also let M and U denote the number of married and " never married " women among whom the latter occur. These numbers are given for each age from 12 inclusive onward, in Table LXXIIL, see columns (ii.J, (iii.), (iv.), (\'i.), and (^^i.), or g, m, u, M and TJ. The numbers are for 8 years, and the mean population from which they are drawn is about 8.0406 times that of the moment of the Censu.s. viz., 3rd April, 1911. Hence the epoch can be regarded as the date of the Census, and the numbers have been divided by 8.0406 to obtain the annual equivalent. 1 If we take 4500 as the number of ex-nuptial births for the age 18, we shall obtain 4.0, 27.7, 145.6, 545.8, 1500.3, 3000.0, and 4500, instead of the numbers ehevm on line 3 in Table LXXII. 2 At Budapest, J. Kdr&si records two mothers at 54, one at 56, and one at 57 in 4 years ; vide, Phil. Trans. 1895, B., p. 794. In Edinbvu-gh and Glasgow Matthews Duncan records for the ages 51, 52 and 57, and for an aggregate of 16,301 married mothers, 2, 4 and 1 respectively, p. 9 of his "Fecimdity, Sterility, &c." 1871 Edit. C. Ansell in 1874, vide his " Statistics of Families," regards an alleged case at 59 as needing verification. Tauffer, of Budapest, in 2083 cases, records one at 54. lu handbooks of Forensic Medicine, Casper -Liman mentions one case at 54 ; one is men- tioned by Hofmann at 55 ; see Phil. Trans, loc. cit. C. J. and J. N. Lewis' " Natality and Fecundity," published 1906, out of 84,971 cases of births in Scotland in 1855, give for the ages 15, 16, 17 and 50 and upwards to 58 ; the following results, viz. : — Ages 15. 16. 17 ; 50. 51. 52. 53. 54. 65. 56. 57. 58. Numbers 3. 23. 132; 16. .5. 7. 1. 3. 2. 1. 1. 2. ' See " Sterility and Abortion," Whitehead, p. 46, or M. Duncan, op. cit., p. 32. FERTILITY, FECUNDITY. AND REPRODUCTIVE EFFICIENCY. 241 The ratio (e) of ex-nuptial to nuptial cases of maternity is found by- dividing the values in column (iv.) by those in column (iii.) in Table LXXIII. That is to say— (457) e = u / m. The ratio of " brides" to " unmarried " females, or to females " never married " given in column (viii.) of the table, may be called the "female nuptial ratio " (g) according to age, and is given by — (458) 9 =g/ U the total number of brides being the same as the number of marriages J in (400), p. 176. Suffixes will denote the age to which the ratio refers. The values a are the probabilities of marriage according to age of the unmarried. This probability corresponds to a mean of the marriage- rates of 0.008403, and to a marriage rate over all the eight years of 0.00842863.^ For any particular year the distribution according to age will therefore approximately be in the ratio of the crude marriage rate for the year in question to that above ; expressed ordinarily, say as — ^*^^^ ^' ^u-qM^ n being calculated as indicated by (400), p. 176. The greatest number of never married appears to be for the year be- tween the ages 16.32 to 17.32, the number being about 43,950. Similarly the greatest number of brides appears to be for the ages 21.90 to 22.90, the number being about 27,955. The curve shewing the number of brides of each age is curve F, Fig. 67, and that shewing the number of the females "never married" is curve G of the same figure ; G' and G" shew the terminal values on a larger scale. The circles with crosses denote the positions of the data when corrected for the error of statement of age at marriage ; see pp. 193-6 hereinbefore. The crude results are shewn by circles on E', G, G ' and G". It will be seen from these terminal values that there is considerable regularity in the curve even for advanced ages (see p. 244). The " male nuptial ratio," according to age, is, similarly to (458) and (459)— (^««) t,=./F;or(461) ^'=y- ^^^Z The values are given in Table LXXIII., the crude results being shewn in column (xiv.). The curve shewing the number of bridegrooms of each age is curve W, Fig. 70, and that shewing the unmarried males is curve V of the same figure. V ' and V ' ' shew the terminal values on a larger scale . The smoothed values of the probability g', and u' are given in columns (xviii.) and (xix.) of Table LXXIII. Expressed per thousand, as is usual, 8.42863. 242 APPENDIX A. TABLE LXXni.-Shewing the Numters of Brides and Bridegrooms and the Cases of Nnptaal and Ec-nuptial ^^^^^ I^ 1907-191^ Australia, and the Numbers of Married and Never Married Males and Females, at t^eCwwus of 3rd AjwU, 1911. Shewing SthrProbabUities of Marriage for Never Married Males and Females, and the Probability of Nuptial and Ex-nuptial Maternity, and Ratios Dependent upon these. >. es ■— ^ o IS e "C*^ s§ CJ T-l ^ u < 1 ■a Unspecified, 111. Total including the Unspecified, 301,922. For notes see next page. FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 243 9. Nuptial and ex-nuptial maternity and their frequency-relations. — The crude rate, according to age, of nuptial and of ex-nuptial maternity is found by dividing the number of cases of maternity of each kind by the number of married or of " unmarried " or " never married " women. That is to say ^j, u, and § denoting the probability of maternity, according to age, respectively of the married, the never married, or of both combined, we shall have : — (462) V =m / M; (463) u =u / U; (464) ^ = (m -\- u) / (M + U). The relation, according to age, between the ex-nuptial and nuptial rates, is — (465)....e = «/i. = ^/^ = -^ . _ These crude rates and their ratio to each other are given in Table LXXIII. for the whole reproductive period in columns (ix.), (x.) and (xi.). The smoothed values are given in columns (xv.), (xvi.) and (xvii.). The graphs of the numbers of cases of nuptial and of ex-nuptial maternity are shewn respectively by curves H and I, on Fig. 68, the dots in the former case, and the crosses in the latter, denoting the crude results. The ratio of the ex-nuptial cases to the nuptial cases are shewn by curve J, and on a larger scale by curve J, ' Fig. 68. The nuptial and ex-nuptial maternity-rates are shewn on the same figure by curves K and L, the dots in the former, and the small circles in the latter indicating the crude results (see p. 244). It should be noted that m and M in (462), etc., are not necessarily homogeneous, since each will contain, though in unequal proportions, primiparous and multiparous women, and these will have been subject to risk for unequal periods. Moreover the multiparse may have given birth to very different numbers of children. If, therefore, the probability of maternity is affected by previous issue, the value of p must be regarded as merely a cr%de probability. An exact probabihty would have to be defined in categories according to the age, the number of previous issue, and the length of exposure to risk. This wiU appear more clearly in the theory of fertility and sterihty. For this reason the values given of jj and u in Table LXXIII, are for the " average risk" of the " average married woman" or the " average never married woman" during twelve months, and takes no account of variation of the " risk" according to the age of the husband. In section 11 hereinafter it will be seen that the maxima vary. Notes to Table LXXIII. on preceding page. • If the corrections referred to in Part XII., § 15 and 16, pp. 193-6, be applied, these numbers become 14,004; 19,580; 23,678; 26,927; see formula (407). This will change the ratios in column (viii.) from .03844 to .04064; .05500 to .05937 ; .06557 to .07674 ; and. 11733 to .09706. t The maximum is for the cenfroZaffe 18.73, that is for the group of ages 18.23 to 19.23, and the amount is 0.4849. t The maximum is for the central age 22.50 ; that is for the group of ages 22.00 to 23.00, and the amount is 0.01885. I The ex-nnptial births are attributed to the " never married," but may, pertiaps, be equally well attributed to the " unmarried," that is the " never married " together Mith the " widowed" and " divorced." 244 APPENDIX A. Terminal Frequencies of Fertility; Frequency of Nuptial and Ex-nuptial Maternity; Probability of Marriage of both Sexes at each Age; etc. Fig. 70. Fig. 67. ,10 so 30 40 60 70 80 llfjl COODD Li h ^ ^ %^ 10 13 14 16 Carves A,B>C. 48 SO 63 64 Caires D,E Fig. 69. Fig. 66. Fig. 68. Fig. 66. — Ourvea A and B shew the terminal age of fertility. Curves D and E shew the initial age of fertility. Curve C shews the frequency of the ap- pearance of menstruation according to age. Fig. 67 — Curve F shews the niunbers of brides at various ages. Curves G, G' and G"'shew the numbers of the " never married " at various ages. Fig. 68. — Curve H shews the number of cases of nuptial maternity, and Curve I those of ex -nuptial maternity at each age. Chirves J and J' shew the pro. portion of ex-nuptial to nuptial cases of maternity at each age. Curve K shews the nuptial and L the ex-nuptial rates of maternity at each age, the ex -nuptial rate being determined by attributing the births to the " never married." Fig. 69. — Curve M shews the ratio of the ex-nuptial to the nuptial rates of maternity at each age. Curve N is the ratio of the brides at each age to the " never married females " of the same ages. Curves O and O' are similarly the ratio of the bridegrooms at each age to the " never married males " of the same ages, curve O' being displaced one division (10 years) to the right so as not to be confused with curve N. Pig. 70. — Curve W shews the number of bridegrooms of each age, and V, V and V" the number of " never married males " at each age. In all the above cases the age is the "age last birthday." Fertility, fecundity, and reproductive efficiency. 245 10. Maximum probabilities of marriage and maternity, etc. — ^The position and amount of the maxima determined from the smoothed results in columns (xv.) to (xix.) of Table LXXIII. are as follow : — Table LJilXIV. — Maximum Probabilities, Marriage and Maternity. Maximum probability of — Year -group from — Ainoi.mt. Age. Age. Nuptial maternity 18.45 19.45 .0486 Ex-nuptial maternity . . . . 22.00 23.00 0.01835 Ratio of ex-nuptial on nuptial Probably no maximum value point of maternity inflexion at — 25 to 26 0.0510 Marriage of women . . . . 24.52 to 25.52 0.12632 Marriage of men .. .. 27.5 to 28.5 0.11320 The maxima are for the two heterogeneous groups " nuptial" and " ex-nuptial" aggregated according to age merely. In the next section it will be shewn that the maxima are dependent upon age at marriage. The largest number of marriages of brides would appear to be for the ages 21.9 to 22.9, and to be about 28,000 in 8 years ; and the largest number of marriages of bridegrooms, for the ages 24.8 to 25.8, the number being about 25,000 in 8 years, the total mean population aggregated for the years in question being 35,821,000 persons. The largest number of cases of nuptial maternity occurred for ages 26.12 to 27.12, the number being about 55,500 in 8 years, and the ratio at the crude maximum con- sequently 0.3182. The largest number of cases of ex-nuptial maternity occurred for the ages 19.5 to 20.5, the number being about 5,400 in 8 years, and the ratio at the crude maximum of cases, therefore, 0.01691. The question of a more accurately defined maximum wiU be con- sidered hereinafter. 11. Probability of a first-birth occurring within a series of years after marriage. — ^To determine the variation of initial fertility with age, the initial probability of maternity may be deduced by ascertaining primarily the number of women at different ages who were married during a given period. Then, tracing these through the first portion of their married life, the respective periods which elapsed after marriage before they gave birth to their first living child may be ascertained. Tor this purpose the six-year period, 1909-14, was brought under observation, the experience being all cases in the Commonwealth of ' Australia within a series of years, viz., 6 after marriage. Owing to mis- statements regarding age, however, the number of brides registered at each age during the several years under observation required correction. It was found that, if the actual numbers of brides registered at ages 18, 19 20 and 21 years were accepted, without adjustment, anomalous results would be obtained. Evidently serious errors existed owing to brides of 18, 19, and 20 years overstating their age as 21, and therefore the numbers of brides upon which the rates of fertility should be founded 246 APPENDIX A. needed correction. A special type of smoothing of the number of brides of 18, 19, 20 and 21 years to remedy the misstatement of age had there- fore to be adopted.^ A similar misstatement of age had evidently occurred in the case of mothers (registered as being 19, 20, and 21 years of age), who gave birth to a first-bom child during the period 1909-14, and the numbers conse- quently had also to be smoothed, so as to eliminate the effect of mis- statements in the age of mothers.^ Tables were compiled shewing the mean number of brides of each age in any year and in the year immediately preceding ; and for the same ages the number of first confinements in successive years of duration of marriage. Assuming then that the migration elements balanced each other, the table gave a series of results shewing for the years 1909 to 1914 inclusive the aggregate number of brides of each age at marriage to which the aggregate number of first confinements could be referred, hence the ratio of the latter to the former gave the probability required.* ' The justification for this smoothing is really that the probability of a mis- statement of age is very great, and the probability of some physiological or other cause, for the anomaly, is relatively negligible. ^ The following are the unadjusted and adjusted figures : — • Age. Nuptial First Births, according to Suc- cessive Years of Duration after Marriage. Number of Brides to whom the Births may- lie ascribed, according to Successive Years of Duration after Marriage. Total 0-1 1-2 2-3 3-4 4-5 5-6 Total 0-1 1-2 2-3 1 3-4 4-5 5-6 18 7,568 5,899 1,291 262 81 29 6 10,159 10,159 8,331 6,513! 4,735 3,039 1,484 7,568 5,899 1,291 262 81 29 6 10,736 10,736 8,802 6,880 5,003 3,213 1,571 19 11,625 9,071 1,943 429 118 48 16 13,838 13,838 11,364 8,899 6,463 4,156 1,998 11,228 8,761 1,877 414 114 46 16 14,902 14,902 12,227 9,557 6,917 4,457 2,177 2U 13,596 10,141 2,618 556 202 56 23 15,496, 15,496 12,737 9,978 7,244 4,657 2,241 14,400 10,741 2,773 589 214 59 24 18,100 18,100 14,860 11,630 8,453 5,475 2,675 21 17,507 12,613 3,699 823 262 81 29 24,850 24,850 20,309 15,838 11,520 7,498 3,702 17,100 12,320 3,613 804 256 79 28, 20,600 20,600 16,848 13,158 9,588 6,264 3,002 The upper number is that furnished by the registration records, the lower is that which was obtained after adjustment. The only adjustment deemed essential as a preliminary is for these ages, viz., 18 to 21. For aU other ages the results are as given by the unadjusted data. » The following illustration of the method of compiling will sufiSce :— Year. Age at Mar- riage. Mean No. of Brides for Year and pre- ceding Year. Number of First Confinements in suc- cessive Years of Duration of Marriage. Duration of Marriage, 0-1. 0-1 1-2 2-3 3-4 4-5 5-6 Age. Brides. Confine- ment. Ratio. 1909 26 27 26 27 26 27 (Frou 1,864 1,563 2,076 1,616 2,268 1,781 1 these th 1,002 835 1,047 853 1,171 967 s totals 443 417 551 444 645 527 on the 160 107 219 149 212 178 right w 71 66 79 73 101 88 ere fon 54 39 46 41 ned.) 15 22 26 27 13,637 11,054 7,279 5,721 0.5338 0.5176 Duration of Marriage, 1-2. L910 26 27 11,068 9,004 3,095 2,566 0.2795 0.2850 1911 Duration of Marriage, 2-3. 26 27 ■ 8,571 6,971 800 619 0.0933 0.0888 I*ERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 247 The probabilities so ascertained are shewn on Table LXXV. up to 6 years. The crude results are shewn by the dots on Pig. 71, on which the curved lines give the smoothed results, the corresponding numerical values appearing on the right hand side of the table. TABLE LXXV.— Probability o£ a Nuptial First Birth occurring within 6 Years of Marriage, Based on Australian Data, 1909 to 1914. Crtjde Results. Adjusted Results. Probability of Giving Birtli to a First Child Probability of Giving Birth to a First Child for Age lor a Duration of Marriage of — a Duration of Marriage of — Age last last Birth- less less less Birth- day. than 1-2 2-3 3-4 4-5 5-6 than than 0-1 1-2 2-3 3-4 4-5 5-6 day. lyr. yrs. yrs. yrs. yrs. yrs. 6 yrs. 1 yr. yrs. yrs. yrs. yrs. yrs. yrs. 11 .0000 .0000 .0000 .0000 .0000 .0000 .0000 11 12 .1308 .0963 .0217 .0066 .0030 .0020 .0012 12 13 .2568 .1881 .0433 .0131 .0060 .0039 .0024 13 14 .3781 .2755 .0647 .0195 .0091 .0058 .0035 14 15 .3324 .1233 .0470 .0353 .0278' .4946 .3585 .0860 .0258 .0121 .0076 .0046 15 16 .4352 .1042 .0424 .0177 .0149 .0075 .6219 .6063 .4370 .1073 .0321 .0150 .0093 .0056 16 17 .4979 .1271 .0413 .0128 .0141 .0053 .6985 .6975 .4985 .1263 .0377 .0176 .0108 .0066 17 18 .5495 .1467 .0381 .0162 .0090 .0038 .7633 .7770 .5485 .1455 .0432 .0199 .0123 .0076 18 19 .5879 .1535 .0433 .0165 .0103 .0073 .8188 .8414 .5800 .1664 .0497 .0229 .0138 .0086 19 20 .5934 .1866 .0506 .0253 .0108 .0090 .8757 .8856 .5950 .1854 .0551 .0252 .0153 .0096 20 21 .5981 .2144 .0611 .0267 .0127 .0093 .9223 .9176 .5958 .2051 .0614 .0280 .0168 .0105 21 22 .5919 .2301 .0675 .0299 .0151 .0122 .9467 .9429 .5908 .2247 .0673 .0306 .0182 .0113 22 23 .5800 .2425 .0783 .0314 .0173 .0094 .9589 .9619 .5819 .2423 .0730 .0331 .0195 .OlSl 23 24 .5545 .2466 .0827 .0344 .0231 .0130 .9543 .9730 .5688 .2569 .0785 .0354 .0206 .0128 . 24 25 .5314 .2636 .0815 .0375 .0235 .0158 .9533 .9771 .5533 .2679 .0831 .0378 .0216 .0134 25 26 .5338 .2795 .0933 .0404 .0254 .0081 .9805 .9750 .5357 .2754 .0872 .0402 .0225 .0140 26 27 .5176 .2850 .0888 .0458 .0252 .0141 .9765 .9667 .5168 .2795 .0903 .0423 .0233 .0145 27 28 .5037 .2677 .1013 .0465 .0260 .0126 .9578 .9530 .4967 .2813 .0922 .0439 .0240 .0149 28 29 .4548 .2774 .0836 .0359 .0198 .0107 .8822 .9330 .4766 .2792 .0929 .0446 .0245 .0152 29 30 .4686 .2421 .0898 .0498 .0224 .0107 .8834 .9075 .4545 .2751 .0930 .0448 .0247 .0154 30 31 .4602 .3084 .1003 .0447 .0238 .0178 .9552 .8745 .4310 .2668 .0923 .0446 .0245 .0153 31 32 .4191 .2464 .0873 .0368 .0220 .0132 .8248 .8381 .4073 .2571 .0907 .0440 .0240 .0150 32 33 .4057 .2422 .0825 .0428 .0217 .0194 .8143 .7938 .3789 .2463 .0883 .0426 .0231 .0146 33 34 .3310 .2526 .0928 .0353 .0232 .0204 .7553 .7411 .3487 .2319 .0843 .0407 .0217 .0138 34 35 .3036 .1950 .0771 .0387 .0113 .0155 .6412 .6748 .3123 .2135 .0784 .0382 .0198 .0126 35 36 .3024 .1820 .0724 .0395 .0236 .0061 .6260 .6063 .2768 .1935 .0718 .0354 .0178 .0110 36 37 .2241 .1910 .0741 .0341 .0173 .0000 .5406 .5367 .2423 .1730 .0650 .0315 .0157 .0092 37 38 .1919 .1576 .0634 .0252 .0105 .0144 .4630 .4662 .2088 .1520 .0573 .0276 .0134 .0071 38 39 .1844 .1391 .0406 .0275 .0087 .0000 .4003 .3946 .1755 .1303 .0490 .0237 .0110 .0051 39 40 .1436 .0986 .0520 .0131 .0049 .0000 .3122 .3245 .1426 .1082 .0415 .0198 .0088 .0036 40 41 .1323 .0870 .0336 .0194 .0076 .0000 .2799 .2558 .1111 .0863 .0333 .0158 .0070 .0023 41 42 .0756 .0627 .0211 .0135 .0000 .0073 .1802 .1951 .0855 .0656 .0254 .0119 .0053 .0014 42 43 .0669 .0665 .0131 .0051 .0000 .0083 .1599 .1411 .0634 .0474 .0178 .0080 .0037 .0008 43 44 .0384 .0462 1 .0064 .0030 .0000 .0000 .0940 .0937 .0441 .0321 .0116 .0041 .0014 .0004 44 45 .0258 .0066 .0086 .0000 .0000 .0000 .0410 .0622 .0296 .0220 .0070 .0022 .0012 .0002 45 46 .0400 .0199 .0147 .0035 .0012 .0006 .0001 46 47 ' * .0252 .0131 .0094 .0019 .0005 .0003 .0000 47 48 ■" .0159 .0093 .0056 .0007 .0002 .0001 48 49 ■■ .0095 .0062 .0029 .0003 .0001 .0000 49 50 .0026 .0031 .0000 .0000 .0000 .0000 .0057 .0053 .0040 .0012 .0001 .0000 SO 51 .0028 .0023 .0005 .0000 51 52 .0013 .0011 .0002 52 53 .0006 .0005 .0001 53 54 .0002 .0002 .0000 54 55 .0001 .0001 55 The probabilities in the table apply to the total number of women married at the given ages, not to the survivors after the series of years under observation. The probabilities are of course cumulative, that is to say 248 APPENDIX A. the probability, qJ)„, that a first birth will occur before the end of the n-th year after marriage, is the sum of the probabiUties that it wiU occur during the fibrst, during the second, etc., up to and including the w-th year. Or (466) . ■oPn = oPi + iPz + -iPn 12. Maximum probabilities of a first birth. — ^From the smoothed results in the table, it will be seen that, as the interval to the first birth increases, the age of maximum increases. Thus the greatest probability of a first birth within the first year from marriage is for age at marriage 21.24, during the year succeeding that of marriage it is at age 28.47, and so on as shewn in the following table, viz., LXXVI. TABLE IiXXVI. — Shewing the Age of Mazimuin Fiobability of a Fiist Biith. AustraUa 1909-1914. Interval Years. 0-1 1-2 2-3 3-4 4-5 5-6 0-1 0-2 0-3 0-4 0-5 0-6 0-7" 0-8 0-9 0-10 0-11 Vertex at (years) Corresponding to Median Age at Marajage Or to Median Age at Birth Probability By Formula (467) I 20.74 27.97 29.62 29.75130.03 30.14 20.74 23.75 24.52 24.91 30.25'30.53 30.64 21.24 I I 33.75 35.03 36.14 21.74 21.24 28.47 21.74 29.97 .5962 .2813 30.12 32.62 .0931 .0448 .0247.0154 .5962 25.06 25.16 25.22 25.26 25.30 25.32 25.33 I I tl I t 24.25 25.02 25.41 I i 25.25 26.52 27.41 .8259.9050 840 .920 9421 .947 25.56 25.66 25.72 25.76 25.80 28.06 28.66 29.22 29.76 30.30 9637 .9772 .9859 .9916 .9953 .960 .968 .973 .977 25.82 25.83 30.82,31.33 .9978 .9998 • The ratios 9050/8259, 9421/9050, etc., are 1.0958, 1.0410, 1.0229, 1.0140, which oontinued.are 1.0089, 1.0058, 1.0038, 1.0025, 1.0020, the factor of the last two figures converging to 52/80. Xhis, however, would give 1.0011 for 0-11. It is more probable, however, that the probability is of the type oP'„ + ^='oP'„ + m)/ (1+ m) where m may perhaps be taken even as unity, implying that the residual chance is reduced about one half each year. The matter requires special investigation. t These correspond to the values of the vertices on Fig. 71. It is worthy of note that the above results for 0-2 up to 0-8 are roughly given by the formula — (467). •oP'« = 1 - 0.16 -1 • in which n is the total duration of marriage. The figure (7 1 ) and table shew cleatly that the maximum is a function of the duration of marriage as well as of age. To find the maximum value for any durations to t the line of vertices C D on Fig. 71 must be followed, or during year-intervals < to i -|- 1, the line C E must be followed. Thus for age 20 last birthday, the duration is to 0.93, the probability is about 0.555. The graphic solution may also follow the method indicated in Fig. 71a, which needs no comment when examined in connection with Table LXXVT. FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 249 Probabilities of a flist-bitth during first 6 years of marriage. Interval from nuuriage. i-O ■'e-i!a 25J%^j >vi7fe icr ^'-'■' ^■^ '4 iti^d^A J/ 1 ,J^.,,,..r X ^ 4 '" a ' iivrX"^ ^\ f" IS Fig. , /!!i--^- ^h y ■ -'i r^^- --i--i ^. . v-i- - i ^ fi U '^'^'^T^tet,^'4,>.3;i A^ l--/--.^,,.. J^ i _____] _/:?^^N, .\ n.^^^-^-i ,^ .5--- -/-^ -.-- 1 \ 1 aijlInoQt 2o;;au5t ,0 --W--\-\- " lis VW - .4-- /. — --1-V __JL. __._ 1 ,___ u. ,, \ \^ :— - ■^ ' ^ \^ i :-x::-:'z:::X:- --^-^- "i i "" \ 1\\ ^^ M j' / VT'^;:7-a,;wArta: ,,7.„.,v, y;^ r^fi;^:;^/ 1 /^ J3--'"' ^^ .i-=;:===^:;====igi ]0 45 25 30 3S 40 Ages of mothers at marriage. Fig. 71. 13. Determination of the co-ordinates of the vertices. — ^The repre- sentatioTi of group-totals by means of integral functions of the values of the central abscissa of the group-base (central value of the interval) has been referred to in Part V., § 10, pp. 72, 73. In curves of the type which has just been considered, the results about the vertices may be closely represented by a curve of the second degree, and the curve itself may be regarded as defining the curve of group-totals for all values of the central abscissa (the abscissa of the middle ordinates of the group). In such instances the co-ordinates for the maximum-group may be very accurately ascertained from the tabular maximum together with the tabular values on either side of it. Let the maximum tabular value denote the point M on the curve, and the adjoining tabular values denote the points A, B, viz., the points on either side. Then, if the difference of the mean of the ordinates of the points A and B, and the ordinate of M be denoted by h, and the half difference of the ordinates of B and A be denoted by I, that is if — (468) A = 2/m - i (2/6 + ya) ; and i = 4 (y^ -y^) ; then we shall have — 12. I (4b9) Vmax ^= Vm H tt" i •^max = *m "T 2h 250 APPENDIX A. The proper maximum is greater than the tabular maximum by the amount l^/4}i, and its abscissa lies between that of the tabular maximum and the next highest tabular quantity distant from the former by the amount l/2h. The positions of the vertices have been computed in this way. It remains to be noted, however, that when the value of the abscissa indi- cates merely the " age last birthday," it is necessary to add the amount J to the value given by the formula in order to refer the co-ordinates to the middle values of the group-abscissae. Thus, in Fig. 71, the curves are plotted with the argument " age," i.e., last birthday, hence the vertex- value 20.74, see curve 0-1, and the maximum 0.5962, refer to the group of brides whose ages ranged between 20.74 years of age and 21.74 years of age. The middle value of the range is 21 .24, but the average value is not that. The probability 0.5962 applies to the brides whose ages were between 20.74 and 21.74. Of 10,000 such, 5,962 would give birth to a first child within one year of marriage. 14. Average age of a gioup. — ^The error of adopting the middle value of any range has been considered in Part XII., § 20, pp. 200-201. It is sometimes preferable to relate the values of the dependent variable, not to the middle values but to the average values of the independent variable. In such a case formula (416), p. 201, may be used. Let A, M, and B be three group totals on equal bases k (equal intervals on the axis of ab- scissae). The values of the co -efficients of a rational integral function of the second degree — the graph of which wiU represent, viz., the areas standing on the equal bases, the group-totals — may be found by the formulse of Part V., §§ 1 to 9., pp. 64-72. The weighted mean abscissa of the middle group may be denoted by x'^. If then we make the origin at 0, so that A is the integral of the equation a -f 6a; + cx^ between the limits and k, M the integral between k and 2k, and B between 2k and 3fc, then we shall have — (470) x„,= - ^' ' ■•^"'~ a + lbk + ^ck^ which may be put in the very simple form — (471) x^^€ =x,^+ ^k(B -A)/M. This is the same formula as (416). In general, therefore, it is sufficient to find the value of the abscissa to which a group may be referred by using the value of the group and of these on either side : see the results as to average interval in §§ 21, 24, etc., hereinafter. 15. Curves of probability for different intervals derived by pro- jection. — ^Reverting to Fig. 71, it may be noted that the probabihties of a first birth between 1 and 2 years, 2 and 3 years, etc., after marriage FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 251 may be derived for each age approximately by projection if the ratio of the aggregates and the position of the maximum are known. For ex- ample the faintly-dotted curve is the curve for the interval between 1 and 2 years after marriage derived by projection from that up to 1 year (0-1). The difference between the two curves is nearly negUgible. The following are thte relations between the curves : — Let X, y be the co-ordinates of any point P on a curve, and let x', y' denote the co-ordinates of what may be called the corresponding point P ', on a curve derived therefrom by drawing the line P P ' Q to cut the axis (OX) of abscissse in the point Q, so as to make the angle of inter- section therewith, XQP, equal to 6, and also the ratio QP '/QP equal to p. Then, if 6 and p be constant, the derived curve will belong to a family of curves of the type of the original, but differing therefrom in " skewness" if 6 be not 90°. The co-ordinates of any point P', viz., of the " corresponding point" on the derived curve are simply related to those of the point P on the original curve from which it was derived, being given by the equations — (472) y'=py\ x'=x-y(l -p)cot.e. Hence if the equation of the original curve be f{y) = F(x), that of the derived curve will be — (473) /(— )= F(x' -ky'}; in which k — cot. 6 {I — p)/p. To determine whether the succession of probabilities for 0-1, 0-2, 0-3, etc., and 0-1, 1-2, 2-3, etc., are rigorously derivable by projection would involve data embracing larger numbers and free from all uncertainty as to the effect of migration thereupon. 1 6 . Numbers of fiist-biiths according to age and duration of marriage. — ^There were in Australia during the years 1907-14 inclusive, 220,021 cases of nuptial first births . The records of these were compiled according to " age last birthday," and duration of marriage." Multiplying the numbers as compiled by a factor, that would make the total 1,000,000, the results are as shewn in Table LXXVII., compiled for single months of duration of marriage from 1 to 12 months, and for single years of duration of from 1 to 26. The table thus furnishes the distribution of 1,000,000 nuptial first births according to age and duration of marriage. The figures for the months are of course only one-twelfth of the figures which would be comparable to the yearly values. This distribution may be called the nuptial protogenesic distribution. 252 APPENDIX A. TABLE LXXVn.— Shewing the Number in 1,000,000 Nuptial First-births of Births occuiring for all Births occuiiing in Australia during the Years [STERVAL AITEK MaEEIAGE DUEINQ WHICH A BlETH OCCURS. AOE OF • MOIHEBS. O-I 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11 11-12 0-1 1-2 a-3 3-4 4^5 5-6 6-7 mths. mths. mths. mths. mths. mths. mths. mtlis. mths. mths. mths. mths. year. years. years. years. years. yrs. yrs. 12 IS " 5 " 5 " "■ " 5 "15 4 14 32 14 23 18 9 9 5 5 9 124 15 45 73 109 68 91 77 68 36 9 27 23 626 18 16 382 423 514 532 568 541 486 295 91 95 73 91 4,091 209 "l8 9 17 959 1,073 1,336 1,663 1,532 1,836 1,859 1,250 532 704 450 304 13,498 1,054 73 5 9 18 1,523 1,773 2,754 3,163 3,613 3,891 4,468 2,950 1,636 2,118 1,359 1,082 30,330 3,417 377 50 "'5 19 1,886 2,227 3,513 4,272 5,127 6,054 6,790 4,609 2,850 3,909 2,972 2,345 46,554 7,794 1,054 155 32 20 1,754 2,236 3,172 4,104 5,086 5,995 7,068 5,590 3,254 6,154 4,777 3,318 52,508 11,921 J'S?^ 382 64 27 9 21 1,877 2,309 3,454 4,640 5,704 6,981 8,331 6,613 4,613 9,122 6,845 4,772 65,261 16,125 2,950 682 227 36 23 22 1,532 1,827 2,682 3,445 4281 5,740 7,254 6,159 5,077 11,981 8,935 i'Z08 65,621 22,225 3,995 1,118 395 132 32 23 1,113 1,523 1,941 2,909 3,854 4,740 6,263 5,413 6,177 11,953 9,226 6,331 60,443 24,316 5,672 1,859 677 236 95 24 986 1,104 1,586 2,086 2,886 3,441 4,959 4,231 4,045 11,467 8,726 6,313 51,830 24,261 6,413 2,268 904 377 191 25 768 818 1,273 1,573 1,873 2,830 3,754 3,250 3,895 10,549 7,986 5,645 44,234 21,988 6,954 i'®l2 1,040 550 277 26 691 677 1,027 1,168 1,500 2,154 2,836 2,732 3,272 9,031 7,222 4,877 37,187 20,670 6,613 1^12 1,268 782 459 27 382 432 718 895 1,136 1,463 2,234 2,143 2,373 7,649 5,909 4,177 29,533 18,419 6,009 ^W hill 800 427 28 491 345 691 736 964 1,232 1,595 1,677 2,091 6,372 5,140 3,450 24,784 15,315 5,508 H^i h^t 695 541 29 305 282 414 600 568 845 1,177 1,245 1,613 4,536 3,659 2,795 18,039 12,281 4,436 2,263 1,218 727 455 30 227 255 282 418 432 677 877 1,034 1,404 4,191 3,336 2,304 15,457 10,221 4,113 2,077 1,182 732 477 31 209 159 241 395 359 450 641 691 945 2,600 2,245 1,782 10,717 7,549 2,936 1,618 868 532 377 32 177 182 227 264 373 405 523 627 786 2,145 2,000 1,354 9,063 6,680 i-^^S 1,300 823 577 441 33 173 168 195 150 273 345 441 436 564 1,654 1,468 1,082 6,949 5,086 2,032 1,154 650 441 395 34 73 105 191 141 218 256 335 327 382 1,264 1,027 773 5,112 3,918 1,712 800 586 373 323 35 105 100 132 127 227 177 268 318 314 1,014 863 559 4,204 3,530 1,314 714 423 282 268 36 95 55 150 132 118 150 223 255 264 750 777 450 3,419 2,654 1,232 577 423 268 259 37 55 91 36 105 64 114 141 155 182 609 527 345 2,424 2,034 1,041 459 282 259 168 38 41 59 59 45 68 86 105 150 136 400 364 332 1,845 1,786 786 477 314 232 159 39 73 32 32 68 82 86 123 109 82 309 259 232 1,487 1,427 641 377 200 114 100 40 41 27 64 39 27 45 64 59 64 168 164 136 918 1,114 523 300 195 109 82 41 14 27 36 27 41 23 59 32 45 109 105 68 586 600 286 173 114 68 55 42 9 14 27 32 45 41 14 27 18 55 50 73 405 386 318 159 91 68 23 43 5 14 18 27 14 18 23 18 14 41 36 36 264 295 132 105 36 64 ?? 44 14 14 4 14 14 14 14 27 9 14 138 209 82 45 50 18 18 45 18 9 9 5 5 27 18 18 109 91 55 50 14 14 3 46 5 9 5 19 27 18 23 9 5 47 4 4 5 9 22 14 18 14 5 5 4 48 5 5 9 5 4 5 49 5 5 5 15 5 5 4 50 51 4 •• .. * S 4 9 ■• 52 •■ ■■ 4 4 •Age at Maximum 20.0 20.1 20.1 21.0 20.2 20.6 20.9 21.2 22.4 22.4 23.0 22.80 23.6 25.1 26.0 27.6 27.4 28.0 Mean Interval 0.0 0.1 0.2 0.3 0.4 0.5 0.5 0.6 0.7 0.8 0.9 1.0 1.5 2.5 3.5 4.5 5.5 6.5 Age at Marriage 20.0 20.0 19.9 20.7 19.8 20.1 20.4 20.6 21.7 21.6 22.1 21.8 22.1 22.6 22.5 23.1 21.9 21.5 do., smoothed 20.0 20.0 20.0 20.1 20.2 20.3 20.4 20.8 21.6 21.7 21.8 21.9 22.0 22.3 22.7 22.7 22.3 22.0 Frequency at Max.t(crude) 1,890 2,291 3,437 4,527 5,800 6,980 7,933 6,565 5,065 12,100 9,226 6,700 24,600 6,960 2,620 1,395 803 541 do.,(3mooth'd 1,890 2,290 3,440 4,530 5,800 6,980 7,930 6,560 5,070 12,100 9,230 6,700 72',520 24,600 6,960 2,620 1,400 800 640 Totals Smoothed . . 16,041 18,452 26,929 33,880 41,156 50,722 63,043 52,472 45,742 111053 86,569 61,789 607848 247676 71,816 29,354 14,908 8,541 5,700 Batio Max. to Total . . .1178 .1241 .1277 .1337 .1411 .1377 .1258 ,1250 .1108 .1089 .1066 .1084 .0993 .0969 .0893 .0939 .0936 .0947 • Age at beguming of year of maximimi. Add 0.5 year for the median age of the FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 2S3 durations of Mairiage up to 26 Years with Women of Ages 13 to 52 inclusive. Based upon 220,021 1907-1914 inclusive. Unadjusted Numbers. Interval after Marriaoe durino which a Birth Ooours. ToTAi Number op First Births. AOB OF 7-8 yis. 8-9 yrs. 9-10 yra. 10-11 yrs. 11-12 yrs. 12-13 yrs. 13-14 yrs. 14-15 yrs. 15-16 yrs. 16-17 yrs. 17-18 yrs. 18-19 yrs. 19-20 yrs. 20-21 yrs. 21-22 yrs. 22-23 yrs. 23-24 yrs. 24-25 yrs. 25-26 yrs. 1-26 years. 0-26 years. Mothers. 14 9 36 82 136 200 232 345 305 391 327 300 323 250 232 195 136 123 77 14 36 36 32 18 5 9 5 4 'i8 59 55 118 145 200 255 864 227 191 191 205 177 164 86 132 59 45 36 23 18 5 14 4 'is 14 65 77 127 127 827 177 191 177 136 150 95 86 123 64 32 41 36 14 19 .5 "9 9 18 18 64 36 82 114 118 805 123 145 105 145 77 50 64 32 27 9 4 14 18 5 27 23 86 45 100 77 132 73 100 77 82 59 32 23 9 9 5 5 4 5 14 18 41 55 32 91 109 100 105 50 77 73 91 27 18 14 4 5 9 14 41 27 68 82 114 45 59 18 45 59 9 9 9 9 5 9 9 41 36 27 41 32 41 32 27 14 I 18 9 4 5 9 23 36 68 41 36 23 23 14 9 18 14 4 9 5 23 18 36 45 9 27 23 9 5 9 14 5 9 9 14 32 64 18 18 18 18 9 4 4 5 14 9 9 27 23 14 23 5 4 "4 5 14 9 9 5 14 9 87 5 4 4 li 14 9 5 4 "5 9 14 14 9 5 •• "4 4 4 5 5 5 14 9 5 5 4 "5 4 "9 5 4 18 236 1,141 3,849 9,035 14,371 20,057 27,906 32,918 34,582 33,600 32,688 30,215 26,703 22,222 19,975 14,860 13,526 10,885 8,836 7,614 6,362 5,034 4,595 3,455 2,710 1,601 1,285 802 537 281 117 74 27 18 9 4 19 124 644 4,327 14,639 34,179 55,589 66,879 85,318 93,627 93,361 86,412 77,834 69,875 59,748 51,487 40,261 35,432 25,577 22,589 17,834 13,948 11,818 9,781 7,458 6,440 4,942 3,628 2,187 1,690 1,066 675 390 136 96 32 33 17 4 4 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 S3 34 35 36 37 38 39 40 41 -42 43 44 45 46 47 48 49 50 51 52 30 7.5 22.5 22.0 391 390 30 8.5 21.5 21.9 264 285 31 9.5 21.5 21.8 227 225 32 10.5 21.5 21.7 205 175 34 11.5 22.5 21.6 132 ■ 140 33 12.5 20.5 21.5 109 115 35 13.5 21.5 21.4 114 93 35 14.5 20.5 21.3 41 75 36 15.5 20.5 21.2 68 60 38 16.5 21.5 21.1 45 47 38 17.5 20.5 21.0 64 36 38 18.5 19.5 20.9 27 27 40 19.5 20.5 20.8 14 20 39 20.5 18.5 20.7 14 15 41 21.5 19.5 20.6 14 11 45 22.5 22.5 20.5 14 8 9 20'.4 9 6 ? 2'o'.3 5 5 ? 20'.2 9 4 •Age at Maximum Meanlnt'rv'l Age at Marriage do., sm'tiied Frequencyat Maxt(crude) do.,(sm'tlied 3,872 2,691 1,991 1,486 1,005 920 820 631 620 332 442 332 322 221 264 232 193 137 140 105 101 59 72 56 51 41 37 23 25 9 16 14 9 392,152 1,000,000 1,000,000 Totals Smoothed .101 .106 .112 .117 .139 .140 .150 .170 .186 .178 .187 .193 .198 .208 .216 .216 .240 .313 .444 Ratio Max. to Totals maximum 12 montlia. t The freijuenoy at tlie maximum is for the age. 254 APPENDIX A. The detailed results for the successive years shew considerable regularity in the frequency of fifcrst births even for individual ages, as for example the births, for ages 23 and 25 during the tenth month and first year after marriage, were respectively as follows : — Interval. Year. 1908. | 1909. j 1910. 1911. | 1912. 1913. j 1914. 1908-1914. Months 10-11 Number (23) Number (25) Corresponcling Marriages . . 239 195 32,480 232 184 32,704 261 302 237 249 34,127 36,953 328 288 39,815 314 296 42,078 354 308 41,808 290.0» 251.0' 37,138* Tears 1-2 Number (23) Number (25) Conesponding Marriages 622 559 31,440 685 631 32,510 688 698 604 654 33,163 '35,183 1 860 757 38,037 888 813 40,814 909 820 41,870 764.3* 691.1* 36,145* * Ayeiage for the period 1908-1914. The significance of these figures, which are taken at random, is seen, when the " corresponding marriages" (i.e., the marriages earlier, by the proper interval, than the year indicated) are taken into account. The interval in question is about 10 J months in the one case, and 18 months in the other. Thus for the two upper numbers the figures adopted for 1908 are those for 1907, plus one-eighth of the difference between them and those for 1908, and so on ; and for the lower numbers the mean of the figures for 1906 and 1907 ; and simil- arly throughout. The ratio of each number to the seventh of the total shews the degree of correspondence since the whole of these ratios are relative, and the vertical columns should be identical for exact correspond- ence . The ratios corresponding to the six lines above are : — Interval. Year. 1908. 1909. 1910. 1911. 1912. 1913. 1914. Number (23) .82 .80 .90 1.04 1.13 1.08 1.22 Months 10-11 Number (25) CJorresponding .78 .73 .94 .99 1.15 1.18 1.23 Marriages .87 .88 .92 .99 1.07 1.13 1.13 Number (23) .81 .89 .90 .91 1.13 1.16 1.19 Yeare 1-2 Number (25) Corresponding .81 .91 .87 .95 1.09 1.18 1.19 Marriages .87 .90 .92 .97 1.05 1.13 1.18 Seeing that the original numbers are very limited, the agreement is remarkably good, and confirms the utility of Table LXXVII., and the utility of the graphs of the protogenesic surface, to which surface refer- ence wiU now be made. FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 255 17. The nuptial protogenesic boundary and agenesic surface.— If the relative numbers of first-births, after different durations of marriage and for different ages of women, given on Table LXXVII., are regarded as vertical (z) ordinates, with the ages of women and duration of marriage as the other two ordinates {x and y), the surface so defined may be called the nuptial protogenesic surface or surface of nuptial primiparity. In the graph of such a surface the area for which the ordinates are zero may be called the agenesic region, or the surface of absolute steriUty ; and the boundary between the two may be called the agenesic boundary. The values of x and y for all points on the boundary between the agenesic region and the protogenesic surface are the ages and correspond- ing durations of marriage which define the existence of perfect steriHty. Thus with a duration of marriage of say 6| years there were no cases of first-births among women of 19^ years of age in the records extending from 1908 to 1914 ; see Table LXXVII. or Fig. 72. The Protogenesic Surface. Australia, 1908-1914. ,£3 I 01 3m e rtjD. 05 18 Olyr 5 lo Duration o£ Marriage. Fig. 72. 15 2Syr5 The contours represent equal frequency of first-births with varying age and duration of marriage. The area outside the contour is the agenesic region. The figures on the contours are per million first births, for all women of age x last birth- day, and for durations of marriage « to * + 1, where t is expressed in months on the left hand part of the figure and in years on the right hand part. The characteristic features of the protogenesic surface are shewn in Figs. 72 and 73. On Fig. 72 this surface is defined, by contours, on extended lateral scale for to 18 months after marriage, and on a smaller 256 APPENDIX A. Profiles of the Frotogenesic Surface. 3 6 9 12ino, 18 2000) loot J^D IDOCO 5000 N ^ «l4Zi 42 18,-32 lateral scale from to 27 years after marriage, and in both cases for the whole nuptial reproductive period, say 13 to 52 first-births. A vertical frequency of 1 on the right hand side of Fig. 72 corresponds to the frequency of 12 on the left hand side. The line of maxima is shewn by a broken line on the figure. In Fig. 73 the vertical sections of the protogenesic surface are shewn for each age from 13 to 27 years (" age last birthday"), and for the 5-year groups 28-32, 33-37, 38-42, 43-47, and for the group for all ages from 13 to 52. The frequencies of first-births, which are identical on any contour, are indicated by figures. These are per million total first births for intervals of a month of duration of marriage on the left-hand side of the figure, and for intervals of a year's duration of marriage on the right hand side. The " age" indicated is always to be taken as the " age last birthday," or what is the same thing, and more general, for the age x to the age x-\-\. It will be seen that these contours constitute a family of curves for which there is no simple mathematical specification. The unique maximum shewn by a small contour hke an " 0" on the left hand side of the figure and by an asterisk on the right hand side. The profiles of the protogenesic surface, shewn on Fig. 73, from to 18 months, are the curves shewing the frequency at various ages, for a total of a million first -births at all ages, and for the first 18 months after marriage. These curves have characteristic similarities, indicated by the points letters o, b, c, d, on the figure. The similarities are important since they shew that there is a remarkable regularity in the interval between marriage and first-birth in women of different ages. The curves drawn are not for instantaneous group-values, viz., for the groups x to «+ dx, but are the values for mensual groups, the abscissae for which are referred to the middle of the month. 100 000 1)000 12 mo. Duration of Marriage. Fig. 73. 18. Curve of nuptial protogenesic maxima. — The curved broken hne on Fig. 72, shewing the ordinates for greatest frequency of first-birth, can be replaced by a regular curve, which will give the actual values of these ordinates with sufficient pre cision. Adopting as the argument the " age last FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 257 Age 20 25 30 35 40 Calo. Value 0.0 2.12 6.73 13.23 21.37 Graph Value 0.0 2.2 7.0 13.0 21.5 birthday," that is the initial value of the age where the range is from x to x+l, and for the corresponding initial value of the duration y, where the duration meant is from 2/ to «/-fl, we have — (474) y = 1.45 |i = 1.45 [x - 20)S ^ being the " age last birthday" less 20. This gives the values on the upper line, while those on the lower are scaled from Fig. 72 : — (Initial value of the duration.) The maximum frequency per million total births, where the age is " age last birthday," and the duration is from y to y-\-\, cannot be ex- pressed by any simple mathematical formula. The values, however, are given at the bottom of Table LXXVII. 19. Ex-nuptial protogenesis. — -The previous issue is not ascertained in the case of ex-nuptial births, and the point of time to which the interval corresponding to duration of marriage should be referred cannot be defined. Hence no comparison can be made with nuptial protogenesis. 20. Average age for quinquennial age-groups of primiparae. — ^The following table gives the average age of mothers of first-births in quin- quennial groups : — TABLE IjXXVIII. — Average Age o£ Mothers, First-births, for Quinanennial Groups. Age-group 14 15-19 19 20-24 25-29 30-34 35-39 40-44 45-49 50-52 Average Age . . 14.36 18.78 18.77 22.61 27.19 32.06 37.08 41.74 40.31 41.84 Middle Age 14.0 17..50 16.51 22.50 27.50 32.50 37.50 42.50 47.50 46.50 Difference + 0.3fi + 1.28 + 2.27 + 0.11 -0.31 -0.44 -0.42 -0.76 -1.19 -4.66 The differences between the middle and average ages are obviously too large to be neglected, and therefore it is always necessary to decide whether the average value or the middle value of the ranges of the argument (age-group ranges) shall be used. In general the middle value is the more convenient. 21. Average interval between marriage and a first-birth, a function of age. — The data furnished in Table LXXVII. shew that the average interval between marriage and first-births is a definite function of age. * 1 T. A. Coghlan, in his " Child-birth in New South Wales, a study in statistics," has given results (see his Table VIII., on p. 26) for the average period from " marriage to birth of first-child" for " post-nuptial conceptions only." He introduces an ad- justment for the non-stationary character of the population from which they are derived, see p. 26. His main result, however, is wholly erroneous, and the true result is inconsistent with his conclusion, viz., that for married women between the ages of 17 and 39 the average period between marriage and a first-birth is only 19.4 months, and the range between 18.3 and 21.5 months. The matter will be referred to more fully later, see pp. 271-2, and particularly the note on the latter page. 258 APPENDIX A. If age-groups of primiparaB be formed, it is found that the mean ages of the groups and the average intervals between marriage and first-births are as shewn in the third colunm of the Table hereunder, viz., LXXIX., see also Figs. 74 and 75. The average values of the ages and of the corresponding intervals are as follows : — TABLE LXXIX.^AveTage Ages and Average Interval between Marriage and Fiist-births. (L) Age of Hairied Women . . Under 20 (U.) Average Age .. .. 18.77 (iii.) Average interval between Hairiage and Fiist-birth (crade data) . . 0.623; (iv.) Average interval by for- | mula (smoothed data) ; 0.604 (V.) Difference (data-calc.) .. +0.019 20-24 22.61 0.994 0.991 + 0.003 25-29 27.19 1.483 1.502 —0.019 30-34 35-39 40-44 32.06 37.08 41.74 2.026 2.862 3.501 2.100 2.766 3.420 —0.074 + 0.096 +0.080 45-49 46.31 4.048 4.209 —0.161 The values on hue (iii.) are fairly well given by the simple formula : — (475) i = 0.0437 x + 0.01221 ;^i-s where i is the average interval between marriage and the first-births, and ;^ is 11 years less than the average age, a 5-year group, that is to say, the age 1 1 is taken as the zero of x ■ This age has not been arbitarily adopted, but, as is shewn by the line OP on Fig. 74, is indicated as the minimum age to which reproductive facts should be referred. (See Table LXXII., p. 239 and p. 268). The small crosses in Fig. 75 are the results for individual years of ao^e last birthday, computed by means of the formula (475) ; see p. 268. There is a fairly definite indication that the continuation of the curve should be as shewn by the broken line in Figs. 74 and 75, terminating therefore at about age 55. There are, however, so few births at ages greater than 45, that this part of the curve cannot be regarded as yet weU determined or determinable : see p. 268. The following Table, LXXX., gives the results in greater detail, and furnishes also smoothed values of the approximate average interval i between marriage and first-births for all first-births ^^•ithin a year of marriage, and for aU ages during the reproductive period. Since formula (475) refers to the average age, it will not give the quantities in the Table LXXX :— !■ The intervals are only approximate. They have been calcvilated by assuming that the births in each month during the first 12 months may be referred to the middle of the months, and those during the intervals of from 1-2 years onwards may be re- ferred to the middle of the year. The change in rapidity of births is so great during the year after that of marriage that a correction is necessary for rigorous accuracy. That the difference is appreciable is obvious from the following results : 1st Births to 9 mths Approximate average interval, age 22 Average interval more rigorously calculated . . 5.52 5.53 1st Births to 12 mths 7.S3 7.54 All First- births. 1 11.70 months I 10.88 months The intervals are found more rigorously hereinafter for birtlis occurring not earlier than nine months after marriage. FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 259 TABLE LXXX. — ^The Fiotogenesic Indices, according to Age. (Appioximate Average Intervals between Marriage and First-births)* Australia, 1908-1914. AVERAOB INIEBVAI. Births Occurring within 12 months Births Occurring after 12mths. Marriage. All First-births. Age of Mother at Birth. after Marriage. Interval for Age- Crude Smo'th'd Crude CrudS Smoothed Smoothed Group. B^sult. Result. Eesult. Besnlt. Result. Result. Crude Eesult. Years. Months. Montiis. Montlis. Montlis. Months. Years. Months. 10 0.00 0.00 11 0.85 0.071 12 1.72 0.143 13 5.i7 2.88 18.00 8.38? 2.61 0.217 4.03 14 3.39 3.49 3.39 3.53 0.294 15 4.41 4.06 18.00 4.80 4.47 0.373 (7.48) 16 4.49 ' 4.60 19.85 5.33 5.44 0.453 17 5.12 5.11 19.15 6.21 6.44 0.537 7.55 18 5.64 5.58 19.54 7.20 7.47 0.623 19 6.07 6.03 19.94 8.32 8.53 0.711 20 6.56 6.44 20.57 9.57 9.62 0.802 21 6.89 6.82 21.19 10.25 10.73 0.896 22 7.53 7.11 21.51 11.70 11.91 0.992 11.93 23 7.77 7.48 22.84 13.08 13.10 1.092 24 8.04 7.77 23.99 14.42 14.33 1.195 25 8.26 8.02 25.29 15.61 15.60 1.300 26 8.36 8.24 26.69 16.94 16.91 1.409 27 8.55 8.43 27.75 18.20 18.26 1.522 17.80 28 8.52 8.58 29,46 19.38 19.65 1.638 29 8.61 8.71 30.95 20.94 21.08 1.757 30 8.75 8.80 33.68 22.81 22.56 1.880 31 8.64 8.86 34.39 23.60 24.08 2.007 25.15 32 8.55 8.89 37.44 25.85 25.65 2.137 33 8.49 8.88 39.41 27.38 27.26 2.272 34 8.46 8.85 42.38 29.96 28.93 2.411 35 8.30 8.78 43.75 31.14 30.64 , 2.553 36 8.30 8.68 45.85 32.73 32.41 2.700 37 8.44 8.55 47.51 34.82 34.22 2.852 34.34 38 8.45 8.38 52.24 39.70 36.09 3.008 39 7.99 8.19 50.15 37.47 38.02 3.168 40 7.71 7.96 49.74 39.09 40.00 3.333 41 7.62 7.70 55.74 42.82 42.03 3.503 42 7.32 7.41 56.15 44.50 44.13 3.677 42.01 43 7.26 7.08 53.32 41.90 46.28 3.856 44 6.87 6.73 59.69 48.99' 48.50 4.041 45 7.46 6.34 55.77 42.13 50.77 4.231 46 8.25 5.92 ?8.92 69.50 53.11t 51.50t 4.426t 4.292t 47 5.90 5.47 64.60 50.55 55.51 50.55 4.626 4.212 48.58 . 48 11.50 4.98 52.00 46.21 57.98 46.22 4.832 3.851 49 7.17 4.47 54.00 33.93 60.52 40.50 5.043 3.375 50 7.50 3.92 66.00 36.75 63.12 30.50 5.260 2.542 51 3.34 18.00 18.00 65.79 18.00 5.483 1.500 28.54 52 9.50 2.73 9.50 68.53 9.50 5.711 0.792 53 71.34 3.00 5.945 0.250 54 74.23 1.20 6.185 0.100 55 77.18 0.00 6.432 0.000 7.491 nonths 29.06 m. 15.95 J nonths. 24.20 years 27.34 yrs. 25.43 3 ^ears. • Based on a total of 220,021 hirtlis. t These values from ages 46 to 55 are merely ' extensions of the curve. t Tliese values are probably iairly reliable. 260 APPENDIX A. The yearly groups may with advantage be referred to the " age last birthday," instead of the middle-age value, which is approximately the "age last birthday plus J." Let then | denote the " age last birthday," less 10 ; the intervals are found to be very accurately given in months and in years respectively by the following formulae, viz. : — (476). . . .i' = 0.83641 +0.01062^ +0.000198P, and for months ;i (476a) ...A" = 0.06971 + 0.000885^ + 0.0000165^, for years : ^ f is of course expressed in years in either case. The values may be readily computed by taking the interval for age 10 as zero, and the smoothed results for 20, 30 and 40, and applying formulae (199) to (199c), see Part v., § 7, p. 69, and remembering that the coefficients b, c, d vary inversely as the variable, and as the square, and the cube of the variable, respectively. To develope the table we may calculate the values for 11, 12 and 13 (i.e., for | = 1, 2 and 3), or calling the leading differences for 10 years as Di, D^, and Dg, the leading diflEerences di, d^ and ds can be found by the formulae^ — (477) di = O.lDi - 0.045I>2 + 0.02852)3 (477a) ....tfe = .. O.OID2 - 0.009X>3 (4776) ....^3 = •• •• O.OOID3 We have also, for the coefficients of the equations above : — (478) b = rfi 1^2+ ^ds (478o) c = .. ^dz— i^ds (4786) .... d = .. .. ^ds The agreement between the crude values and the values by formula (476) for the average interval between marriage and first-birth is remark- ably close throughout, the curve applying as far as age 45. Beyond this age the values for the extrapolated curve are given as well as those of the probable value of the interval. 22. The protogenesic indices. — -The average interval, calculated as shewn in the preceding section (viz., by formula (484) in the section next following, § 23, but omitting the correction term e) is not rigorously 1 These formulae are for the " approximate" average iut«rval ; aoe the preceding note. » See Text Book. Institute of Actuaries, Part II., Chap. XXIII., Art. 22, p. 443, Edit. 1902. FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 261 exact, but is sufficiently approximate to be used as an index of the fre- quency distribution throughout the interval. We shall call the interval so calculated the protogenesic index for married women of the age in question, and for all ages, the general protogenesic index} Table LXXX. is thus a table of protogenesic indexes rather than a table of average intervals, though the intervals are approximately correct. We shall now consider methods of correctly estimating the interval. 23. Exact evaluation of the average interval from a limitefd series of group-values. — The average interval may be determined with a higher degree of approximation from the series of group-values for equal ranges themselves by formulae developed as follows : — Since the group-values can often^ be reproduced with sufficient accuracy by a rational integral function we have, in such cases, for the value (a;^) of the interval (the distance to the centroid vertical) : — (479) . , ^xydx ^aa;2+i6a;3-|-eto. ^ bx^+cx^+ ^doi^-{-^ex^-{-. jydx =*x-' ax+ibx^+etc. ^ 12a+Qbx+4cCX^+Mx^+^hx*+ in which last expression we may substitute, by means of formulae (195) to (197), see Part V., § 5, pp. 67, 68, the values of the groups themselves for a, b, c, etc. This will give a series of formulae according to the number of groups taken simultaneously into account. We may take the common value and the ranges as unity : if it be k the value deduced will then be multiphed finally by A. It will be convenient to call the group values A, B, G, etc., hence if n of these are included, n will be the value of x. That is to say, in formulae in which D appears, x will be 4. From (479) we thus obtain the following series of formulae, viz : — (480) a;^ = l +\- ~f_^B ' *""" ^ ^ ^'' 9 A + G (481) a;™ = li + -g . ^ j^ b + G ' ^°^ ^ ^ ^ ' .482^ X -2+^ -19A-BB + 3G + 19D _ (***^) a:™ -^+45. ji ^ B + G + D ' for a; = 4 ; and 125 -5A-2B+2D+aE . (^83) '''^ = ^^ + m- A + B+G+D+E :'^''''' = ^ * To fully define the term it should be preceded by the term " nuptial " ; but for obvious reasons this may be always understood. * But not invariably : see latter part of Section 24. 262 • APPENDIX A. If the common range be k, these expressions should of course be multiplied by that quantity. From these formulae multiply-infinite series of formulae may be developed, and such development can be effected by processes similar to those indicated in Part VI., § 2, and Table VT., pp. 75 to 77. A practical way of applying the formulae is to calculate by an approximate method and make the necessary correction, if it be sensible. Thus :— _ (^+3^+5g+7Z>+etc.) ^ ^ ' '^'" ~ 2{A + B + + D+etc.) + ^ where e is a small quantity. For the value of e, we have, from (480) to (483) :— (485) . . €2 = ^ . A I T> ' when there are two quantities only. I j4 + C (486) . . es = g-. Aij^ip > when there are three quantities only. (487). .., = - . ^^s+C + D there are four quantities only. when i±iiSi\ 1 - 49 ^ + 385 - 38Z) + 49^ , ^^ (488).. C6 =288- A + B + C +D + E ' ^^en there are five quantities only. i7„ denoting the sum of n successive groups, A, B, etc., these expres- sions may be put in the arithmetically more convenient form hereunder, viz. : — (489).... 62 =0.16(S - A) /Zz; es = 0.125((7 - A) / S^; ei= { 0.18{D -A)—0.2S{G - B)\ /Si (490). . . .£5 = I 0.17014 (E~ A)- 0.13194 {D - B)^ / S^ Whenever each group-value in a series is not greater than say 2 to 2J times an adjoining group-value, the preceding formulae give fairly' good results, and may be used for a succession of three, four, or five groups in a way which will now be indicated. 24. Evaluation of group-intervals for an extended number of groups.— To apply the preceding formulae to a large number of groups it wiU be convenient to adopt the following notation. Let A, B, 0, etc., be denoted by Ai, A%, ^3,etc.,andletaIsoa;' = x^ for A]_ to A^ reckoned from the beginning of A^, x" ^ x^ for say ^^^ito ^™ , etc., reckoned FERTILITY, FECUNDITY. AND REPRODUCTIVE EFFICIENCY. 263 not from the beginning of A^.^i, but from the beginning of Ai ; and so on. Let also A', A", etc., denote the totals of the various series of groups in question ; that is, let A' = Ai -{- Az + etc. ; A" = A^^x + ^A+2 + sto- ; 3'nd so on.^ Then the value for the entire series is : — (4yi)....9a:„- -^' ^ ^"_^ ^»' ^etc. ~ E A' Consequently, if a; ' =w' -\- e', where w ' is an approximate value of x ' and e ' is the correction to make it exact, we shall have for the true value of (492) X - ^^^'""'^ + ^^^'''^ y^^^i o'*'«> — E A' E A' in which E(A'e') = A'e' + A"e" + etc. Let the factors 1/6, 1/8, 1007/90, 3051/90, 49/288, 38/288, on formula (485) to (488) be denoted by ai, az, etc., and generioally by a', a", etc. Then, since when e', e", etc., are multiplied respectively by A', A", etc., their denominators disappear, we have, for the total correction e^ say, the sum of the numerators divided by the sum of all the groups. Thus a A', a" A", etc., denoting the numerators, we have :^ v*^"*;' o-'^m ~ E A' E A' that is to say, the approximate value of the average interval, found by multiplying each group by the middle value of its interval, and dividing the sum of all the products by the sum of all the groups, merely requires the correction found by multiplying each group by its correction co- efficient (a), and dividing by the sum of the whole of the groups. Hence formulae may be developed to embrace the corrections by multiplying the individual groups by factors, and these factors are readily found by summations. Thus we obtain the following, viz. : — (494).. oa;m =(0.375^1+1.5.42+2.625 43 +3.375^4+ 4-5^6 + 6.625^8 +eto.) / EA. the series of coefficients being in threes ; thus the coefficient for the third term from any term of the series is 3 greater than that of the term from which it is reckoned. Further, : — (495). -oXm =(0-31 Ai + 1.73 Az + 2.26 A3 + 3.68 Ai + etc.) / EA ; and (495a). .oCBm = (0.32986.41,+ 1.63194^2 + 2.5^3 + 3.36806^4 + 4.57014^6 + etc.)/ 27 4; the series of coefficients being respectively in fours and in fives : thus the coefficient of the fourth term in the one case, and of the fifth term in the other, from any term in the series, is 4 greater in the former case and 5 greater in the latter, than the coefficient of the term from which it is reckoned. 1 It is of course immaterial what nvmiber of groups are combiaed. 264 APPENDIX A. 25. Average interval for curves of the exponential type. — In cases where A^ is very small (or very large) compared with Ai, the preceding formulae are not very accurate.^ In general, if the curve giving the groups be approximately of the tyipe e^™*, and the groups be also very different in magnitude, it is preferable to proceed as follows : — ^ Let Ai, Az be two adjoining groups ; these can be satisfied by the equation : — (496) y = Be'"', or «/ = e» + ''-" ; in the former of which, therefore, B = e*. Similarly three adjoining groups, Ai, A2 and A^ may be satisfied by the equation : — (497) y = A + Be'"" Putting Ai the group for the range to 1 ; A^ the group with the range 1 to 2 ; A^ the group with the range 2 to 3 ; we have from these equations the following, viz. : — From (496) : — (498) ^ = i^ = e» ; or 6 = 2.3025851 log^^ ^ and this applies to a whole series of groups if the ratio A^^i / A„ be constant. Also : — ^*^^> ^ - e& _ 1 - (e6 _ l)e» ^ (e» - 1) e"" -^*"'- the final equation in (499) being true only if A3/ A^ = A^/ Ai = n, say. From (497) we have, similarly to (499) : — (500). .Ai = A+B{e'>-l)/b; A^ = A -\- B (e» - 1) e'>/b ; A3 = A + B{e'> —1) e^'> /b; and consequently : — (501).... (^3 - A^)/{Az -^i)=e», or 6=2.3025851 log.iQ{(A3 ~ Az)/{A2. - ^1)}; etc. 1 For example, if there be two groups, on equal bases 0-a;, x-2x, one of which is three times greater than the other, the straight line (which in such a case would be the assumed curve, giving areas equal to the groups), would start at the terminal (or 2a!) of one of the groups. If one is greater than 3 times the other, it will fall within one of the rectangles. The question has been exhaustively considered by Prof. Karl Pearson, see Biometrika, Vo. I., pp. 265-303, Vol. II., pp. 1-23. ' As the formulae of this section are of general application x has been used for the independent, and y for the dependent variable. FEBTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 265 Writing n for e*, we have also : — (502)..B=b(Az-Ai)/(n-l)^ = b{A3-2Az+Ai)/{n-l)^,eto.; and /^rwov A A B, ,, . Az- Ai Aiu— Ao (503) A = Ai— -J- (n-1) = Ai r-^ = -^ q-^ ,etc. n-1 w— 1 Thus the constants b, B and A in (497) are determined. In applying these formulae to ascertain the average interval, four bases will require specially to be con^ddered, viz., when the factor b is positive, and when it is negative, the range being either to 1, or 1 to 2 in both cases. For the ascending and descending branches respectively, these cases correspond to the curves Be'"' and Be~'"' For the purpose in view (496) is suitable, and the results, to be tabulated for various ratios of Ai/ A^ or Az/Ai, will be the groups B jy", B j'^, B^e-^", and B fe~'"'. The mean interval lies between the centre of the group- range and the side on which the groups have higher values. For the more general case, that is when three values are satisfied, we should have to determine /xydx _/x(A + Be'"')dx _^Ax^+B{{bx—l)e'"'-{-l}/b^ (504:).. x^= jp^ = y(^_|_5e»«)(^a; = Ax + B (e»* - I) / b If A, however, be taken as zero, this last will become (bx — 1) e"^ + 1 xe^" 1 (505) o^m = 6 (e»* - 1) e** — 1 b which function is the basis of the tabulation hereunder for ratios of Az to Ai and for ranges of x=l and 2, by applying (498). It may be noted that the value of (605) = for a; == 0. In the table hereunder, LXXXI., the four cases above referred to are as follows : — Case I. „ II. „ III. ,. IV. ^i> -^i*; Origin 0; Bange — k ; Tabular Interval computed from 0. ; „ k—2k; „ „ „ „ *. ^a< Ai; „ ; „ 0— i ; „ „ „ „ 0. ,> ; „ }i—2k; „ „ „ „ *. These four cases are illustrated by Fig. 78, hereinafter. The necessary formulae for calculating the required values are simple it we put .^2 = « -^i, viz. : — (506). . „<=I+ ^-|; i»^2=2+^j-i; . .^.,x', = p+^-l ; formulse which are convenient for computing tabular values. For negative values of 6, in which case Az is less than Ai, it is arithmetically convenient to use the ratio Ax/ Az= m, so that m=\/n, and put p = — b, then the preceding formulae become : — „ , »w . 1 ■ „ „ m 1 „ m , 1 It may be easily verified that p-ia;'j, + p-ia^'j, = 1. 266 APPENDIX A. By means of the preceding formulae Table LXXXI. has been computed : it will serve for readily estimating the position of the centroid vertical for any group by means of the relative magnitudes of the adjoining groups. The determination of that vertical from the relative magnitudes of the groups on either side of any group in question gives results of a fair degree of precision. To satisfy three groups by means of (497) we have for the value ai A in terms of Ai to A^, : — (508) . A = AiAs - Al Ai + A3 2Ao instead of (503) : hence we can subtract this quantity from the groups and we then obtain : — (.509). A'l = Ai— A; A' 2 = Az A ; etc. ; etc. these reduced groups, denoted by accents, conforming to the relation A's/A'2 = A'2/A\. The value of the average interval is therefore : — i(l+3 + ..2p — 1}A + A 1 o*'i+ ^'212^2 + to p terms (510)..„x'p= A + ^2 +. .to p terms. Results so computed have a high order of precision. If A, and A'l, etc., be expressed in ratios to ^1+ etc., as unity, the denominator of course disappears. TABLE LXXXI. — Abscissae of the Centroid Verticals of Gronps Bounded by the Carve Bel": and Be-''^. For the Computation of Average Intervals, etc. Ratio Ratio Ratio A,/Ai Case I. Case in. A./A, Case I. Case ni. A,/A, Case I. Case III. or or or Ai/A, Ai/A, Ai/A, 1.0 .50000 .50000 4.0 .61199 .38801 9 .66988 .33012 1.25 .51857 .48143 4.5 .62085 .37915 10 .67682 .32318 1.5 .53370 .46630 5.0 .62867 .37133 11 .68297 .31703 1.75 .54639 .45361 5.5 .63563 .36437 12 .68848 .31152 2.0 .55731 .44269 6.0 .64189 .35811 13 .69346 .30654 2.25 .56685 .43315 6.5 .64757 .35243 14 .69800 .30200 2.5 .57531 .42469 7.0 .65277 .34723 15 .70216 .29784 2.75 .58290 .41710 7.5 .65754 .34246 20 .71672 .28328 3.0 .58976 .41024 8.0 .66196 .33804 25 .73100 .26900 3.5 .60177 .39823 8.5 .66606 .33394 50 .76479 23521 For case II. add unity to the value lor case I., and lor case IV. add unity to the value lor case in Applying the various formulae to the results given on the penultimate line on Table LXXVII. for all first-births, 12 months or more after marriage, the following results are obtained : — FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 267 By formula (484), neglecting the correction e, 29.06 months (Index). „ (494), applied through same range, 28.18 „ (Interval) (495) „ „ „ ^ 28.00 „ (506) „ • „ „ 27.721 „ By graduating and using monthly values for the groups up to 48 months 27. 70^ ,, ,, 26. Positions of average intervals for groups of aU first-births. — The positions of the average intervals (abscissae of the centroid verticals), computed on the basis of the results shewn on the penultimate line of Table LXXVII., wiU probably be found approximately true for any popula- tion. By means of Table LXXXI., they may be readily found. TABLE LXXXn. — Average Intervals'" in Months for First-biiths, for Various Ranges of Inteival. Australia, 1908-1914. Bange Aver- Bange Aver- Bange Aver- Bange Aver- Bange Aver- Bange Aver- ol age oi age oJ age of age of age of In- age Int'rval Value. Int'rval Value. Int'rval Value. Int'rval Value. Int'rval Value. terval Value. mouths. months. years. months. years. months. months. months. months. months. years. months. 0- 1 .051 0- 1 7.51 12-13 149.76 . 0-3 1.70 0- 6 3.68 - 1 10.34 1- 2 1.52 1- 2 16.35 13-14 161.69 a- 6 4.65 0- 9 6.41 - 5 17.35 2- 3 2.53 2- 3 28.95 14-15 173.67 6-19 7.40 0-12 7.51 -10 19.60 3- 4 3.62 3- 4 39.22 15-16 185.73 9-12 10.34 years -15 20.43 4- 5 4.52 4- 5 53.39 16-17 197.74 years 0- 5 13.07 -20 20.72 5- 6 5.52 5- 6 65.53 17-18 209.69 0-1 7.51 0-10 14.63 -25 -26 20.79 6- 7 6.50 6- 7 77.61 18-19 221.68 1- 5 22.36 0-15 16.19 20.80 7- 8 7.49 7- 8 89.63 19-20 233.67 6-10 81.12 0-20 15.38 - 5 22.36 8- 9 8.55 8- 9 101.67 20-21 245.66 10-16 142.96 0-25 15.43 1-10 25.82 9-10 9.53 9-10 113.71 21-22 257.67 15-20 203.05 0-26 15.43 1-15 27.13 10-11 10.62 10-11 125.73 22-23 269.65 20-25 261.40 1-20 27.59 11-12 10.63 11-12 137.77 23-24 24-26 281.58 293.49 25-26 305.43 1-25 1-26 27.71 27.72 • These will be sensibly true for any distribution at all similar to that shewn in Table LXXVII . and in Table LXXXIII. hereinafter. The above results have been computed by using graphic graduation^ v^here necessary, by means of the values given in Table LXXXI., and by formula (416), p. 201. In general the computed values proved to be sensibly identical. A result intermediate between the extreme values has always been taken, regard being had to the general probabilities of each case. 1 These last results are the most accurate ; the value for the month 11-12 is taken into account in the graduating ; in applying formulte (494) and (495) and (506) it is not considered. 2 It is impossible in the absence of monthly data to determine the position of the centroid vertical with great precision. By graphic graduation conforming to the 11 to 12 months group and to the 1-2, and 2-3 years groups, the result, 16.46 was obtained. By extrapolating the 10-11, 11-12 months group -results, adopting this extrapolation for the year -group 0-1, and conforming to this fictitious year- group and the actual year -groups 1-2 and 2-3, the result is 16.25 by formula (510). Adopting the extrapolated result and the group 1-2 only, gives 15.91 ; while the exponential curve conforming to the group 1-2 and 2-3 only, gives the result 16.79. The groups 1-2, 2-3 and 3-4, treated by formulse (508) and (510) give 16.63. After consideration of all the circumstances I have adopted 16.35 as the result which I believe to be nearest the correct value. Similarly the results 28.95, 28.93 and 29.11 were obtained for the group 2-3 ; of these the first was adopted. 268 APPENDIX A. Average Issue and the Frotogeaesic Indices. Fig. 75. Durations of Marriage (Interval between marriage and first birth). iutervaL 0^5 10 15 20 25 30 35_ 1 L. y ■■' .--' 1 . -y^ .-"-.i'- .•• .•' Ai 1 _.. ^y\r y ^K /-^y^' t 1 -.,.:.-,.' -,.i-.^-:; ^l::^- '^'v:: 1 -.ii./_.S— 4- y ^^- 7j- . . « -i;:.-.:: -.^^-— .r - ^''-,c2: \-i a -. _^2^_ _. _,^^^ y / -Ml 1 .J:i_..^:_.,^ L y y 4 ¥ g.^- t\ 1..:/ X iJ'- ^^ ^i ' o3 ._i:'o ^'''' _^^^ ,''■■ ^•'f [. . 5; = li -1-^.^ -_.: _.j^ _ J : """"^ , ::. ._ f ij.i __,.:_. .,/_V • .J:.._...4 1 Jj-...^:__,.^^.3i____ • ..i^ ^\ i 3.Jo._,^Z_^L_ ,,i^' L_..i ■| 1. .!..:::_..: y-k^ 1 ! .L,a._.r: .^^^^^ \___\ 2 '-■' - "tx^ --i- 1 __,..a ,.J|?:^ ._! S 1 ^ .-J^?^" ^ -- . o,^>-^*^' - — \J < A.ges. 10 15 20 25 30 35 40 Age of Mothers at birth of child. 45 50 55 Fig. 74. FiG.74. — The lower curve OPQ is the curve of the protogenesio indices (or approximately computed average intervals between marriages and the first-births) according to the age of mother at the birth of the first child. !FiG. 75. — ^The upper series of lines are graphs of the average number of children bom to aU mothers under 20 years of age, to mothers of from 20 to 24, 25 to 29, etc., and to mothers of all ages — ^who come xmder observation — according to duration of marriage. The fine dots give the crude results. The parallel broken lines indicate that the average raie, of increase is nearly independent of the age of the mothers, and is dependent on the duration of the marriage. 27. The unprejudiced piotogenesic interval. — The protogenesio interval gives unequivocally a measure of what may be oalled the modified- fertility of married women, that is fertihty as modified by physiological and social conditions, by Malthusianism, etc. It is evident that first births are likely to give the best available indication of the physiological element in fertihty ; that is to say, the ratio of cases of nuptial-maternity at any age to the total number of nuUiparous women, is a better indication of variations with age of physiological fertihty, than would be the indi- cation given by later births. But what have been called "prejudiced cases" should obviously be excluded, viz., cases where maternity, being expected, leads to marriage. For this reason the interval obtained by excluding such cases is not only appreciably longer, but also gives a truer idea of the normal probability of maternity, other things being equal. Results were published in New South Wales in 1899, purporting to shew FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 269 that, when prejudiced cases were excluded, the^ " average period from marriage to the birth of a first child" was, for unprejudiced mothers of from 17 to 39 years of age, about 19^ months, individual cases ranging between 18.3 and 21 .9 months. ^ In order to definitely ascertain whether there was any justification for the statement, the New South Wales statistics, upon which they were based, were examined and recompiled ; the data are given in Table LXXXIII. hereunder. Table LXXXIII. — ^Interval between Marriage and First Births occurring later than 9 Months after Marriage.— New South Wales, 1893-98. Interval (mths.) Interval Years.) ^ 9 10 11 1 2 3 4 5 6 7 8 9 10 IT 1213 14 15 16 17 18 20 22 t,o to to to to to tn to to to to to to to to to to to to to to to to 10 11 12 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 21 23 13 14 15 3 1 1 5 16 U 11 7 20 1 54 17 f,9 51 42 132 9 286 18 IS-i 137 lOP 380 37 i: 1 800 19 273 256 159 641 73 17 1 •• 1,420 ?,n 3!?0 337 209 772 129 36 9 6 1,818 ?,1 47(1 425 292 1,026 172 41 ,11 1 1 .. 2,439 ?,?, Mi 521 365 1,181 210 61 22 4 6 1 2,915 ■PA 48S 498 357 1,205 249 79 37 17 7 1 1 2,934 Zi 453 431 265 1,031 245 96 40 13 9 5 1 ■• ■• 2,589 9.fi 419 382 246 925 205 85 41 17 10 6 5 2 1 2,344 9fi 34? 294 240 8U1 205 83 41 17 19 4 5 1 1 2,054 9,1 ?,43 264 . 185 650 176 74 38 34 15 lU 2 7 2 1,700 9.H ?,3? 185 153 549 142 86 32 27 15 18 4 5 3 1 1 1,453 29 141 145 103 417 122 52 43 13 12 9 6 8 3 2 2 1,078 an 133 131 83 343 124 46 37 32 16 14 6 6 3 5 3 1 983 !t1 68 83 62 248 76 32 20 lU 13 7 4 7 6 5 1 643 W, 5?, S3 48 209 78 27 12 19 9 9 14 7 8 3 3 4 1 556 »» 48 47 41 142 66 25' 13 8 8 8 6 6 4 3 3 1 2 431 34 33 45 31 117 43 16 13 5 6 1 2 2 4 1 4 4 4 1 1 333 3!> 33 29 25 95 34 16 11 6 2 4 8 4 5 3 4 1 1 281 3A 29 K5 12 90 32 7 12 6 4 1 4 7 3 4 4 I 2 1 1 246 37 1?, 22 9 59 20 8 8 6 2 2 1 2 3 . , 2 1 1 1 159 38 10 1?, 8 58 13 14 3 7 3 1 3 1 2 1 1 1 138 39 11 8 13 47 15 9 5 1 1 1 2 2 1 1 .. .. 1 118 4Qi 3 5 37 15 6 2 3 1 1 1 2 1 2 ..3 1 83 41 2 9. 10 7 3 1 1 1 1 ... 1 31 ^9. 3 6 3 17 12 3 1 1 2 1 . . . . . , 49 43 1 3 2 H 3 2 1 1 1 1 1 . . ' . . 24 44 2 6 1 1 1 1 12 4.'i 1 1 9 1 ,. 12 46 3 1 1 34 30 13 15 "5 2 5 2 4,561 4,407 3,075 11229 1 2,515 928 453 256 L58 107 76 70 44 6 6 1' 1 2 27,993 Further, to ascertain whether any material difference existed between the results for New South Wales for the period 1893-8, and for the whole of Australia for the period 1908-14, the latter were also computed, and are shewn in the same table. On Fig. 79 the intervals for successive ages are shewn by a light zig-zag line, and for the Commonwealth by a heavy zig-zag line. The two are evidently substantially identical, as the figures in Table LXXXIV. also shew. 1 See note on page 257, hereinbefore. 270 APPENDIX A. Table LXXXIV. — Protogenesic Interval or Average Interval elapsing between Marriage and First-birth, for all First-births occurring not Earlier than 9 months after Marriage. New South Wales, 1893-8 ; and Australia, 1908-14. Agel of ' ' INTERVAL. Age INTEEVAL. Age Interval. Age Interval. of Mother of Mother of Mother Mother 1 last last last last EBirth- N.S.W. lAust. Birth- N.S.W. Aust. Birth- N.S.W. Aust. Bh1>h- N.S.W. Aust. day. day. day. day. years. months. months. years. months, months. years. months. months years. months. months -13 — 13.83* 23 16.10 16.25 33 28.87 30.70 43 — 49.38 14 — 10.14* 24 16.88 17.20 34 30.10 33.64 44 — 57.35 15 — 12.09* 25 17.30 18.23 35 32.21 35.04 45 — 47.49 16 — 13.65 26 28.28 19.49 36 35.91 36.92 46 — 58.70 17 13.48 12.72 27 19.71 20.54 37 32.34 38.55 47 — — 18 13.93 13.46 28 20.91 21.92 38 33.00 43.62 48 — — 19 14.40 14.05 29 22.07 23.39 39 28.47 42.14 49 — — 20 15.03 14.46 30 24.65 25.42 40 43.88 43.77 50 — — 21 14.71 14.76 31 25.15 26.28 41 48.57 51 — 22 15.04 15.02 32 30.09 28.94 42 49.83 52 * Depend upon 9, 14, and 68 cases only. The above table and Pig. 79 indicate that there has been no materia] change in the interval between marriage and first-birth during the elapsed 15 years, and also that the average period is not constant but is a function of the age when tabulated according to "age of mothers," that is, according to age at maternity. It will be shewn later that when the TABLE LXXXV.— Approximate Protogenesic Index for (These results are only approximate, the table being constructed from the data in Table Ages of Number of each Duration of Marriage, the total being 1,000,000, Mothers at Mar- riage. 0-9 months. 9-12 months. 1-2 yrs. 2-3 yrs. 3-4 yrs. 4-5 yrs. 5-6 yrs. 6-7 yrs. 7-8 yrs. 8-9 yrs. 9-10 yrs. 10-11 yrs. 11-12 yrs. 12-13 yrs. 13-14 yrs. 14-15 yrs. 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 "lO 110 576 3,832 12,040 25,771 37,328 38,259 44,522 37,997 32,933 25,336 20,054 16,057 11,798 9,822 7,049 5,614 4,090 3,564 2,745 2,048 1,768 1,442 943 749 687 450 304 227 151 88 46 5 13 5 4 5 14 50 259 1,458 4,559 9,226 14,249 20,739 27,624 27,510 28,506 24,180 21,130 17,735 14,962 10,990 9,831 6,627 6,499 4,204 3,064 2,436 1,977 1,481 1,096 800 468 282 178 113 50 63 14 9 5 'I 4 4 18 209 1,054 3,417 - 7,794 11,921 16,125 22,225 24,316 24,261 21,988 20,670 18,419 15,315 12,281 10,221 7,549 6,680 5,086 3,918 3,530 2,654 2,054 1,786 1,427 1,114 600 386 295 209 91 27 14 9 5 4 18 73 377 1,054 1,968 2,950 3,995 5,672 6,413 6,954 6,613 6,009 5,508 4,436 4,113 2,936 2,532 2,032 1,712 1,314 1,232 1,041 786 641 523 286 318 132 82 55 18 18 5 9 5 50 155 382 682 1,118 1,859 2,268 2,550 2,482 2,650 2,454 2,263 2,077 1,618 1,300 1,154 800 714 577 459 477 377 300 173 159 105 45 50 23 14 5 9 32 64 227 395 677 904 1,040 1,268 1,373 1,432 1,218 1,182 868 823 650 586 423 423 282 314 200 195 114 91 36 50 14 9 5 4 5 27 36 132 236 377 550 782 800 695 727 732 532 577 441 373 282 268 259 232 114 109 68 68 64 18 14 5 5 4 9 9 23 32 95 191 277 469 427 541 455 477 377 441 395 323 268 259 168 159 100 82 55 23 32 18 5 5 4 14 9 36 82 136 200 232 345 305 391 327 300 323 250 232 195 136 123 77 14 36 36 32 18 5 9 5 4 18 59 55 118 145 200 255 264 227 191 191 205 177 164 86 132 59 45 36 23 18 5 18 14 55 ■ 77 127 127 227 177 191 177 136 150 95 86 123 64 32 41 36 14 19 6 • 9 9 » 18 18 64 36 82 114 118 205 123 145 105 145 77 50 64 32 27 27 5 9 4 14 18 5 27 23 86 45 100 77 132 73 ion 77 82 59 32 23 9 9 5 5 4 5 14 18 u 32 91 109 100 105 50 73 91 27 18 14 4 5 9 14 41 27 68 82 114 45 59 18 45 59 9 9 9 I 5 9 9 41 36 27 41 32 41 32 27 14 5 18 Totals 348,437 259,411 247,676 71816 29354 14908 8,541 5,700l 3,872 1 2,691 1,9911,486 1,005 920 631 ' 1 332 PBRTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 271 tabulation is according to " age at marriage," there is a great approach to constancy of the interval, though the distribution according to interval is very different for different ages. 28. Frotogenesic Index based on age at and duration of marriage. — The protogenesic indexes as determined in the preceding sections, viz., §§ 21, 22, 26 and 27, are based upon the ages at maternity. For certain purposes, however, they might with advantage be based upon the ages at marriage, and for exact results the evaluation of the index would of course require a compilation according to those ages, and cannot be quite satisfactorily deduced from the results given in Table LXXVII. A very fair approximation, however, can be obtained by reconstructing that table (see pp. 252-3), and the simplest fgrm which this reconstruction can take is to treat the results in columns 1-2, 2-3, etc., years as re- spectively applicable to " ages at marriage, 1 year, 2 years, etc., earUer than that in the age-column. Such a compilation will be sufficiently accurate to disclose the general characteristics of the protogenesic indices for ages at marriage. This has been done in Table LXXXV. hereunder, which is self-explanatory when compared with Table LXXVII. Australia, 1908 to 1914 based on Age at Marriage. LXXVII. by moving the successive columns upwards, 1, 2, 3, etc., places respectively). iaoludlng those Born within 9 Months of Marriage. Protogenesic Index, or Proto- genesic Quad- ■ ratio Index. .(Crude). Ages of 15-16 16-17 yrs. 17-18 yrs. 18-19 yrs. 19-20 yrs. 20-21 yrs. 21-22 yrs. 22-23 yrs. 23-24 yrs. 24-25 yrs. 25-26 yrs. 9 mtha. ■ to 26 yrs. Approximate Average Interval. Moth'r at Mar- riage. Crude. Smooth'd 9 4 5 23 36 68 41 36 23 23 14 9 18 14 4 9 5 23 18 36 45 9 27 23 9 5 9 14 5 9 9 14 32 64 18 18 18 18 9 4 4 5 14 9 9 27 23 14 23 5 4 4 5 14 9 9 5 14 9 4 '6 4 5 18 14 9 5 4 '6 5 9 14 14 9 5 4 4 4 5 5 5 14 '6 9 5 5 4 '6 5 4 9 5 12 41 106 601 2,167 7,190 16,424 27,406 39,196 54,645 64,922 65,125 61,525 56,667 50,539 42,252 35,220 27,283 22,621 17,248 14,060 10,728 8,882 7,045 5,526 4,454 3,405 2,427 1,565 863 623 410 181 117 38 18 10 10 8 4 134.0 80.9 66.0 57.4 33.3 29.5 24.8 23.5 22.9 22.3 21.1 21.0 20.8 20.7 21.2 21.2 21.1 21.4 21.3 21.1 21.7 21.2 21.6 21.9 21.4 21.5 21.2 19.8 21.8 19.5 20.7 19.9 20.8 19.5 20.0 14.3 14.3 10.5 14.3 10.5 134.0 88.0 67.0 58.0 34.8 28.5 25.3 23.6 22.4 21.6 21.2 21.0 20.9 20.8 20.9 21.0 21.1 21.2 21.3 21.4 21.4 21.5 21.5 21.6 21.6 21.5 21.4 21.2 21.0 20.7 20.5 20.4 20.0 19.3 18.3 If.O 15.4 13.5 11.3 8.8 6.0 159.0 101.9 82.5 81.8 47.8 44.3 35.8 34.0 32.6 31.6 29.4 29.2 28.0 27'8 28.4 28.5 27.7 28.1 27.9 26.6 27.8 26.7 26.5 26.7 25.9 25.8 25.2 22.9 25.2 22.0 23.9 22.4 24.1 24.6 22.9 14.7 14.7 10.5 14.7 10.5 12 13 14 15 16 17 18 19^- 20 21 22 2i 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 62 332 221 232 137 105 59 56 41 23 9 14 651,563 Totals 272 APPENDIX A. Much more accurate results would be secured by that reconstitution of the data, which would be possible if monthly or quarterly graduations for at least the first 3 years after marriage were used. Such gradua- tions would have to be both for the horizontal and vertical values, and when effected, the sub-divided numbers would admit of a new table being compiled, giving with considerable exactitude the required numbers of births occurring after various durations of marriage, borne by women of various ages at marriage (instead of ages at maternity). The general characteristics of the values determined from such a table will, however, not differ materially from those in the table pp. 270, 271. In the final columns of Table LXXXV. are given the crude and smoothed protogenesic indexes or approximate of protogenesic intervals according to age, with the argument " age at marriage." These are quite different in form from those deduced with the argument " ages at maternity." The values exhibit considerable regularity and require relatively little smoothing. As might be expected a priori, the interval decreases rapidly as the age at marriage increases, until the age 20 is reached, when it is 21 months. It remains sensibly constant tiU age 46, and then rapidly diminishes. It is evident that it must necessarily have a small value at the end of the child-bearing period. The protogenesic index, or the protogenesic interval, determined according to " age at marriage," is perhaps to be preferred to one or the other based on the "age of mothers" {i.e., age at maternity). The average " period elapsing between marriage and the birth of the first child of post-nuptial conception" is evidently not the same for all women marrying at ages below 40 years, as had been stated,'^ but is a function of age, and is very nearly constant for a long period, viz., from about 22 to 45 years of age. The maximum frequency is about age 23.4 or 23.5, but cannot be very accurately ascertained without a special compilation. 29. Protogenesic quadratic indices and quadratic intervals. — The fact that the protogenesic indexes or the protogenesic intervals are sensibly identical through a wide range of ages, notwithstanding the "scatter" of the distributions varies enormously, necessitates the adoption of a second and different index, or of a second and different type of " interval." This wiU of course be of the nature of a higher moment since the higher the power the greater the influence of the distribution on the product. It will in most cases be sufficient to employ the second power of the "duration of marriage," and to use the quadratic index, viz., that » T. A. Coghlan, " Childbirth in New South Wales," 1899, p. 26, says : " . but where a marriage proves fertile, as the following table shews, the period elapsing from marriage to the birth of the first child of post-nuptial conception averages the same for all women marrying at ages below 40 years. This average period is 19.4 months, ranging between 18.3 and 21.5 months." In the table referred to the results are grouped under " age of mother," not under " age at marriage," but the text might suggest that what is implied is " age of mother at marriage" (age of brides). The table shews that from age at marriage 21 to 45 the average interval is sensibly constant, and only slightly larger than that deduced by Coghlan if in his Table VIII. " age at marriage" be substituted for " age of mothers." FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 273 analogous to the radius of gyration in mechanics. That is, we shall require the value of G where its square is given by : — (511) (?2 = /^^ /(^) ^^- J f {X) dx When /(a;) is a rational integral function {a -\- bx + etc.), this gives — (512). ...oGl = T*'+ ^^-^ '"" ,, ^ 12a + 66x + icx^ + 3dx^ + J|- ex* + .... a formula which is appropriate when the graphed areas extend from the origin. The values of b, c, etc., can be ascertained from the group-totals, see, for example, by formulae (195) to (197d), etc., pp. 67, 68. When the gra,ph-totals are not continuous to the origin, the solution is a matter of integrating between the same limits in both numerator and denominator. If the limits be x — ^k to x-\-^k, that is, if the middle of the group-range be taken as the value x in the formula, then it is easy to shew that (513). . . . Gl = — ^ -, ^ ^° a + bx + c{x^ + ^F) + d{x^ + ^Bx) + etc. Gm being the radius of gyration of the figure standing on the range referred bo, viz., x ± ^k. This formula can be readily recast into arith- metically convenient forms. When the function is a simple exponential one (5e*^), we have : — n / 2 \ 2 (514). . . .,Gl = ^-— ^ (^1 _ _J + — ; or generally (5l5)....,G;=^^[p-f^)+l^ in which n = e*. These are also suitable only for the figure starting from the origin. When the limits of the integral are p and q, we shall have (516).. ...«.• = '"' (^-|)-'K^ -I) +f.='g--^'+ ^- w* — «.'' in which last expression s = p — 1/6 and < = g — 1/6. When the values of the squares of the several " radii of gyration" have been obtained, the radius of gyration of the whole series of groups is given by : — (517).. Gl=={AiGi+A2G2 + eto.)/{Ai+A2+eto.)=i:{ AG) /a A Ai denoting the number in group 1, .^2 in group 2, and so on. The protogenesic quadratic index is computed in a manner analogous to that for computing the simple protogenesic index : that is by multiply- ing the square of the middle value of the successive yearly ranges of * This may be seen by adding l/b' to the first term, thus making the terms in brackets perfect squares when multipUed by q and p respectively ; and then multiplying both numerator and denominator by e- «. 274 APPENDIX A. duration by the number in the group : that is in formula (491) a;'^, a/'*, etc., is written instead of x' , cb", etc., a;', etc., here denoting the durations of marriage. 30. Correction of the protogenesic interval for a population whose characters are not constant. — When a population is increasing, all other facts remaining the same, the first-births, after a given duration of marriage {%), are drawn from a smaller population than are those for any lesser duration and presumably also from a smaller number of marriages. For comparative purposes, therefore, they need to be " corrected" so as to agree with what would be shewn by a constant population. Thus, were the ratio of first-births to marriages constant, it might very properly be assumed that the number of first-births to be expected would vary roughly as the ratio of the total marriages (marriages at all ages) for the period i years earlier, to the total number for the period being compared. Thus, if J-i be the total number in the former case, and J the total number in the latter, the correction to be applied would be^ : — (518) 1 -t- Ci = J /J.j a quantity ordinarily greater than unity, i.e.., Cj is ordinarily a positive factor since populations generally are increasing. We may, however, envisage the problem more rigorously as follows : Let M, with suffixes shewing the age, denote the number of mothers of first-born children, and / the number of women marrying, from which they were derived. Then in the case of a " constant population," in which also the relative frequencies of nuptial first-births were constant, the former number would bear a constant ratio to the latter, for any age in question ; that is to say, for any age and at any time we should have M/ J = fi, 3. constant. Actually this ratio, however, is not quite con- stant, hence, rigorously, the number of nuptial primiparse must be taken as : — (519) M^ = ^fifJ^ = J^ .f{x,t) In short we cannot take the marriages as the basis of the correction, but we should take what may be called their Malthusian equivalent ; that is the number of marriages so reduced (or increased) as to be of equal productive efficiency : thus, ju, J must replace J, and fj, is not a constant . The character of ^ may not be simple ; it is probably a function also of the interval elapsing before birth, i.e., (520) /x = f {X, i, t). The form and constants of this function can be ascertained only by computing jx for differing ages with different intervals and at different times. Thus, instead of (518) we should write : — (521) 1 + Ci = fi'J/{iM'.i. J.i) 1 This was pointed out by Sir (then Mr.) T. A. Coghlan, Childbirth in New South Wales, 1899, p. 26. He used this correction, which, however, would not be completely satisfactory if the " Malthusian coefficient" were increasing. FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 275 in which ;u,' denotes : — (a) the value of /x for a given age and interval, (when J and care to be ascertained for a given age and interval), or (6) : — its value for the total for all ages and for a given interval, (when J and c are required for the total of all marriages). For Australia the ratio M/ J is known only since 1893. During the period 1893-1914 it ranged between .790 in 1903, and .901 in 1912, for first-births and women of all ages (see hereinafter). As this average 0.0156 per annum for the 9 years interval between the years mentioned, it is of the same order as the yearly increase of population, and in the case cited would increase the correction. It may fall or rise 0.03 in one year. This term may be negleeted, however, because its effect is relatively negligible when the correction is large, so that it has very little influence on the result computed by ignoring it. This is shewn by the results in the following table : — TABLE LXXXVI. — -Correction to the Computed Average Inteival between Maiiiage and First-biith when Population is Increasing. Factors to be multiplied into the When the increase per unit per annum is, computed average interval be tween marriage and first-birth when the correction for increase is ignored 0.010 0.015 0.020 0.025 0.030 See (511) to (514). Multiply the compi ited interi tbX by the factor : — (a) When the first-births after 12 months are taken into account 1.0195 1.0294 1.0395 1.0500 1.0604 (6) When the first-births after 9 months are taken into accoimt 1.0132 1.0199 1.0267 1.0338 1.0408 (c) When all first-births are taken - into account . . 1.0083 1.0125 1.0168 1.0213 1.0257 It is to be remembered that the epoch to which the results refer is (sensibly) the middle of the year of observation, and that the intervals are 0, 1, 2, etc., years. Since the relative numbers for different intervals will probably differ from those of Australia but slightly for most countries, we obtain the following very simple rules : — (i) If the ratio of first-births to marriages increase continually at the rates indicated in Table LXXXVI., or (ii.) if that ratio be constant, and the number of marriages increase con- tinually at the rates in the table, or (iii.) if the sum of the ratios in ques- tion be as indicated in the table, then — The correction to the interval for all first-births occurring more than twelve months after marriage is For all first-births occurring more than nine months after marriage the correc- tion is . . For all first-births occurring marriage, the correction is after in which r denotes the rate of increase. Twice the rate of increase. 1 + 2r The rate of increase plus one-third. 1 +llr The rate of increase less one-sixth. 1 + %r 276 APPENDIX A. 31. Proportion of births occurrii^ up to any point of time after marriage. — The rate of occurrence of first-births, for different intervals after marriage, is well shewn by giving the proportion of the whole which have occurred up to any given time. The following table furnishes the proportions in question : — TABLE LXXXVU. — Shewing Fiopoition of Nuptial First-births occurring up to any point of time after Marriage. Up TO EHD OF MONTH. AGE OF MOTHEBS. 1 2 3 4 5 6 7 8 9 10 11 12 15 .0699 .1832 .3525 .4581 .5994 .71891 .8245 .8804 .8944 .9363 .9720 .9798 20 .0262 .0597 .1071 .1685 .2445 .3341' .4398 .5234 .6721 .6641 .7356 .7851 25 .0099 .0204 .0367 .0569 .0810 .1176 .1659 .207« .2577 .3932 .4958 .5683 30 .0070 .0140 .0229 .0362 .0493 .0685 .0945 .124C .1631 .2761 .3677 .4346 35 .0080 .0156 .0284 .0395 .0562 .0723; .0958 .1216 .1485 .2338 .3083 .3577 40 .0120 .0201 .0324 .0466 .0606 .0748! .0977 .1161 .1340 .1884 .2376 .2781 45 .0128 .0281 .0510 .0536 .0714 .0714 .0867 .102C .1122 .1658 .1888 .2194 13-52 . . .0160 .0345 .0614 .0953 .1365 .1872 .2502 .3027 .3484 .4595 .5461 .6078 13-52 i j 1 1 Proportion ol first year's 1 ' births dur- ing month .0264 .0304, .0443 .0557 .0677 .0834 .10371 .0863 .0753 .1827: .1424 .1017 Proportion of ' t first year's ' births up to end of i months . . .0264 .0.)68 .1011 .1568 .2245 .3079 .4116 .4979 .57321 .7559 .8983 1.0000 V P TO E ND OF YEAK. AGE OF MOTHBE.S. 1 i 1 2 3 4 5 6 10 15 20 26 15 1.0000 '.. (7. 20 .9634 : .9928 .9985 .999. > ' .9999 1.0000 (I!) 25 .8608 , .9402 .9729 .986. S .9933 .9996 1.0000 (i8) 30 .7278. .8458 , .9043 .936' ' 1 .9565 .9928 .9998 1.0000 (22) 35 .6455 .7631 , .8223 .86K i .8869 .9562 .9932 .9997 1.0000 40 .5704 .7052 ' .7842 .8314 I 1 .8585 ! .9202 .9676 .9916 (=5) 1.0000 45 .4974 .6276 .7270 .790f ; .8316 .9107 .9541 .9745 1.0000 13-52 . . .8555 ' .9273 .9567 .97ie .9801 ' .9943 .9988 .9998 1.0000 13-52 Proportion of first year's births dur- ing mont'^i 1 Proportion of i first year's 1 1 births up to end of 1 month 1.4075 1.5254 1 1.5739 ' 1.5984 1 1.6125 1 1.6359 1.6431 1.6448 1.6451 1 This table is interpreted as follows : — Taking the upper line, 13-52, 0.0160 of all nuptial first-births occur within one month of marriage, 0.3484 occur before the end of the ninth month after marriage, and 0.6078 before the end of the twelfth month. Again, of the nuptial first- births occurring, with women of all ages, during the year of marriage, 0.5732 are bom before the end of nine months, and all births exceed those bom during the first twelve months by only 0.6451. This is shewn on the last line of the table. 32. Range of the gestation period. — In order to accurately estimate the cases of first-births properly attributable to pre-nuptial insemination, the range of the normal gestation-period must be taken into account as FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 277 well as the frequency of premature live births. Contrary to popular opinion this gestation-period has a considerable range .^ The following data represent the best available results : — TABLE LXXXVIII.— Relative Freqiuency of Births after Different Periods, between the last Menstruation and Parturition. Authorities. Duration Days. Reid.* Hannes.f Hannes.J Hannes. Various! Reid, with 500 Cases. 561 Cases. 314 Cases. 875 Cases. 51 Cases. Hannes. 241-250 56 36 16 28 1 41 251-160 59 37 13 29 20 44 251-170 150 141 111 130 210 140 ^71-280 317 325 366 340 510 329 Maximum (days) (277.77) (277.73) (277.02) (277.42) (274.64) (277.58) 281-290 269 271 258 267 160 268 291-200 97 121 118 120 100 109 301-310 24 50 76 59 ? 41 311-320 18 14 22 17 ? 18 321-330 10 5 19 10 ? 10 Total Average! Duration 1,000 1,000 1,000 1,000 1,000 1,000 277.2 279.2 281.9 280.3 276.5 278.8 Note. — ^The oases for 241 to 251, 316-330, have been obtained by extrapolating Eeid's curve. * See Hart, Edinburgh Medical Journal, 1914, New Ser. XII., p. 401 ; also Journ. Edin. Obstetr. Soc, XXXVIII., pp. 107-134; 1912-3. Biometric analysis of some insemination-labour and menstrual-labour curves in certain mammalia. The distribution of Eeid's results according to the normal curve of probability for a table of frequency is unquestionably unsatisfactory, as an examination of the original data will shew. The distribution does not conform to the normal curve. The average is given as 278.3 ; it should be 278.84 ; there is an arithmetical mistake in the original calculation. t Zeit. f. Geburt und Gynak. LXXI., 1912, p. 524. Die korperliche En- twioklung der Frucht in ihrer Beziehung zur bereohneten Schwangerschaftsdauer. Walther Hannes. Children 3000 to 4000 grammes weight. X Same authority, children above 5000 grammes weight. § Interval reckoned from coitus, certain. These i .elude 51 cases reported by Desormeaux, Girdwood, Montgomery, Rigby, Lockwood, Lee, Dewers,' Beatty Skey, Mcllvain, Ashwell, Clay and Reid. The average durations indicated are not exactly identical with the maximum frequency, since the frequency curves are very sHghtly asymmetric. If Hannes' cases are combined with Reid's, a total of nearly 1400 is obtained. If the result be " smoothed," so as to agree with the final column of Table LXXXVIII., the result shewn in Table LXXXIX. on next page is obtained.^ 1 Other values are as follows : — Hippocrates, repl dxTa/i-^vov, generally within 280 days ; Hansen, 128 cases, 272.5 days after coitus ; see Handbuoh der Physiologie by Hermann, VI., 2., p. 73, 1881 ; M. Zbllner, after menstr., &st-births 279.1, second births 282.0 ; see Zur Kenntniss und Berechnung der Schwanger- schaftsdauer, Jenenser Dissertation, 1885, p. 6. Hasler, 195 cases, 281.0 ; after coitus 665 cases, 272 days ; Glusing, after menstr., 279.6 ; Wiirzburger Dissertation, 1888, p. 15 ; Voituriez, 274-8 after menstr. Thgse de Paris (Lille), 1885, p. 62 ; Winckel, 274.8, Lehrbuoh d. Geburtshiilfe, p. 78, 1889 ; Ahlfeld, 270.4 after coitus, Monatsohr. f. Geburtskr u. Frauenkr., XXXIV., p. 304. 1869. 278 APPENDIX A. TABLE LXXXE.— Shewing the Frequency per diem per 100,000 Births occurring- between the 240th and 332nd day after the Termination of the Menstrual Period. i Batioot Ratio of Eatio of Batioof Eatio ol Day No. Aggre- gate. Day No. Aggre- gate. Day No. Aggre- gate. Day No. Aggre- gate. Day No. Aggre- gate. 240 297 .00297 260 675 .09012 280 3,429 .56930 300 657 .93538 320 Ill .99424 241 303 .00600 261 733 .09745 281 3,318 .60248 301 597 .94135 321 101 .99525 242 310 .00910 262 807 .10552 282 3,196 .63444 302 546 .94681 322 91 .99616 243 318 .01228 263 911 .11463 283 3,014 .66458 303 496 .95177 323 81 .99697 244 327 .01555 264 1,052 .12515 284 2,847 .69305 304 455 .95632 324 71 .99768 245 1 338 .01893 265 1,305 .13820 285 2,676 .71981 305 420 .96052 325 61 .99829 246 349 .02242 266 1,548 .15368 286 2,504 .74485 306 389 .96441 326 51 .99880 247 : 361 .02603 267 1,784 .17152 287 2,332 .76817 307 361 .96802 327 40 .99920 248 1 374 .02977 268 2,015 .19167 288 2,160 .78977 308 334 .97136 328 30 .99950 249 ! 388 .03365 269 2,246 .21413 289 1,988 .80965 309 304 .97440 329 20 .99970 250 404 .03769 270 2,470 .23883 290 1,816 .82781 310 277 .97717 330 15 .99985 251 420 .04189 271 2,689 .26572 291 1,644 .84425 311 252 .97969 331 10 .99995 252 437 .04626 272 2,913 .29485 292 1,477 .85902 312 227 .98196 332 5 1.00000 253 455 .05081 273 3,132 .32617 293 1,320 .87222 313 207 .98403 333 254 474 .05555 274 3,420 .36037 294 1,189 .88411 314 188 .98591 240 255 496 .06051 275 3,455 .39492 295 1,077 .89488 315 171 .98762 to looiooo 256 521 .06572 276 3,501 .42993 296 976 .90464 316 156 .98918 333 257 551 .07123 277 3,511 .40564 297 885 .91349 317 143 .99061 258 ; 587 .07710 278 3,506 .50010 298 804 .92153 318 HI .99192 259 : 627 .08337 279 3,491 .53501 299 728 .92881 319 .99313 Maximum frequency occurs on tlie 277.67th day. Average (240 to 332 days) = 279.28 days.* • If tlie average date be found in the usual way (t.«., from the weighted mean), it will prove to be 278.78. But the births occurring on the nth day range between n and n + I, hence the average is about n + i, consequently the 278.78th day Is from 278.78 to 279.78 ; hence the average interval U> 279.28 about. It would appear from these results that the most frequent interval between the termination of menstruation and parturition, and the average interval, may be regarded for practical purposes as identical, and may be taken as 278 days on the average for births of children of ordinary weight, and that only 2 or 3 days need to be added in the case of the birth of heavier children. For first-births the interval is about 3 days shorter. From insemination to parturition the interval is slightly shorter, perhaps 5 or 6 days on the average. In view of social custom, however, the interval for first-births may be taken as say about 14 days longer than the 278, or about 292 days in all. Making allowance for live births occurring after 210 days from insemination, and for the fact that 40 per cent, of births occur between the 261st and 278th day from the last menstruation, (see Table LXXXIX.), we may take 274 days, or 9 months, as the period to be rejected as uncertain as regards post-nuptial conception. The frequency-curve for the interval between the termination of menstruation and parturition is curve E on Fig. 76, see later, page 284. 33. Piopoition of births attributable to pre-nuptial insemination. — It is evident, from the preceding table, that there is a certain period during which it is not possible to ascertain what proportion of births should be regarded as attributable to pre-nuptial insemination.^ The numbers ^ T. A. Coghlan in 1899 based his computations on the assumption of a 9- months iaterval, see Childbirth in New South Wales. He points out that ia the years 1893-8, the nuptial first-births registered were 41,384, of which 13,366, or 32.3 per cent., were " due to pre-nuptial conception." It may be observed that pre-nuptial insemination may have characterised some cases where birth occurred in the tenth or even eleventh month after marriage, and a small nvimber of births may be attribut- able to cases of post-nuptial insemination from 200 to 240 days after marriage, and a considerable number from 240 to 270 days. However, the jjercentage he deduced for New South Wales in 1893-8 seems, on the whole, to be confirmed by the present investigation for Australia, 1908-14. FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 279 per million nuptial first-births for women of all ages bom during various intervals after marriage are shewn on the penultimate line of Table LXXVII. By plotting the groups of first-births occurring monthly from 1 to 12 months, and drawing a continuous curve giving the same totals, re- sults are obtained analogous to those shewn on Pigs. 76 and 77. On the former figure the part of curve A, marked f, g, g', h, denotes the boundary of the groups, which may be attributed to pre-nuptial insemination. The curve i, i', j shews the boundary of the groups which may be at- tributed to post-nuptial insemination. On Fig. 77 the curve k, 1, 1 ', m, denotes the pre-nuptial insemination quota, and the curve n, n', o, p, the post-nuptial quota ; see page 284. By fixing the position of that part of the curve shewn by the dotted lines in the figures referred to, it would appear that about 0.634 of the births occurring during the 9th month after marriage are to be attributed to pre-nuptial insemination. Thus, about 0.952 of the first-births occur- ing within 9 months of marriage are due to pre-nuptial insemination. This is equal to 0.546 of all first-births occurring during the year of marriage, and 0.332 of all first-births, in every case for women of all ages. These ratios, it will be seen from Fig. 73, are a fairly definite function of the age of the mothers ; and this function could be ascertained by treating the group-results given in Table LXXVII. in the manner above described.^ 34. Issue according to age and duration of marriage. — The recording of the number of children borne by married women of various ages, and after various durations of marriage, furnish data of value in any attempt to ascertain the law of increase " according to age and duration of mar- riage." But it is to be kept in view that the immediate results from such data apply only to those who thus, through maternity, come under observation, and does not aipply to married women generally. That is to say, if averages be formed these averages are not averages for all married women of the given ages and durations of marriage. During the seven years, 1908-1914, 805,015 mothers came under observation in Australia, their total issue being 2,675,291, or an average of 3.3233 each. The results are shewn in Table XC. hereunder, the averages being found as follows : — Let jm"a, denote the mothers of age-group x — k/2 to a; -|- k/2, and of duration of marriage i — 1 to i, and let the total issue of these be iCx', then the average, ^Ca, is given by : — (522) ca, = i,Gx / im^'x and these are the averages which have been tabulated.^ 1 The attributing of the whole of the births occurring during the 9 months after marriage to pre-nuptial insemination, gives a, result somewhat too great. Nevertheless it is clear that for practical purposes it is a satisfactory rule for eliminat- ing the so-oalled " prejudiced" from the " unprejudiced" oases, to assume that, on the average, births occurring less than 9 months after marriage are " prejudiced." ' The original data will be found in the Population and Vital Statistics of AustraUa for the years 1908-1914, Bulletins 14, 20, 25, 29, 30, 31 and 32. 280 APPENDIX A. TABLE XC. — Shewing the Average Number of Children Bom to those who Bear during Varying Intervals after Marriage, based upon the Experience of Australia during the Years 1908-1914. Dura- Age-groups. (Age at Birth of Last Child.) ation of -19. 20-24.' 25-20. 30-34. 35-39. 40-44. 4.5- 1 AU Ages. 1 : Totals, All Ages.' Mar- riage. Jlothers. Issue. Years. VVEBAGE NUMBER OF CHILDREN. n-1 1.006 1.010 1.016 1.030 1.051 1.029 1.142 1.013 134,171 135,996 1-2 j 1.250; 1.157 1.085 1.087 1.089 1.113 1.151 1.125 61,213 68,906 2-3 1 1.9251 1.882 1.747 1.700 1.627 1.454 1.545 1.802 64,229 115,759 3-4 2.145' 2.171 2.087 2.039 1.997 1.923 1.786 2.107 70,317 148,160 4-5 2.4661 2.622 2.520 2.441 2.401 2.207 2.041 2.525 59,407 150,009 5-6 2.701 3.020 2.919 2.825 2.803 2.870 2.153 2.906 53,275 154,836 6-7 •2.750 3.401 3.339 - 3.194 3.216 3.038 3.000 3.290 47,250 155,476 7-8 3.000 3.776 3.731 3.576 3.544 3.447 2.846 3.655 41.713 152,461 8-9 . . 1 4.105 4.126 3.954 3.883 3.820 3.142 4.018 37,115 149,129 9-10 . . ' 4.292 4.514 4.330 4.271 4.149 3.940 4.374 32,170 140,725 10-11 .. '4.347 4.910 4.705 4.600 4.619 4.318 4.726 29,607 139,942 11-12 1 4.950 5.256 5.122 4.965 4.954 4.931 5.091 25,887 131,795 12-13 4.571 5.541 5.513 5.329 5.319 5.037 5.443 23.372 127,226 13-14 i 5.790 5.868 5.725 5.608 5.761 5.718 20,339 117,691 14-15 } . . 6.131 6.269 6.091 6.056 5.721 6.156 17,572 108,160 15-16 ■ •6.24 7.434 6.453 6.324 6.493 6.494 15.217 98,827 16-17 i . . 5.59 6.967 6.859 6.688 6.844 6.844 13,271 90,836 17-18 j ;; 5.16 7.239 7.401 6.985 7.282 7.193 11,617 83,539 18-19 5.00 •7.371 7.679 7.431 7.291 7.575 10,073 76,308 19-20 •• 7.480 8.018 7.865 7.775 7.926 8,520 67,530 20-21 7.111 8.418 8.282 8.168 8.329 7,424 61,839 21-22 6.192 8.824 8.750 8.449 8.751 5,988 52,403 22-23 5.60 9.154 9.230 8.962 9.191 4,726 43,437 23-24 9.609 9.503 9.171 9.483 3,561 33,770 24-25 16.00 9.265 9.973 9.700 9.884 2,664 26,330 25-26 •9.053 10.450 10.500 •9.932 1,809 17,967 26-27 9.105 10.730 10.773 10.16 1,146 11,637 27-28 7.000 10.860 11.150 10.54 643 6,781 28-29 11.260 11.480 10.71 383 4,102 29-30 11.210 11,840 10.75 192 2,064 30-31 12.00» •12.220 12.51 77 963 31-32 13.00 11.770 9.51 45 428 32-33 10.00 12.460 12.94 17 220 33-34 14.80 7.80 5 39 AllTJura- tions 1.202 1.760 2.643 3.837 5.341 6.997 8.565 3.3233 Totals all dur'tions i Mothers 29,371 185,694 239,066 181,191 118,310 46,705 4,678 805,015 805,015 Issue . . ' 35,292 326,868 i 631,954 695,220 626,641 326,095 40,181 2,675,291 2,675,291 Owing to the limited data, the values are not reliable for the age-group 45, nor for the values shewn by the asterisks and those for greater durations of marriage. The table shews that, for all ages, the average total issue of married women, with various durations of marriage, who each year appear in the Australian maternity records, increases approximately at the rate of one child in 2.745 years, or 0.3643 'of a child per annum. The results are graphed in Fig. 75, p. 268. The parallel dotted lines in the figure shew that the rate of increase of the total issiie according to the duration of marriage is identical for all ages, at least for the greater part of the range of duration. That the graphs approximate so closely to straight hnes, and, moreover, to parallel straight hnes, is remarkable.^ These hnes may be defined by equations : — (523) c"x =aa:+ bi = 0.6667 -|- 0.3643i, approximately ; 1 This characteristic can no doubt be deduced, but no explanation of an elementary nature can be offered. FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 281 in which only ax is dependent on the age of the mothers, being about |, and b is constant for all ages. The more exact values of a are given in Table XCI. hereinafter. The results ^hewn in Fig. 75, p. 268, and detailed in the table referred to, can be referred in a general way also to the age-groups, that is to say, if yx denote the average issue for mothers of a given age-group for all dura- tions of marriage, then the number is as shewn in Table XCI. The average ages for these age-groups, as shewn in the table, are found on the sup- position that the distribution of the cases of nuptial maternity occurring during the period 1907-1914, in Australia, apply. This distribution is given in Table LXXIII., p. 242, and the average ages of each age-group have been calculated strictly^ : these are as given hereunder. TABLE XCI. — Shewing the Total Issue foi Mothers in various Age-groups, for All Durations of Marriage ; the Constants of Formulae for Computing this Number, and the Differences between the Observed and Computed Numbers. Australia, 1908-1914. Age-group Average age . . -19 18.92 20-24 22.87 25-29 27.46 30-34 32.35 35-39 37.29 40-44 41.91 45- 46.29 13-52 Average number of children, all dura- tions of marriage Smoothed resultt 1.202 1.242 1.760 1.751 2.643 2.636 3.837 3.895 5.341 5.413 6.997 6.994 8.565 8.764 3.3233 The above crude and smoothed (Crude) results are equivalent to dura- (Smooth- tions for ed) all ages of: 1.37 1.48 2.90 2.88 5.33 5.31 8.60 8.76 12.73 12.93 17.28 17.27 21.58 22.13 Crude Smoothed Values of Ax for age- group Value of 6 .6515 .3643 .7909 .3643 .7778 .3643 .6921 .3643 .6646 .3643 .5977 .3643 .4939 .3643 .7029 .3643 Calculated Values of . Ax+bu when u = and the value of ft g 1 1 2 ■a 3 ^4 ^ 5 1.016— .010 1.380— .130 1. 744+. 181 2.109 + .036 2.473— .007 e 1.155— .145 1.520— .363 1.884— .002 2.258— .087 2.622— .000 1.052— .036 1.416— .331 1.781— .034 2.155—068 2.519 + .001 1.056— ,026 1.421— .334 1.795— .095 2.159—120 2.524— .083 e 1.029 + .021 1.393- .304 1.758— .131 2.122— .125 2.486—085 0.962 + .067 1.326- .213 1.691— .237 2.055— .132 2.419^.212 0.858 + .284 1.223— .072 1.587— .042 1.951— .165 2.315— .274 1.067— .054 1.432— .307 1.796 + .006 2.160— .053 2.524+. 001 t The smoothed result conforms to a rational integral equation of the fourth degree. i e is the quantity which, added to the tabular value (calculated), makes it identical with the data. The smoothed results for the average number of children, according to age, for all durations of marriage, are given by : — (524) yx = l +bx+cx^ + dx'' + ex* ; in which x =* — 13, and the values of which for 2 J years' intervals are as follow : — TABLE XCII.— Shewing the Effect of "Age of Mothers " upon the Total Issue for All Durations of Marriage. Australia, 1908-1914. Ages at bkth of last child, in years . . Children* .. 13 1.000 15.5 1.019 18 1.160 20.5 1.413 23 1.770 25.5 2.221 28 2.760 30.5 3.378 33 4.070 35.5 4.829 38 5.650 40.5 6.528 43 7.460 45.5 8.441 48 9.470 Difference for 2i yrs 0.019 0.141 0.S53 0.357 0.461 0.539 0.818 0.692 0.759 0.8S1 0.878 0.93S 0.981 1.029 • That these are given by a curve of the fourth degree, can be readily seen by taking the values for 13, 18, 23, etc. ^ That is, the numbers are referred to the exact average for the year of age, not merely to the age for the middle point. 282 APPENDIX A la the above table the differences for 2^ years shew that for all durations of marriage, differences of age have much less infliience than differences in duration. To obtain this relationship exactly, it is necessary to compile for each age, and for given durations of marriage the total issue. For all age-groups the general result is 0.3643 a child per year, that is 0.9107 for 2| years. Prom the above table, however, it would appear that this value is not attained for " all durations of marriage" until, almost exactly, age 40. Such results as are referred to, are dependent upon the combination of two things, viz. : — (a) The age-effect proper, and (6) the fact that for the higher ages the average of the durations of marriage are greater, and thus, throughout the range of observation, the conditions are not homo- geneous. 35. Initial and terminal non-linear character of the average issue according to duration of marriage. — An inspection of Fig. 75, p. 268, and the results given in the preceding table, shew that there is a more or less systematic departure from hnearity at the terminals of the graphs repre- senting " issue according to duration of marriage." The table reveals the fact that the character of the differences, according to age, and for various durations of marriage, between the values according to formula (523), and the individual results are as follow : — (i.) For the first year of duration of marriage, the computed total issue for ages under 35 is too great, and for ages over 35 is too small, (ii.) For the second year of duration of marriage, the computed total issue is invariably too great, the maximum difference being at about age 24. (iii.) For the third year of the duration of marriage, the computed issue is less than the actual for the younger ages, but soon becomes greater, the maximum difference occurring at about the age 43 or 44. (iv.) The same remarks apply to the fourth year of the duration of marriage with the exception that the age is later than 45. (v.) In the fifth year of the duration of marriage, the differences are small until the age of 40 is reached, when the computed result becomes markedly greater than the actual. The relatively large differences for the various age-groups character- ising the second year of the duration of marriage are due to the fact that the length of the period, which must necessarily intervene between a first and second birth, does not admit of so wide a " scatter" of the cases of maternity as to make the result uniform ; thus the average for the second year is in defect. This consequence is one which will (and does) tend to vanish for longer durations of marriage, owing to the fact that any want of coincidence of the intervals between birth and birth must more markedly characterise the points of time in proportion to their remoteness from the first year of duration of marriage. Owing to the fact that the period of FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 283 gestation alone is three-fourths of a year, and the period of lactation a considerable part of a year, and to the fact that so great a proportion of births appear in this year, it follows that the second year of duration must necessarily disclose a falling off in the apparent average. As time goes on, however, this apparent defect will tend to disappear, as will be clearly seen by a reference to Fig. 75, p. 268. The character of the curves at their terminals for the longer durations may be fairly well ascertained by combining the terminal values. This has been effected as follows : — In the series shewn on Fig. 75 the two differences between the three last averages of the issue of curve for under 20, are taken, and similarly the four differences between the five last averages of the issue, etc., the number of values (averages of issue) being respectively 3, 5, 8, 9, 10, 12, and 12. The means of the differences, the numbers of which are respectively 7, 7, 6, 6, 5, 5, 5, 4, 2, 2, are taken, the results being as follows : 1-0.230, +0.285 + 0.582, +0.106, + 0.153, —0.105, — 0.162, + 0.060, —0.246, +0.845, —0.489. The accumulated results compared with the successive multiples of 0.3643 furnish the co-ordinates of the average terminal shape. This gives : — .364, .230 .729 .515 1.093 1.097 1.457 1.203 1.822 1.356 2.186 1.251 2.S50 1.089 2.914 1.149 3.279 .903 3.643 1.748 4.007 1.259 Diff. Smth'd .134 .025 .214 .100 — .004 .225 .254 .400 .466 .625 .935 .900 1.461 1.225 1.765 1.600 2.376 2.025 1.895 2.500 2.748 3.025 The differences shew the amounts by which the successive points fall short of the line defined by the formula (523) . As is shewn by the smooth- ed values, the defect from the linear condition, once it initiates, increases, on the average, very approximately as the square of the duration from the initiating point onward. This average defect ij is expressed by the equation : — (525). 7] == 0.025 P I denoting the duration reckoned from the initiating point. This point may approximately be found as foUows : — Average age at, birth Initiation of droopf Difference . . 18.9 6.0 12.9 22.9 10.0 12.9 27.5 15.0 12.5 32.4 18.0 14.4 37.3 24.0 13.3 41.9 29.0 12.9 46.3 33.? 13.3 Aver. 13.1 • ».«., Age oi mother at birth of children. f Years of duration of marriage. In these results the first line gives the average age of women at the time of maternity, and the second line gives the points where the droop from the linear relationship commences : the positions of these points being estimated from the graphs. Fig. 75, p. 268. The differences give a sensibly constant age, which is seen to average 13.16, hence the droop implies that the fecundity of those who are characterised by early marriage and late motherhood is less than the average for those who may be regarded as falling into the normal place. 284 APPENDIX A. Fig. 76. Fig. 79. 140.000 2 16 Fig. 77 ._los. Curve ' C, , 18 monlbs after marriage years Cntvea C,C Fig. 78. Fig. 76. — Curve A denotes the frequency, according to duration of monthly- groups, of first-births, viz., the number of cases in a total of 1,000,000 first-births for all durations of marriage (see Table LXXVII., pp. 252-3). ■ T?he curve f, g, g', h, denotes the relative numbers attributable to prenuptial insemination, and the curve i, i', and j, etc., the relative numbers attributable to post-nuptial insemination. Curve B denotes the frequency, according to duration, of yearly groups, with a less extended lateral scale, the point g" thereon corresponding to g on Curve A. Curves B', B" and B'" are plotted on a larger vertical scale, y' and y" being the same point as y, and z ' and z ' ' the same point as z. Curve E is the curve of relative frequency of birth, according to the interval after the last menstruation, see Table LXXXIX., p. 278. Fig. 77. — Curve C shews the relative maximum frequencies according to age (i.e., for any age). The points 1, 1' and m, and n, n', o and p have the same signific- ance as points g, g' and h, and i, i' and j in Fig. 76, curve A, and the point k corresponds to f. Curves C ', C", are an extension of curve C, the lateral scale being altered. The point p' is the same as p, q' as q, etc. Curve D denotes the ratio, according to age, of first-births, to married women. It appears to be compounded of two curves, viz., u, u' and v, v', w, s. Curve D ' is plotted on a larger scale, the point s ' being identical with s. Fig. 78 illustrates the formulae for determining the exponential curves so as to make the shaded areas equal to the areas of the rectangles Aj and Aj, in order to determine the positions of the centroid verticals, etc. See formidse (496) to (510), pp. 264-5. Fig. 79 is the graph of the approximate average intervals to between marriage and the " unprejudiced" first-births for New South Wales, 1893-1898, and for the Commonwealth, 1908-1914 ; the light zig-zag line marked W denoting the result for the former, and the heavy zig-zag line marked T denoting that for the latter. The figures denote months, and the lateral divisions denote two years' duration. FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 285 36. The polygenesic, fecundity, and gamogenesic distributions. — As we have seen, there are two ways in which records of issue, according to age and duration of marriage, come to hand, viz. : — (i.) When, at the registration of births, the age, duration of marriage, and "previous issue" are also registered ; and (ii.) When, at a Census, the age, duration of marriage, and total issue are ascertained. There are certain differences between these. In (i.) the total age-range covered is that of the child-bearing period only ; in (ii.) the age-range is from the earliest age of maternity to the end of life. In (i.) the cases come under observation ditrmgr a period of time ; in (ii.) they come under observation at a given moment. Hence, to deduce (ii.) from (i.) it is essential that the necessary records of births, migration, and deaths should extend over a long period of time, and even then, the deduction of (ii.) from (i.) is by no means simple. Both records are, however, of value statistically and both yield appropriate measures of fecundity, though on the other hand both require corrections if they are to represent what would have been furnished by a " constant population." If, on a plane, the ages of mothers [x) be plotted as abscissae, and their duration of marriage {y) be plotted as ordinates, and if then verticals to this x^z-surface be drawn denoting the number of cases of maternity, corresponding to each age and duration, the surface so defined may be called the genesic distribution at maternity, or simply (i.) the polygenesic distribution. Similarly if the verticals denote the number of children recorded at any moment as having been borne by women of any age and duration of marriage, the distribution may be called the general genesic dis- tribution, or (ii.) the fecundity distribution.''- The fecundity-distribu- tion-contours, or lines denoting equal issue for various ages and dura- tions of marriage, can be drawn by means of formula (523), together with the values of the constants given in Table XCI., the values of the durations (according to age) where the linear condition ends, see § 35, and formula (525). If 11 be assumed to be, the earliest age of what may be called " extraordinary marriage," and 14 be assumed to be the earliest age of " ordinary marriage," and if also the generally approximate result, be adopted, viz., 0.6667 + 0.3643 i, the plan of the polygenesic surface will have for a limiting boundary the line y ^ x — 11; its surface will, for the major part, be (approximately) a plane, steepest at right angles to the axis of abscissae (age), and making an angle 6 with the xy plane, the tangent of which angle is 0.3643. For any age X, .the line on the surface denoting increasing durations of marriage, 1 The assigning of the word " polygenesic" to the one, and " fecundity" to the other distribution, is, of course, somewhat arbitrary : the terms might, of course have been interchanged. 286 APPENDIX A rises uniformly till it attains the value y = x — li. For greater durations than this the surface will droop. Between the axis and the contour- line representing say the third or fourth child, the surface is somewhat irregular. If the distribution is based on the ages at marriage and the duration of marriage, it may appropriately be called the gamogenesic distribution. The abscissae then are the ages of mothers when married (i.e., " ages at marriage"), and the ordinates, as before, are the duration of marriage. 37. Diminution of average issue by recent maternity. — ^Returning to the results shewn in Tables XC. and XCI., for the second and subsequent years of duration of marriage, it may be noted that they are important in any attempt to ascertain what may be called the unmodified fertility - ratio. When the fertiUty-ratio is found by merely dividing the total number of cases of nuptial maternity at any age by the number of married women at the same age, the quotient is " modified" by the fact that they are not ail at equal risk. If the fertiUty-ratio is to shew what is due to change of age alone, or rather, to change of age, unmodified by the effect of a recent birth, but unaffected as to all other factors, a certain proportion of the married women should be subtracted from the total. We shall first consider the question of estimating the diminution of average issue by recent cases of maternity. Formula (523), shewing the general rate of increase in the average issue, (since it is derived only from aU cases of maternity coming under observation for each duration), gives what may be called " the unmodified rate of increase" for what also may be called " the fertile section only" of the whole body of married women ; see § 34, hereinbefore. Con- sequently the differences of average issue for successive durations of marriage, although an indication of, do not give a very exact measure of the proportions of women who are virtually removed from risk. These proportions are doubtless better defined by the differences between the observed average and the average issue computed upon the assumption of constant average rate of increase per year of duration. Hence the ratio of the diminution in the cases of maternity for any given age-group and for any given duration of marriage may at least approximately be foundjas follows : — Let c" be the average number of children (or average issue) on the supposition of a uniform increase, and c the actual number, each with suffixes to denote the duration of marriage and age . Then the diminution- ratio, that is the amount by which any previous births wUl have dimin- ished the actual record of cases, will presumably be c/c" But this diminution-ratio appUes only to the cases in which maternity has occurred. FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 287 Consequently if the values of this fraction be formed, for successive years of duration, commencing not from marriage, but from the number for th^e first year of duration of marriage, it will furnish a rough estimate of the correction necessary, if it be desired to ascertain, from the number of cases actually occurring, the number of cases that would have occurred had the whole of the women in any age-group been at full average risk. If to the values of c ", f or duration to 1 , given in the top line of Table XC, successive multiples of 0.3643 be added, and the sums, so formed, be subtracted from the values on the second, third, etc., lines of that table, we shall obtain the figures shewn on Table XCIII. on next page. These figures afford a fairly good indication of a systematic effect, according to duration, that is, of an effect which varies with age. This variation is not the same for each duration, and appears to change somewhat irregularly with age. The mean of the changes gives a fairly regular curve (see the upper part of Table XCIII. ).^ The individual graphs for the various durations, however, appeared to shew that the adoption of this general average for each series, was of doubtful validity, and for this reason a different linear change according to age was adopted for each duration. In any attempt to estimate the diminution of the numbers at risk by means of the falling off in the average issue, according to duration, it is probably desirable to take the adjusted results in the upper part of Table XCIII. This will give — .186/.364, + .177/.729, etc., for age 18.92, — .217/.364, +.177/.364, etc., for age 22.87; and so on. The results are shewn in Table XCIV. If we call the tabular value c"', the ratio p of the altered risk to the average risk is given by : — (526) p = I + c'" / 0.3643 = 2.745 (0.3643 + d"). The value of 1 — p will be required ; it is consequently : — (527) 1 -p = - 2.745 c'". Since c"' is negative, if for any duration of marriage fewer women than the average have given birth to children (owing to a recent birth, etc), then this last expression is positive. Table XCIII. shews the deviations, according to age and durations of marriage up to four years ; from the general rate of increase. '■ The curve can be very closely represented by the curve a+ bx + cX" , where n is greater than 1. Smoothed, the values would be about + .000, —.031, —.072, -.124, -.183, -.265, -.422. 288 APPENDIX A. TABLE XCIII.— Shewing the Average Effect of a recent Maternity upon the Average Issue (Number of Children) Corresponding to Various Durations of Marriage, and of a Consequent Correction. Mothers Excess ( + ) or Defect ( — ) in the Average Number of Children, on an Average (Linear) Increase according to Duration of Marriage. group. Years. -19 20-24 25-29 30-34 35-39 40-44 45- Duratlon of Marriage. Crude Besults. •1-2. 2-3. 3-4 -.120 .217 .295 .307 .326 .280 -.355 + .190 .143 .022 —.059 .153 .304 —.326 + .046 .068 —.022 .084 .147 .199 —.449 4-5. Mean. —.007 + .027 .210 —.054 + .047 .062 —.046 .124 .107 .183 .279 .265 —.558 .422 Aver. Difference for an age-difference of 10 yrs. — .077 Mothers Age- group. 19- 20-24 25-29 30-34 35-39 40-44 -45 Adjusted Results. tl-2. -.186 .217 .252 .292 .328 .363 -.396 2-3. 3-4. + .177 + .078 .106 .031 .023 —.024 —.070 .086 .154 .141 .237 .197 —.815 —.249 + .049 — .014 .086 .168 .241 .313 —.382 —.120 -.158 Aver- age Age. 18.92 22.87 27.46 32.65 37.29 41.91 46.29 Average Increase in the Average Number of Children. Crude Results.t Adjusted Eesults.§ .244 .675 .220 .321 .245 .690 .240 .453 .147 .725 .289 .451 .147 .680 .275 .446 .069 .662 .340 .433 .088 .650 .310 .429 .057 .613 .339 .402 .053 .600 .34.5 .402 .030 .538 .370 .404 .032 .530 .380 .365 .084 .341 .469 .284 .019 .440 .415 .328 .009 .394 .241 .255 .011 .330 .450 .271 Age. 17.5 22.5 27.5 32.5 37.5 42.5 47.5 Values of 1 — P = — 2.745 c'" Moth- ers' Age. jt=l 12 13 14 15 16 17 18 19 20 21 22 23 24 25 30 35 40 45 50 + .364 .385 .407 .428 .449 .470 .491 .512 ,534 .555 .576 ..597 .618 .639 .745 .850 .956 1.062 + 1.167 *=2. J;=3. 4=4. — .777 .728 ,679 .629 ..580 ,531 .482 .433 .384 .335 .286 .236 — .187 + .059 .304 .550 .800 + 1.041 .376 .343 .310 .277 .244 .211 .178 ^148 .113 .080 .047 .014 .151 .316 .480 .645 .810 — .304 .261 .218 .174 .131 .088 .044 — .001 + .043 .086 .129 .346 .563 .780 .997 + 1.214 * These results are found by adding multiples of 0.3643 to the figures in the first row of Table XC, and then subtracting them from the figures for the corresponding duration in the successive columns. t These results are the linear smoothings of the crude results. The linear adjustments are made by using the " average" ages, and can be regarded only as fairly satisfactory. The total number of cases of maternity analysed is, however, large ; viz., 805,015. t These rows are the differences of the columns in Table XC. § The adjustments follow no general law : tlie first is on a curve jle' **, the second is A' — Bx', the third. A" + B'x, and the fourth A"'—E'x — Ca;', the intervals x^—x^, etc., between the age groups being taken as always ol equal value, i.e., the adjusted values are for 17.5, 22.5, etc. The above table appears to shew that the period of time over which the influence of a case of maternity extends on the average, follows no simple law, and is by no means negUgible for some years, especially as regards the later portion of the child-bearing period. The whole method is not quite satisfactory, but is the best available, until the record of the procreative history of a large number of married women is to hand, giving the intervals between marriage and the births of successive children preferably compiled for intervals of single months from at least one to sixty, and for somewhat larger intervals (quarters, half-years, or years), to the end of the child-bearing period. Such statistics would reveal accurately the characteristic of the frequency of maternity according to duration of marriage, and would allow of the ratio p referred to in formulae (528, 529) hereinafter being more exactly ascertained .* ^ As far as I am aware such a statistic has aot been compiled, although it is of considerable ituportanoe. FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 289 38. Crude fertility according to age corrected for preceding cases of maternity. — The ratio [m/M), between the number of nuptial mothers (m) of a given age -group during a given period of time, to the total number [M) of married women of the same age-group, is not the true monogen- ous-fertility -ratio, inasmuch as the M married women are not homogene- ous as regards the maternity-risk (p ) to which they are subject. Obvious- ly m/M is too low a value for women whose fertility remains in abeyance, and is too high a value for women who have just borne children. The survivors after the lapse of k years of the married women of age x last birthday are 2/j,+ft / L^.^ Consequently if p^. is the average risk for the jfcth year after a birth (calling the year of birth 0), the corrected fertihty ratio (p") is given by the eq^uation : — (528) 1^ '^ *^ ilf^- |m^_i .y^ (l-pi) (i-r^_i)-fm^_2.^ (l-p2)(l-2r,_2+etc. } (1— fcr^..^) denoting the rate at which the mothers of age x — k have increased in k years. This may perhaps be ordinarily taken as the same at all ages, and as the rate of the population increases. The above formula may be put in the following form, viz. : — (529) . V --W "■■'-i^'iT;<'-"'"-"'+-+™i?-&"^'""'-'"'+-} and the ratios of the m/M quantities in the denominator do not need to be very exactly computed. It will always be abundantly accurate for the purpose in view to assume that : — (530) i/^/Vft = 1 - P fe + ^x-k) a formula which is satisfactory through a fairly large range for ^.^ Since the quantity between the braces in (529) is positive and small, its effect is to increase the value of p" The correction is important in any attempt to ascertain the age of greatest fertility, con- sequently the values given in Table LXXIII., p. 242, are those with which we are mainly concerned ; see columns ix. and xv. therein. The values of the factors (k) of m/M in the denominator of (529) can be readily tabulated for say r =0.01 and 0.03. ^ Iix denotiixg the mean population living in the year of age x : as in the ordin- ary actuarial notation. 2 For example from Australian Life Tables for 1901-1910, Report of Census, Vol. III., pp. 1217-8, we have for ages 40 and 30, from the L values 0.93986, and from the a values 0.93815, i.e., for so large a value of h as 10, the error is less than 0.002. 290 APPENDIX A. The value of the L, p and r terms are as follows for Australia : — TABLE XCIV. — Shewing the Factors Beauired to Correct the " Grade Feitility- ratio," for Preceding Cases of Maternity. AustraUa, 1908-1914. Values of (1— *r) L^/L^.^ Values of « when r = .01. Age of Mother. r = .01«; i = lto4. r = .01 and .03 ; * = 1 to 4. 1. 2. 3. 4. 1. 2. 3. 4. 15 20 25 30 35 40 45 .9879 .9868 .9858 .9849 .9840 .9830 .9821 .9761 .9739 .9720 .9701 .9683 .9663 .9845 .9843 .9813 .9584 .9556 .9529 .9500 .9472 .9527 .9590 .9450 .9414 .9378 .9340 .9304 + .423 .414 + .527 .516 + .630 .617 + .733 .718 + .837 .820 + .940 .921 + 1.043 1.022 —.662 .635 —.422 .405 —.182 .175 + .057 .055 + .294 .282 + .531 .509 + .787 .736 —.331 .311 —.171 .160 —.013 .012 + .144 .135 + .301 .282 + .456 .428 + .611 .573 —.290 .288 —.084 .077 + .122 .112 + .326 .299 + .528 .484 + .728 .687 + .927 .850 * To find the values for any other value, r ' say, of r, multiply the tabular values by (r '— r) / r. t To find the values for any other value of T, multiply by 0—rk) / (1 — .Olt). Thus, for r=.02 the multipliers of the successive columns are 0.9899, 0.9796, 0.9891, 0.9583 ; and if r= .03 the successive multipliers are 0.9797, 0.9592, 0.9381, 0.9167. The above values are very approximately given by : — (531) (l—kr) L^ /L^-^. = 1 — 0.000188A; (47.7 + x) ;i and those for the correcting factors e by : — (532) . . . . ei = 0.02070 {x + 5.43) ; (532a) (5326). ...es= 0.03140 (a;— 25.54) ; (532c) Formula (525) may thus be written : — (533) €2 = 0.04763 (a;— 28.91); €4 = 0.04057 (a;— 22.15). 1 -^ (/carnal + +«rtw^t) k being the tabular value given in Table XCIV. (in which r = .01 and r = .03), and the probabihty of maternity ascertained by this last formula, will be free from the effect of recent cases of maternity : that is the crude probability must be multiplied by the fraction foUowing m/M. 39. Age of greatest fertility.— When the probabilities according to age of maternity have been corrected so as to represent what would be given if aU women were at equal risk, then the age of greatest probabihty may be regarded as the age of greatest fertihty. Applying formula (533) to the data in Table LXXIII., p. 242, we have the following results about the maximum : — 1 More exactly the valvies of the constant to be added to x are 47.60 46 81 47.63, and 48.63, and of the coefficients to be multiplied into k are 6 OOOlQ^iq' 0.0003866, 0.0005700, and 0.0007433. "-uuuiMrfrf, FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 291 TABLE XCV. — Shewing Corrections to the Fertility-ratio for Ages 13 to 23, when Allowance is made for Preceding Cases of Maternity. Age of Values ol k when 4 = 1 to 4. Factor Kin = ?" Fertility-ratio. Mothers. 1. 2. 3. 4. Crude. Smoothed. Crude. Corrected. 13 + .374 —.727 —.371 —.342 1.039 1.001 .5? .52? 14 .394 .681 .341 .304 1.011 1.013 .2055 .2076 15 .414 .635 .311 .266 1.013 1.024 .2269 .2299 16 .434 ..589 .281 .288 1.012 1.036 .4063 .4112 17 .466 .543 .251 .190 1.048 1.048 .4316 .4521 18 .475 .497 .221 .152 1.066 1.059 .4776 .5093 19 .496 .451 .191 .114 1.077 1.071 .5022 .5409 20 .516 .406 .160 .077 1.092 1.083 .4540 .4956 21 .536 .369 .130 .039 1.074 1.094 .4375 .4700 22 .656 .313 .100 —.001 1.106 1.106 .4167 .4596 23 + .677 —.267 —.070 + .037 1.123 1.117 .3813 .4283 Although the values of k are of the same order of magnitude, yet within the range shewn, the values of the successive ^m-terms rapidly diminish, so that although there is no theoretical justification for stopping at & = 4, the inclusion of later terms would but slightly afiect the result (at least in the second place decimals). The factors K shew that about the age of maximum fertihty the correcting factors to give the fertility, unprejudiced by previous cases of maternity, increase linearly with age, and are represented very ap- proximately by thrB formula : — (534) K = 1 + 0.01163 {x - 12.91). The values for these factors, so computed, are the smoothed values in the preceding table. A smoothing, independent of that already given in Table LXXXIII., gave, as the maximum for the uncorrected fertiUty-ratio, 0.483 ; and a similar smoothing of the corrected values gave 0.517, the maxima and corresponding ages being : — Uncorrected, age, 18.8,i 0.483 ; corrected, age, 19.0, 0.517.2 In the method outlined, of correcting the crude fertiUty-ratio (proba- bility of maternity), equal " weight" is attributed to the values of k. An examination of Fig. 75 shews, however, that the " weight" to be attributed should probably decrease with increase in the value of k (that is with the number of years elapsed since a previous birth). Moreover, the change in the numbers of married women and cases of maternity is so rapid at the ages of maximum fertility that the age divisions should be less than one year, and the ages need to be very exactly given, which unfortunately they are not. For these reasons great exactitude in regard to the correction is at present impracticable. 40. Fecundity-correction for infantile mortality .^The frequencies of child-bearing as between two populations are, like their birth-rates, rigorously comparable as accurate measures of fecundity, only when their infantile mortahty-rates are identical, and the crude frequencies require, 1 The result in Table LXXXIII. was 18.23 years. ' The factor, according to (534) above, gives, on multiplying into, 0.483, 0.5168. 292 APPENDIX A. therefore, a correction, to reduce the risk of maternity to an equality ;^ see Part XI., §§ 4-6, pp. 145-152. It has been shewn that the infantile mortaUty correction to birth-rate is, on the whole, about ^g = ^ (l-fO.OSS/i) ; see p. 145. If, therefore, there were two equal populations of say married females (M), of equal fecundity (/), but with different rates of infantile mortaHty, we should have for the cases of maternity (m) occurring therein, respectively : — (535) mi = fM (1 + kiiJLi), and m^ = fM (1 + hfiz) ; whence it follows that (536). •/ = nti mz M(l+ Vi) -^ (1 + V2) Thus the correction is always very small, and, in general, is practically negUgible. 41. Secular trend 0! reproductivity. — ^The crvde reprodiictivity may be measured by the ratio of the number of confinements to the number of persons at uniform risk ; thus the Nuptial and Ex-nuptial Maternity- Ratios, etc. .008 Curve A is the ratio of nuptial con- finements to all married women. Ciirve B is the ratio of ex-nuptial confinements to "unmarried" women of 12 years of age and upwards. Curve C is the ratio of the ex- nuptial to the nuptial confinement rates, the range being between .038 and .059. Curve D shews the variation in the average number at a birth. Curve E shews the variations in the survival factor for the first year of hfe. crvde nuptial reproductivity is the ratio of nuptial confinements to the total number of married women, and similarly, Uie crude ex-nuptial reprodiictivity is the ratio of ex- nuptial confinements to the total " unmarried," which here wiU include the " divorced " and " widowed." The ratios are " crude," since no corrections have been applied for age-differences in the female population, and it is obvious from columns ix., X., XV., and xvi. of Table LXXIII., p. 242, that fertility greatly varies with age. For this reason, whenever the age-distribution is not identical, the results are not strictly comparable : they do not rigorously measure the degrees of reproductivity, or of malthusianism, operating. Con- sequently, for strict comparisons, a properly determined index of initial reproductivity would have to be computed, see §§3 to 6, pp. 235-239. Neglecting this, however, for the present, and restricting the consideration to the crude initial nuptial 1 It may be noted that after deducting the period of gestation and the puerperal period, there remains about one-sixth of a year during which mothers of the first- sixth of any year of record may give birth to a second child even in the same year and the chance of this occurring is increased by the death of the child born ' FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 293 and ex-nuptial reproductivities, the results are set out in Table XCVI. hereunder ; see columns (ii.), (iii.), and (v.) thereof. The results are shewn also by curves A and B of Fig. 80, the former curve denoting the nuptial, and the latter the ex-nuptial .frequency of maternity. The figure shews that while the nuptial and ei-nuptial rates by no means run identi- cally, they yet exhibit, on the whole, similarity of trend, the ex-nuptial rate being roughly 0.05 of the nuptial. The exact fluctuations of the ratio of the ex-nuptial to the nuptial rate are indicated in column (v.) of Table XCVI., and are shewn as curve C in Pig. 80. The dotted lines on curves A and B shew the general trend of the phenomena. TABLE XCVI.— -Shewing the Secular Changes of Nuptial and Ex-nuptial Reproductivity. Australia, 1881 to 1914. Ratio of Nuptial Confine- ments to Married Women.* Batio of Infantile Year. Ex-nuptial Confine- ments to Number of Unmarried Eatio of Births to Total Con- finements. Eatio of Ex-nuptial to Nuptial Bates. MortaUty (Ratio of Deaths of Children during first Survival GoefflcientB for end of First Year Women.t 12 Months)! (i-) , (ii.) (iii.) (iv.) (V.) (vi.) (vii.) 1881 . . .2285 .00950 1.00865 .0416 .1165 .8835 1882 . . .2206 .00891 1.00779 .0404 .1357 .8643 1883 . . .2245 .00870 1.00847 .0388 .1222 .8778 1884 . . .2305 .00893 1.00875 .0380 .1260 .8740 mean 1-4 .2269 .00901 1.00842 .0397 .1251 .8749 1885 . . .2301 .00918 1.00873 ■ .0399 1292 .8708 1886 . . .2274 .00946 1.00866 .0381 .1271 .8729 1887 . . .2285 .00957 1.00852 .0419 .1164 .8836 1888 . . .2271 .00983 1.0102i .0433 .1164 .8836 1889 . . .2206 .01008 1.00989 .0457 .1319 .8681 Mean 6-9 .2267 .00962 1.00920 .0418 .1242 .8758 1890 . . .2216 .01021 1.01005 .0461 .1082 .8918 1891 . . .2181 .01026 1.01030 .0470 .1155 .8845 1892 . . .2133 .01060 1.00865 .0497 .1058 .8942 1893 . . .2072 .01034 1.01008 .0499 .1149 .8851 1894 . . .1947 ■.00961 1.00931 .0494 .1031 .8969 Mean 0-4 .2110 .01020 1.00968 .0484 .1115 .8886 1895 . . .1916 .00947 1.01008 .0494 .1012 • .8988 1896 . . .1788 .00935 1.00900 .0558 .1126 .8874 1897 . . .1770 .00914 1.01066 .0517 .1048 .8952 1898 . . .1700 .00879 1.00997 .0586 .1272 .8728 1899 . . .1697 .00894 1.01086 .0527 .1167 .8833 Mean 6-9 .1774 .00914 1.01011 .0636 .1125 .8875 1900 . . .1691 .00905 1.01078 .0535 .1002 .8998 1901 . . .1668 .00865 1.01095 .0519 .1037 .8963 1902 . . .1625 .00826 1.01060 .0508 .1071 .8929 1903 . . .1513 .00807 1.00997 .0533 .1105 .8895 1904 . . .1554 .00859 1.01079 .0553 .0825 .9175 Mean 0-4 .1610 .00852 1.01062 .0630 .1008 .8992 1905 . . .1524 .00861 1.01076 .0565 .0819 .9181 1906 . . .1527 .00868 1.01112 , .0568 .0836 .9164 1907 . . .1527 .00864 1.00962 .0566 .0814 .9186 1908 . . .1506 .00857 1.00969 .0569 .0780 .9220 1909 . . .1506 .00837 1.01024 .0556 .0718 .9282 Mean 6-9 .1518 .00867 1.01029 .0565 .0793 .9207 1910 . . .1511 .00801 1.01040 .0530 .0751 .9249 1911 . . .1541 .00818 1.01033 .0531 .0680 .9320 1912 . . .1632 .00821 1.01037 .050S .0708 .9292 1913 . . .1609 .00805 1.01025 .0500 .0720 .9280 1914 . . .1598 .00766 1.01038 .0479 .0713 .9287 Mean 0-4 .1678 .00802 1.01036 .0509 .0714 .9286 * That is, to all married women, irrespective of age. t That is, to " never-married," " widowed," and " divorced," of 12 years of age and upwards, taken together. t The infantile mortality as given is not the ratio of deaths registered as under one year of age, in any year, to the births registered in the same year, but are those given in a paper " On the im- provement m infantile mortality, etc.," read before the Australasian Medical Congress in September, 1911 (see p. 672 Journ.), and are related to the number of births of the "equivalent year." 42. Crude and corrected reproductivity. — It has been shewn in Part XI., § 6, see Table XXXV., that the crude birth-rate gives only the initial reproductivity, and that, owing to the measure of infantile 294 APPENDIX A. mortality, the residua], after the first 12 months have elapsed, is more sigmfic8.nt than the birth-rate as regards the increase of the population. The necessary correction is secured by multiplying by a " survival factor." The principle may be extended for various purposes. Thus survival factors (cr) maybe calculated for the commencing school-age, the ages of puberty or nubility, the commencing age of miMtary service, the age of highest average economic efficiency, and so on. In actuarial notation these factors are denoted by Ix/lo^ *nd for brevity's sake may be denoted by ax- To compare two populations for survivals, S, up to any age x, we have, therefore, B denoting the births : — (537) Sx= Bk:/lo = Bax= B — Dx in which Dx denotes the aggregate of the deaths (of the native-born) up to age X. When x = 1, the values of a are unity, less the rate of infantile mortality taken for the "equivalent year." For rates, these quantities must be divided by the mean population of the period covered by the births.* The more rigorous treatment of this question has already been dealt with in Part XI., §§ 7 to 9, pp. 152-180 ; see also Tables XXXVI. and XXXVII. The infantile mortality varies, however, considerably from year to year, see column (vi.) in Table XCVI., which gives the rates calculated approximately for the "equivalent year."^ If y denote the infantile mortahty (see p. 151, hereinbefore), a being the survival factor, then we have : — (538) a = 1 — y; ory = l — a; as on (352), p. 151. This, of course, differs according to sex, with time, as is shewn in Table XCVI., and according to locality. The highest value of the survival-factor for Australia was 0.9320 in 1911. For the period 1901-10 for the Commonwealth of AustraUa it was 0.90490 for males, and 0.92047 for females,^ corresponding to infantile mortahties of 0.09510 and 0.07953. We thus arrive at the conception of a survival- value for a birth-rate, that is, the birth-rate reduced to its value at age x, and this survival-value may be averaged for the whole of life, i.e., integ- rated for all ages. Such an integral will constitute the best general measure of the reproductivity. It is equal to the average period lived multiplied by the birth-rate. Or if o) denote the greatest possible age, then : — (539) 2*0 =-p\axdx o and Eq is the reproductivity of the population taken as a whole. If o-q be unity, and the unit of x be, one year, then the value of (538) will be the ' Vide a paper (by the author): " The improvement in infantile mortality ; its annual fluctuations and frequency according to age, in Australia." Journ. Aus- tralasian Medical Congress, Sydney, Sept. 1911, pp. 670-679. • See Life Tables, Census Report, Vol. III., pp. 1215 and 1217. FERTILITY, FECUNDITY, AND REPRODUCTIVE EFFICIENCY. 295 birth-rate multiplied into the number of years expressing the length of life lived on the average ; consequently the product of the birth-rate into the " expectation of life at age 0," may be taken as the most service- able expression of the reproductivity.i The value given by (538) may be regarded as the crude reprodwctivity. The birth-rate j8 is ordinarily computed as for the total population, but may also be based upon the total female population, upon the female population of child-bearing ages, or upon the married of child-bearing ages plus a reduced number of the unmarried, equating them to the nuptial condition. Let the ratio of the fertihty of women at full risk (or otherwise if desired), at any age x, to the fertility at the age at which it is a maximum be denoted by fx : then the actual number of married women of all ages may be reduced to an equivalent number of women at the age of maximum fertihty by multiplying by this quantity. With these can be included also the unmarried, with whom in Austraha the fertility is about one -twentieth of that of the married. The corrected reproductivity may be given in the form of a birth-rate, viz., j8e : — (540) I3e = B /S {fxMx + .A Ux) in which 2 denotes " sum, "/and/' are the ratios for the fertilities of the married and unmarried respectively, referred to the greatest fertility of the married, and M and U are respectively the numbers of the married and the unmarried, who together give birth to B children. This measures the ratio of the actual births to a fictitious number of mothers of highest fertihty, and hence birth-rates so computed shew the variations of the extent to which potential fertility is actualised. These, of course, may be further reduced to their survival values. The mode of comparing reproductive efficiency by means of an index, viz., the genetic index or first natality index, has already been indicated ; see § 5, p. 237, hereinbefore. 43. Progressive changes in the survival coefficients. — The survival- factors are by no means constant, as is shewn in column vii. of Table XCVI. As tabulated, they are merely unity, less the ratio of the deaths under 12 months to the births in the same year. This, as shewn before. 1 Actuarially, the quantity :- ex = T^x / Ix = \ Ix dx -i- Ix when a; = 0, may, when multiplied by the birth-rate, be adopted as the measure of the reproductivity of a popjilation. Since this is obtained from the mortalities at suooeasive ages, it ia not quite homogeneous, as it is aSeoted by the vitality of migrants, and, moreover, the mortality of the older part of the population is affected by their earlier history, and may not therefore represent future experience. If J„ = I. then e„ = To = So / P- 296 APPENDIX A. is not quite correct, see pp. 155-160, but the correction is of no moment for the present purpose. It is worthy of note that the infantile mortality is roughly about 0.5522 of the rate of confinements of married women, as is shewn by comparing the means. The means (see Table XCVI.) 0.2269, 0.2267, etc., multiplied by the above fraction gives the follomng results : — Period 1881-4 1885-9 1890-4 1895-9 1900-4 1905-9 1910-4 Infantile mortality As^ computed from the nuptial confinement rate Survival factor divided by ratio of nuptial confine- ments .1251 .1253 .5513 .1242 .1252 .5479 .1115 .1165 .5284 .1125 .0980 .6342 .1008 .0889 .6261 .0793 .0838 .5224 .0714_ .0871 .4525 The ratio is therefore not uniformly constant. The infantile mortality is decreasing, but nevertheless shews a fairly definite fluctuation, see curve E, Fig. 80, which shews it on a large vertical scale ; its Hmiting value is, of course, unity. ' XIV.— COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY. 1. General. — In dealing with the more complex elements of fertility and fecundity, it will generally be necessary to distinguish between the nuptial and ex-nuptial cases, and since their frequency is very different, some simple method of correlating and comparing the two wijl have to be devised.^ Often it is necessary to distribute unspecified cases, since, in double- entry tabulations, the cases are often partially specified, and the neglect of partially-specified and wholly-unspecified cases will often lead to material error. There is another general matter of importance, viz., the corrections required in statistics of duration, if they are required to represent the results which, other things being equal, would have been furnished by a constant population. This will receive attention in § 3, pp. 298-9. 2. Correspondence and correlation. — It is often possible to see the essential identity of two curves by mere change of scale, or by systematic deformations (anamorphosis) of one in order to bring it into agreement with another. This fact is of value in the graphs of various vital phenomena. For example, any attempt to make the widest possible comparisons of population phenomena requires the construction of world-norms for the human race. But such an attempt involves the consideration of physiological and general correspondence of human developments. In connection with marriage, fertility, fecundity, etc., and their signific- ance, for instance, this demands the consideration of the following, viz. : — (a) The average ages of puberty, nubility, etc. (b) The frequency-distribution about those ages ; (c) The fertility and fecundity at different ages ; (d) The characteristics of the decay of fecundity at the end of the fertile period. * The determination of a type-formulae to be adopted for any two curves, the ascertaining of their constants, and of the " skewness" of each curve will serve to exhibit their degree of correlation. This can also be expressed by a correlation coefficient ; see " Statistical Methods," by C. B. Davenport, 1904, and the mono- graphs of Prof. Karl Pearson, W. F. Sheppard, G. U. Yule, De Vries, W. Pahn Elderton, Gini, Savorgnan, and others. 298 APPENDIX A. r ^ /p^ ^ o^yir. k^ v^^ ^:x Suppose, for example, curve A, Fig. 81, represents the average fertility according to age of women of one part of the world and B that of another part. Let x, x' , x", etc., denote the abscissa, of the initial point, that of the mode, and that of the terminal point of the curve A, or of curve B, the particular curve being indicated by the sufiix a or b. Then the simplest correspondences are those where xjxi, = x Jx i,= xl'^/o^'j,, etc., or where xj, — x^ = xfj, — a-'^^ etc., i.e., where the abscissae of the correspond- ing critical points of the curves are in a constant ratio, and the ordinates are also in a constant ratio, or where the abscissae of the critical points differ by a constant. Correspondence of this character may be called planar, because the curve B can be derived from the curve A by parallel linear projection on to a plane inclined to that on which A lies. If the two curves in question be represented by y^ = Fa (x) ; Vb = Fb {^) then planar correspondence may be defined as follows :■ — The points on curve B are in planar correspondence with those on A when — Say Age. Fig. 81. (541). ■Vb = kFaimXa +q) k, m, and q being constants : when k ot m or both are functions of x^, then the correspondence is nort-planar. If these functions of x^ are not simple, the correspondence becomes less significant. This method of envisaging the problem has advantages over the system of determining a mere numerical " coefficient of correlation,""^ because it is often possible to construct one curve from the data of the other. Moreover, it is not without value to examine how far the graphs of phenomena, which might have been imagined a priori to be identical, or convertible by oblique projection with change of scale, differ. Later nuptial and ex-nuptial fertility, according to age, will be compared. 3. Conections necessary in statistics involving the element of dura- tion. — ^The type of corrections necessary to be applied to the data of statistics involving the element of duration, depends upon the purpose in view. Two types are of special importance, that which aims at presenting the results, in the form in which they would have been given by (a) a constant population, and (6) by a population increasing according to some definite law, which for general comparative purposes is preferably • See Galton's graphic method, F. Galton, 1888, Proe. Roy. Soc. Lond., XLV., 136-145. Davenport, Statistical methods, p. 44, 2nd Edit., Lond., 1904. See also Pearson's, Yule's, and other papers on the subject. COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY. 299 the norm of increase, i.e., the characteristic of the increase of the whole of the populations to be compared. The latter involves the smaller corrections, and has the advantage that for many purposes the corrections will be negligible. Let it be supposed that the population is an increasing one : the data will then be characterised as follows : — (i.) The data for longer durations, drawn therefore from a smaller population, will be smaller (all other things being equal) than would characterise a constant population of the size from which the more recent data are drawn. Hence the necessary correction is a factor 1+e, where e is positive. (ii.) If the numbers of individuals have been taken into account for earlier dates, they can be deduced from the survivors, provided (a) that a correct mortality table is available, and (6) that migration has introduced no (material) modification. (iii.) If the data are related to events occurring with a varying rate (as in cases of birth, marriage, death, etc.), the rate at which they occur must be determined according to the duration in question. The type-formula for correction is as follows : — Let N denote the number given at any point of time, that is, let N denote the survivors after the duration i, from N' persons ; then if, in origination, N may be presumed to vary with the population, we shall have, on making allow- ance for the fact that these are only survivors, and that what is required is a result which shall either coincide (i.) with the final magnitude of the population, viz., at the date from which i is reckoned, or (ii.) with a definite rate of population growth (the rate of normal increase) : — (542).. J\r' = Nei'iL^.i/Lx = Nei^^ll + i(g'a:.i+g«)],* approximately. "■ See formula (530), p. 289. The notation is the ordinary actuarial notation. It is fairly obvious that Lx-i/Lx must equal 1 + ^ {^x-i +qx ) i approximately. It will be found that, through a large range, this latter and arithmetically more convenient form is sufficiently accin:ate for correction purposes to the data of statistics of duration. For example, if 12 be taken as the lowest age (it is the age of least mortality for Australian females), and successive intervals of 10 years from this be also taken, the following results are obtained, viz. : — x-i and X 12-22 12-32 12-42 12-52 Exact formula 1.03114 1.0933 1.1861 1.3133 Approx. formula . . 1.03110 1.08^3 1.1500 1.2888 Even the final difference is ordinarily of no moment, since, as a rule, the numbers to which it would have to be appUed are very small. 300 APPENDIX A. In this p will denote in case (i.) the absolute rate of increase, and in case (ii .) the excess over the normal rate of increase. Ceitain events, however, for example births, marriages, and deaths, migration, etc., occur with a rapidity which fluctuates on either the positive or negative side of the general rate of increase of the population, in which case it inay be necessary to introduce, into equation (542), a factor depending on the fact in question. 4. Distribution of partially and wholly unspecified quantities in tables of double-entry- — If a series of quantities. A, B, C, etc., and A', B', C, etc., fuUy specified so as to permit of proper double-entry, and others, a, a', etc., and a, a ', etc., specified so as to permit only of single entry, and again a third set at not specified, so as to permit of entry under either of two series of headings, be tabulated or arranged as hereunder, and totalled, the result will be as shewn syrubolically in the following table : — TABLE XCVII.— Scheme ot a Donble-entry Tabulation of Defectively Specified Data. Arguments y y' 2/" y'" etc. etc. Specified as regards x only. Totals. X A B C D etc. etc. a (6) S + o (S + 6) x' A' B' c D' etc. etc. a' (6') S' + a'(S' + 6') 3i' A" B" C" D- etc. etc. a" (6") S" + a' (S" + 6") etc. etc. etc. etc. etc. etc. etc. etc. etc. Specified as regards y only "W a' (;3') a"(^') a"'(r') etc. etc. a-(O) [a + a'+ ..] + <- (;8 + ;8'+ ..)+ Totals T +« T' + a' T" + a' T"'+a"' etc. etc. [a+a'+..] + w (6 + 6'+.. ) + SS + Sa + Sa + w ST + Sa+2o + u In this type-table, the horizontal and vertical totals of the fully-specified quantities are respectively S, S', etc., and T, T', etc., but the aggregates of the rows are S + a, etc., and of the columns are T + a, etc. (i.e., for^ the fully specified quantities together with those specified as regards one particular only). The totals T + a are specified as regards the " argu- ments" in the horizontal headings, and the totals S + a are specified as regards the " arguments" in the vertical headings. Thus the grand total is i7S ( = Z'T) -\-Za-\- Sa + a> , and this is the sum of either of the series of totals, viz., that of the final column or that of the final row. In order to distribute the quantity wholly unspecified, it is necessary to add a portion of oj to the" (vertical) columns, and a portion thereof to the (horizontal) rows, so that the corrected values of A, B, A', B' etc., shall equal the grand total, and so that the adjustment COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY. 301 shall be the most probable. Such adjustment can be effected as follows : It is assumed that the division of the quantity w into two p'arts, viz., CO ' and w", proportional to the aggregates of the a and a quantities respectively, is the most probable apportionment of the doubly-unspeci- fied quantity among the two, and further, that if these divisions, u> ' and w", be again subdivided proportionally to the individual values of a, a', etc., and a, a', etc., the result will be the most probable sub- division. Let — (543) CO = w' +co" ; a.nd Q = la + Ea; then (544) w ' = Za .oi/Q\ and m" = Ea . oj / Q ; consequently the amounts of the corrections to the a and a quantities are ascertained by multiplying each of them by the ratio oy/Q, or what is the same thing, the required result is attained by multiplying by this factor increased by unity. Calling the adjusted numbers b and j8 respectively, their values are : — (545) b = a(\ + u) / Q); /3 = a (1 + oj / i3>. Similarly, if these 6 and ^ quantities are distributed proportionally to the A, B, C, etc., quantities, and the A, A', A", etc., quantities respectively, the required corrections are : — (546)..A+a = A(l + |-+|-); B + b = B (1 + -|-+-|-' );etc. (547). .A' + a' = A' (1+ |i+| ) ; B'.+ b'= B' (1+1'+|) ; etc. and so on. The additive quantities, A6 / S, Aj8 / T, etc., are most readily computed separately, and are then added to the fully -specified quantities. By the process indicated, both series of singly -specified quantities, and the unspecified quantities are suitably distributed, the adjusted table consisting of the values A + a, B + b, etc. ; and A' + a', B'+b',etc. The process indicated is also valid when the distribution should be made on other bases. Let a = aj + a2 ; b = bj -f b2 ; etc., a' = a'l -j- a'2 ; etc., etc., the subdivisions being the values of A6/S, Aj8 /T, etc. Then, if the fundamental supposition that the corrections are proportional to A, B, etc., A', B', etc., be not satisfactory, any function of these quantities 302 APPENDIX A. may be substituted, in which ease S and Twill be 2ers Batios .. 113,900,167 34,208,424 53,955,512 16,204,832 3,329,594 1,000,000 102.02 12,064 665,919 32,636 9,802 118.25 6,527 276 82.9 .00846 1 53.2 5 1.50 .00015 .0188 1 Year. Authority. Total Births. Con- finements Cases of Twins. Cases of Triplets. Cases of Quad- ruplets. Cases of Quin- tuplets. 1871-80 1872-80 Neefet Prinzlngt Enlbbs 50,000,000 63,000^00 German Em- pire 1,000,000 1,000,000 1,000,000§ 12,080 11,677 12,856 156 143 124 1.8 1.3 1.33 0.2511 • Sum of the mean annual populations of the Australian States for which the necessary birth statistics were taken o*t. t ZuT Statistik der Mehrgeburten. Jahr. f . Nat. u. Stat., 1877, Bd. XXVni., p. 174. } Medizinischen Statistik. H, Prinzing, p. 65. § Confinements 12,013,134 ; Twins 154,444 ; Triplets, 1489 ; Quadruplets, 16 ; in the German Empire. t Based on 15,965,391 children born, excluding still-births about 15,758,822. 306 APPENDIX A. Quintuplets have been reported by Volkmann,^ Dasseldorf ; by A. Bemheim,* Philadelphia; by Horlacher,' Wiirttemberg; by Nyhoff,* Groningen ; in 30 cases collected by the last-named, the majority were bom at between 4 and 5 months. Sextuplets are reported by Vassali,* and Vortisch, Alburi,' and sextuplets at Hameln in Westphalia in 1600' ; no cases, however, so far as I am aware, have been reported in Australia. The observed frequency of multiple births is as follows : — TABLE CI. — Relative Frequency of Twins in Various Countries.* Coiintry. Period. Frequency. Country. Period. Frequency. Australia Switzerland . . 1881-1900 .0126 Spain 1863-70 .0087 Germany 1901-1902 .0127 Roumania 1871-80 .0088 Baden 1891-1900 .0128 France . . 1899-1902 .0109 Prussia J, .0129 Belgium . . 1890, 5, .0111 Netherlands . . .0129 1900 Hungary .0131 Italy 1891-1900 .0117 Wiirttemberg 19 .0132 Kuasia 1887-91 .0121 Norway 1876-1880 .0133 Bavaria . . 1891-1900 .0123 Sweden 1871-80 .0146 Saxony . . .0123 Finland 1891-1900 .0147 Austria . . 1896-1900 .0126 * The results other than for Australia are given in H. Prinzing's " Handbuoh der medizinischen Statistik, p. 64. The frequencies, however, have wide ranges of values. Thus, in Italy, they ranged in the period 1892-1899 through .0080 for BasiUcata, to .0148 for Venice. For rough approximations the order of frequency with which twins, triplets, etc., occur, is as follows : — 8. Uniovular and diovular multiple births. — Observations as to the frequency of what may be called uniovular and diovular production of twins shew (i.) that the sexes are the same where the twins are produced by the division of a single ovum ; (ii.) that this occurs in about one-iifth or one-fourth of the cases, these being recognised by the fact that they have common chorion ; and (iii.) that where the twins are produced from two ova, the sexes may be identical or otherwise, these being recognised by the fact that the chorion is divided. Zentral bl. f. Gyn., 1879, p. 17. Deutsche med. Wocheuschrift, 1899, p. 274. Horlacher, Wfirtt., Korr. Bl. 1840. Zeitsohr. f. Geb. u. Gyu., 1903, Bd. Iii., p. 173. • Anatom. Anzeiger, Bd. x., No. 10. Munch, med. Wochenschr., 1903, No. 38, pp. 1639-40 a photograph is given. Date of birth, 9th January, 1600. Deutsche med. Wochenschr., No. 19, 1899, p. 312. COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY. 307 Statistics for an examination of this question are not available in Australia, but are available for the German Empire. The data for 1906 to 1911 inclusive are as follows : — TABLE on.— Frequency oJ Multiple Births (German Empire, 1906-11). Confinements. 2 Males. Pairs. 2 Fe- males. 3 males. 2m.,lf. lm.,2f. 3 Fe- males. Males Born. Females Born. Total Quin- tuplets.* 12,013,134 49,426 58,382 46,637 343 390 395 361 28 36 Children Born. Total Cases of Twins. Total Cases of Triplets. Total Cases of Quadruplets. 12,170,604 10,000,000t 154,444 "128,563 1,489 1,239.5 18. 13.3 3. 2.54 • This is based upon 15,965,391 children horn ; or about 15,758,800 conflnements in 1872 to 1880, during which time 4 quintuplets were born, t This would give the proportion 3.05. As is evident for the number of children, the twins must be multiplied by 2, the triplets by 3, etc. The proportion (^) of uniovular cases can be deduced at once from the preceding figures. Let /x denote the masculinity, defined as the ratio of the difference of the pairs of males and pairs of females to their sum ; see (335), p. 132. Obviously, the uniovular cases are in the ratio (l+/x) pairs of males to (1 — ju,) pairs of females. Th^ diovular cases are in the same ratio as regards the same pairs, and the mixed pairs are equal to both combined, that is they are : — TABLE cm. — Theoretical Distribution oJ Diovular and Uniovular Cases Among Cases of Twins. Total T 2 males : Male and female + Female and male : 2 females 1 + M : 1 + 1 : 1 - At l+yu.: + : I — iJ. Of the total there are ^ uniovular and (1- quently — -^) diovular cases : conse- (551). .^ = JI/ + F — P Jf + .F + P and IJ- = M M -\- F M denoting the number of pairs of males, F the pairs of females, and P the cases of one of each sex. The above results thus give f = 0.24397 and n = 0.029023. Direct observations according to Weinberg^ and Ahlfeld^ gave respectively for the relative frequency of uniovulate cases 0.21 and .0172, but it would appear from the preceding result that a sufficiently extended number of cases could be expected to give a higher ratio. * Beitrage zur Physiologie und Pathologie der Mehrlingsgeburten beim Menschen. Arohiv f. ges. Physiol., 1901, Bd. Ixxxviii, p. 346 ; Neue Beitrage zur Lehre von den Zv^illingen. Zeit. f. Geb. u. Gyn., 1903, Bd. xlviii., H. 1. » Zeit. f. Geb. u. Gyn., 1902, Bd. xlvii., p. 230. 308 APPENDIX A. A similar investigation may be applied to the more limited results for triplets. Neglecting the masculinity tendency, it is obvious that for the triovular and diovular cases the proportions of cases in each category will be respectively : — TABLE CIV. — ^Theoretical Distribution oi Diovular and Triovular Cases Among Triplets. Total T 3 males : 2 males and 1 female : 1 male and 2 females : 3 females (M) (P) (Q) (F) T(\- r) Ti' .125 .375 .375 .125 .25 .25 .25 .25* * It is assumed that when the births m.f.m and f.m.f occur, the chance of the two males or two females being uniovular is zero. If this condition were not physiologically impossible, it is easy to see (by exhaiostive enumeration) that the probabilities of the four cases would be 0.2 : 0.3 : 0.3 : 0.2. An examination of the individual figures for each year shews that the differences are too great to give any ground for deducing masculinity to be other than zero. Hence we may take means adopting : — 352 : 392.5 : 392.5 : 352 instead of 343 : 390 : 395 : 361. and this gives for the series of triovular and diovular births respectively : 20.25 : 60.75 : 60 : 75 : 20.25 and 331.75 : 331.75 : 331.75 : 331.75, or 162 triovular and 1327 diovular births in all ; or ratio of diovular cases of no less than 0.8912 of the total, the triovular being 0.1088. Thus it follows that triovulation is a mtich rarer occurrence than the pro- duction of uniovular twins, that is, the ratio of triovulation in triplets to diovulation is 8.20. From the above we obtain by symmetrically in- cluding all the data : — (552) i'= i^Z{M + F)-{P + Q)] /{M + P + Q + F). Thus, according to the recent experience of the German Empire, we have for 10,000,000 cases of confinement, 31,365.5 cases of uniovulation production of twins among the twins, and 1104.6 cases occurring among the triplets. We may assume at least the same ratio for the cases of quadruplets and quintuplets, which will give, say, 14.1 for both combined.^ Hence the ratio t, of occurrence for all cases of uniovular production of twins (i.e., appearing as twins or as portion of triplets, &c.) : — (553) t, = 0.0032484. or, say, 13 cases in 4000, or 1 case in 308. 1 In quadruplets there are 16 possible orders in which births may occur, and in these 24 possible cases of uniovulation. Since, however, the number of males and females are unequal — 28 and 36 — the possible cases have not occiu'red, and hence we may regard the 16 quadruplets and 3 quintuplets as roughly expressing the probable number of cases. Sohroeder (Lehrbuoh der Geburtshulfe, 10° aufl.) gives for twins 1 : 89, triplets 1 : 7910, quadruplets 1 : 371126. COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY. 309 9. Small frequency of triovulation. — ^The preceding analysis appears to shew that the triovular cases are only 162 in 12,013,134 confinements. The probabihty of triovulation, ^ ', therefore, would appear to be : — (554). .r = 0.00001348. or, say, I case in 74,000 confinements, though triplets occur at the rate of 1 case in 8068 confinements in the German Empire. This subject might well form the result of more definitive study when the data are adequate. 10. Nuptial and ex-nuptial probability of twins according to age. — The probability, in any nuptial or in any ex-nuptial confinement, of the occurrence of twins has been ana- Frequency of Twins according to Age. .010 .006 .000 .010 .005 .000 Age ^ C^ •^ ^ ^ i A^*. °r* k K ^ >f*^ V <• / y .\ ~^ 10 so 30 Fig. 83. 40 60yrs. lysed from an aggregate of the Aus- tralian data- from 1908 to 1914, both inclusive. It must, of course, be in substantial agreement with the result found for e in § 8. Table CV., columns (ii.) and (vi.), give the number respectively of nuptial and ex-nuptial confinements (totals 897,618 and 54,913) occurring in AustraUa ia 8 years, and the num- bers of twins corresponding to each, viz., 9187 and 422. These are shewn by curves A and B, Fig. 83, the dots denoting the individual results for nuptial cases, and the firm Unes the smoothed results ; the values for the latter being given in column (v.) of the table. The ex- nuptial cases are denoted by circles, and where the numbers were small, the quinquennial aggregates only were graphed. The rate of increase per year of age up to age 37 is for nuptial and ex-nuptial cases respectively. Curve A represents the ratio of the number of cases of at least two births to the number of nuptial confinements. Curve B represents the same ratio for ex -nuptial confinements. Curve C represents the number of cases of three or more at a birth to the number of cases of two or more. (555). = 0.000632 {x — 12) and e', = 0.000668 (a; — 12) X being the age of the mother. Beyond the age in question the results can be taken from the table. The ratios for all ages are— nuptial, 0.010234, and ex-nuptial, 0.00768. The general result is (i.) that with increase of age (and possibly duration of marriage) the frequency of twins increases linearly, till the end of the ordinary child-bearing period is approached, and (ii.) this increase is slightly greater for ex-nuptial cases, viz., about 5.7 per cent, greater. The ex-nuptial relative frequency of 310 APPENDIX A. twins for all ages combined is exactly 0.75 the nuptial relative frequency. Since in the ex -nuptial cases the confinements are probably on the whole not repeated, the result would appear to be due to age. This matter will be further considered later. TABLE CV. — Shewing Probability according to Age of the Occunence of Nuptial and Ex-nuptial Twins, and of Triplets, based on 8 Years' Australian experience, 1907-1914. Age. Nuptial Con- fine- Cases of Nuptial Frequency of Nuptial Twins. Ex- Nuptial Con- fine- Cases of Ex- Nuptial Frequency of Ex-nuptial Twins. All Twins. All Iriplets Batio of Triplets to Twins. ments. Twins. Crude Smo'th- ments. Twins. Crude. Smo'th- ed. ed. (i.) (ii.) (iii.) a (iv.) (v.) (vi.) (viii.) (vu.) (ix.) (X.) (xi.) (xii.) 12 .0000 5 .0000 .0030 13 4 .0006 21 .0007 .0035 14 30 34 .0013 126 152 "0 .0660 .0013 .0039 ie 170 "o .0019 537 "1 .ooio .0020 .0044 16 1,138 2 .oois .0025 1,500 1 .0007 .0027 .0049 17 S,962 12 .0030 .0032 2,980 9 .0030 .0033 .0054 18 9,761 36 .0037 .0038 4,604 16 .0036 .0040 .0058 19 18,071 94 .0052 .0044 6,317 23 .0043 .0047 .0063 33.102 144 14,838 50 .0337 i94 "1 00.52 20 25,159 147 .0068 .0051 5,272 27 .0051 .0053 1 .0068 21 35,326 202 .0057 .0067 6,008 33 .0066 .0060 3 .0072 22 43,353 254 .0059 .0063 4,231 34 .0080 .0067 1 .0077 23 50,322 329 .0065 .0069 3,848 32 .0083 .0073 6 .0082 24 53,176 392 .0074 .0076 3,182 24 .0075 .0080 3 .0086 207,335 1,324 21,541 150 .0696 1,474 14 .0095 25 64,269 452 .0083 .0082 2,548 21 .0082 .0087 1 .0091 26 65,006 434 .0079 .0088 2,161 19 .0087 .0094 3 .0096 27 53,735 487 .0091 .0095 1,785 26 .0100 8 .0101 28 53,244 506 .0095 .0101 1,699 23 .0107 5 .0105 29 49,200 538 .0109 .0107 1,410 8 .0114 2 .0110 265,444 2,417 9,603 96 .oioo 2,6i3 19 .0076 30 47,980 548 .oii4 .oii4 1,356 14 .0i20 7 .0115 31 40,199 485 .0121 .0120 851 13 X127 4 .0119 32 41,528 659 .0135 .0126 956 11 .0134 8 .0124 S3 37,426 505 .0136 .0133 812 15 .0140 3 .0129 34 34,362 488 .0142 .0139 779 15 .0147 4 .0133 201,495 2,585 4,754 68 .0i43 2,663 26 .0098 35 31,349 436 .0140 .0145 688 17 .0i54 10 .0138 36 29,399 488 .0166 .0162 636 9 .0160 12 .0143 37 26,213 414 .0158 .0168 544 7 .0167 6 .0148 38 24,664 377 .0153 .0161 665 8 .0168 4 .0152 39 20,790 324 .0156 .0156 436 6 .0163 3 .0167 132,415 2,039 2,859 46 .oiei 2,085 34 .0163 40 17,023 226 .0133 .0143 383 6 .0150 6 .0162 41 12,262 171 .0140 .0127 201 3 .0134 2 .0166 42 11,012 123 .0112 .0112 208 .0118 2 .0171 43 7,457 85 .0114 .0099 156 1 .0104 .0176 44 4,746 36 .0076 .0086 85 1 .0090 1 ? 52,490 641 1,029 11 .0167 662 U .0169 45 2,756 21 .0076 .0074 58 .0077 7 46 1,389 10 .0072 .0062 36 1 .0064 ? 47 684 4 .0060 .0051 17 .0052 ? 48 310 1 .0032 .0041 12 .0042 ? 49 106 .0031 7 .0032 ? 5,244 36 130 1 .0077 ■37 .0000 SO 34 .0022 6 .0023 ? 61 12 , .0016 1 , , .0016 ? 52 6 1 .17' .0009 .0009 7 53 4 .0005 1 .0005 7 54 3 59 "l .0002 ■7 .0002 "1 7 .0000 Not" Stated Totals 897,618 9,187 .01023 54,913 422 .00768 9,609 105 .01093 11. Probability of triplets according to age. — ^The results of the 8 years, 1907-14, gave the following results for nuptial and ex-nuptial twins and triplets, viz. : — COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY. 311 Twins. Triplets. Nuptial. Total. Ex-nuptial. Nuptial. Total. Ex-nuptial. Numbers Ratio 9,187 1.0000 (9,609) 422 .0459 98 1.000 (105) 7 0.071 The numbers are too small, however, to establish that the frequency of the occurrence of triplets ex-nuptially is between 50 and 60 per cent, greater than nuptially. If the frequency be related to the number of twins, it is roughly given by the smoothed results in column (xii.) of Table CV. We shall call the probability Pg /P2 say, t^. Thus we shall have : — (556) . . . .T3= 0.0030 + 0.00047 {x — 12); or = 0.00047 {x — 5.6) the second form, however, being without meaning till the age of child- bearing. The firm line, curve C, on Fig. 83, denotes the increase ; the crosses represent the group results used in deducing this. 12. Probability o£ twins according to duration of marriage.— Given a birth, the probabihty of a second child being born is found by dividing the number of twins, including triplets, by the number of confinements tabulated according to duration of marriage. Thus, column (v.) in Table CVT. is found by dividing the figures in column (iii.) by those in column (ii.). The crude results are shewn by the dots in Fig. 84, and the smoothed results by the firm line, curve A. For the form of the initial part of the curve see § 14, and also Fig. 85 hereinafter. 13. Probability 0! triplets according to duration of marriage.— The probability of a Jhird child being born may, as before, be referred to the number of cases where a second child has been born. This probability is found by dividing the number of triplets by the number of twins, in- cluding the triplets, etc. But the numbers to be dealt with are so small and irregular that the expedient was adopted of forming groups of eleven. As no correction was apphed for the systematic error of the grouping, the curve represents the ratio of 11 -year groups of duration of marriage, the argument being the central years of the group. The results are shewn on Fig. 84, curve B, and the data are shewn in Table CVI., and seem to indicate the change with duration of marriage is sensibly a Unear one through for the major part (presum- ably) of the child-bearing period. .030 .010 .000 .010 .000 ■y^ ■'•B \, r^ •\ V. / ><^ A-^ ^ V' \' •\ 10 so Duration of Marriage. 30 40 [years. Fig. 84. Curve A denotes the frequency of the birth of two or more children to the number of confinements. Curve B denotes the ratio of 11-year means of the number of triplets to the number of cases of two or more children. 312 APPENDIX A TABLE CVI. — Probability of Twins* and Tripletsf according to Duration of Marriage. Australia, 1908-1914. Dura- Ratio of Twins to Ratio of Triplets to tion Con- Twins Confinements. Twins (Groups of U). of Mar- finements including Triplets. Triplets. riage. Crude. Smoothed Crude. Smoothed (i.) (ii.) (iii.) (iv.) (V.) (vi.) (vii.) (viii.) 0-1 134,171 1,129 9 .0084 0084 .0073 1-2 61,213 460 3 .0075 .0075 .0073 2-3 64,229 465 4 .0072 .0072 , , .0073 3-4 70,317 564 3 .0080 .0080 .0073 4^5 59,407 551 2 .0093 .0090 , , .0073 5-6 53,275 504 4 .0095 .0098 .0074 .0073 6-7 47,250 468 1 .0099 .0106 .0072 .0075 7-8 41,713 492 3 .0118 .0113 .0078 .0080 8-9 37,115 466 7 .0125 .0120 .0077 .0087 9-10 32,170 417 3. .0130 .0126 .0088 .0095 10-11 29,607 404 5 .0136 .0132 .0112 .0102 11-12 25,887 328 2 .0127 .0138 .0115 .0109 12-13 23,372 352 5 .0151 .0143 • .0125 .0117 13-14 20,339 273 2 .0134 .0148 .0130 .0124 14-15 17,-572 281 6 .0160 .0152 .0128 .0131 15-16 15,217 228 9 .0150 .0154 .0138 .0138 16-17 13,271 196 2 .0148 .0152 .0139 .0146 17-18 11,617 159 1 .0137 .0149 .0153 .0155 18-19 10,073 139 .0138 .0145 .0152 .0158 19-20 8,520 117 2 .0137 .0139 .0164 .0158 20-21 7,424 89 2 .0120 .0132 .0149 .0149 21-22 5,988 76 .0127 .0124 .0087 .0121 22-23 4,726 46 1 .0097 .0114 .0083 .0095 23-24 3,561 35 .0098 .0103 .0105 .0068 24r-25 2,664 34 .0128 .0092 .0043 25-26 1,809 22 .0122 .0080 .0028 26-27 1,146 8 .0070 .0067 .0016 27-28 643 2 .0031 .0054 .0010 28-29 383 4 1 .0104 .0041 .0006 29-30 192 .0028 .0003 30-31 77 .0016 .0002 31-32 45 .0010 .0002 32-33 16 .0006 • .0001 33-34 5 .0004 .0001 34-35 .0003 .0001 35-36 1 .0002 .0000 TotaU 805,015 8,308 77 .010320 .00927 •■ * That is, of two or more occurring at a birth, two are bom. t That is, of third child in any case where 14. Remarkable initial fluctuation in the irequency of twins, accord- ing to interval after marriage. — -There i» no known ground for supposing that the ratio of the number of twins to the number of confinements in which they occur, can in any way depend on the interval after marriage, at leaist, if that interval be small. The results in Tables CVII. and CVIII. COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY. 313 hereunder for the years 1908 to 1915 and 1908 to 1914 respectively, shew, however, that apparently the dependence exists. The average for the first three months after marriage equals that of the third three months, and both are very much above the average. The second and fourth periods of three months are about equal. These results are shewn by curve C on Fig. 85. TABLE CVn. — Shewing Variation in the Fiec[uency of Twins during the First 24 Months after Marriage. Australia 1908-1915. Twins Born during Interval after Marriage of Months Confinements during Intervals Ratio of Twine during Intervals Year. after Marriage of Months after Marriage of Months 0-3 3-6 6-9 9-12 12-24 0-3 3-6 6-9 9-12 12-24 0-3 3-6 6-9 9-12 12-24 1908 16 24 34 56 \ 60 1,533! 3,152 ,4,006 7,007 6,298 .0104 .0076 .0085 .0080 .00951 .0068] 1909 21 2« • 44 62 48 1,799 3,556 4,139 7,307 6,973 .0116 .0073 .0106 .0085 1910 19 29 58 as 48 1,888 3,659 4,474 7,500 6,919 .0101 .0079 .0129 .0079 .0069 1911 15 31 49 64 56 1,987, 4,075 5,220 7,877 7,400 .0076 .0076 .0094 .0081 .0075 1912 27 82 60 61 60 2,119 4,458 5,827 8,899 8,518 .0127 .0072 .0103 .0069 .0071 1913 17 34 61 66 65 2,107, 4,502 5,916 9,301 9,142 .0081 .0076 .0103 .0071 .0071 1914 14 32 j>8 60 63 2,080 i 4,268 5,897 9,185 9,247 .0067 .0075 .0098 .0065 .0069 1915 28 46 51 76 82 2,023, 4,149 5,828 8,795 8,953 .0099 .0111 .0088 .0086 .0091 Totals 157 254 415 504 482 15,536 31,819 1 41,307 65,871' 63,450 .01010 .00798 .01005 .00765 .00760 Thus the proportion of twins for all pre-nuptial conceptionB is high. It is to be noted, however, that the proportion of ex-nuptial twins over all is low (see Table CV.), and it is not unUkely that the initial high rate, and, in general, the higher rate for the cases due to pre-nuptial insemination is due to the transfer, owing to the peithogamic influence, of what might have been ex-nuptial to the nuptial cases. To obtain the fluctuation more exactly, the results were taken out monthly, from 1908 to 1914, according to interval after marriage. TABLE CVm. — Shewing Variations in the Freaueney of Twins foi each Interval of One Month after Marriage (First Births only), and of Triplets. Australia 1908-14. Interval* . . Twins Confinements 0-1 39 3,529 1-2 40 4,059 2-3 50 5,925 3-4 55 7,455 4-5 70 9,055 5-6 83 11,160 6-7 85 13,870 7-8 109 11,545 8-9 170 10,064 9-10 195 24,434 10-11 146 19,047 11-12 87 13,595 0-12 1,129 133,738 Katio .0110 .0098 .0084 .0073 .0077 .0074 .0061 .0094 .0169 .0080 .0076 .0064 .00844 Intervalf Twins Confinements^ 1-2 400 54,497 2-3 141 15,801 3-4 58 6,458 4r-7 59 6,413 7-11 17 2,209 11-26 7 905 1-26 682 86,283 0-26 1,811 220,021 Interval' Triplets Twins 0-1 8 1,12s 1-26 6 682 Batio .. .0073 .0089 .0091 .•0092 .0077 .0078 .00790 .00823 Ratio .0071 .0088 • Months. t Years. First births. 314 APPENDIX A. tl s V, s)\ ■\3- —- — / / \ 0-3 c , J J \ A LJ n t"" -— - _ .020 d I " .010 .000 Q 150 I 100 |Zi 50 S 10 15 20 25 30 35 40 Duration of marriage, [months Fig. 85. Curve A denotes the actual number of twins in Australia during 7 years' experience. Curve B denotes the ratio of cases of births of 2 or more children to oases of confinement. Curve C denotes, similarly to curve B, the group ratios for three months, however, instead of one. The ratio for 1-4 i^i .0078, and for 5-26 is .0087. The numbers for the lesser subdivisions are doubtless too small to rely on the results. The results shewn are for first births only ; but for the smaller durations the distinction is without meaning. Fig. 85 shews the results, curve A denoting the actual number of twin births, and curve B the frequency with which twins occur. 15. Frequency of twins according to order of confinement. — ^From the frequency of the occurrence of twins according to previous issue, an estima- tion according to order of confine- ment can be made by taking account of the probability of twins or triplets, &c. From the frequency according to previous issue, it may be deduced that the probability of twins is approximately as follows : — Previous Confinements Probability (about) .0082 1 .0096 2 .0107 .0117 4 .0124 5 .0180 6 .0134 7 .0136 .0138 .0139 10 .0140 We have also from the general result that the frequency of single births, twins, and triplets in Australia was, for 1908-14, 799831 : 8247 : 77 1 1 : 0.010311 : 0.000096 | or roughly, say, 10,000 : 100 : 1 The probability of twins occurring twice, 2P2, ^, therefore, approxi- mately identical with that of the occurrence of triplets, ^3, that is : — (557). !!?'2 = fl = Pa' approximately. The number entered under will be correct. That is, the cases " accord- ing to previous issue," and " according to previous confinements" are identical. But in every case where there were twins or triplets, etc., at the first birth, the cases would be tabulated under " previous issue," 2 or 3, etc., respectively, instead of under 2 ; and similarly mutatis mutandis for all later columns in the " according-to-previous-issue" COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY. 315 tabulation. We therefore must add the appropriate numbers, and deduct equal numbers from later columns. The precision of the result will, of course, never be of a high order. The data are given in the upper part of Table CIX., and the approxi- mate restatement according to the order of confinement forms the lower part of the table. TABLE CIX. — Frequency of Multiple Births according to Previous Issue. Australia 1908-14. Previous Issue (upper table), or Order of Confinement (lower table.) Numbers. 1 2 3 4 . 5 6 7 8 9 Cases ot at least 2 children . . Cases ot at least 3 children .. Mothers of at least 1 child Ilatio ot twins to mothers 1,811 12 220,80? .0082C 1,357 10 167,091 .008121 1,325 7 125,779 .01053 1,094 7 92,116 .01188 834 8 65,343 .01276 591 5 46,156 .01280 477 9 31,733 .01503 306 2 21,918 .01396 22" 14,72' .01541 ! 127 i 1 r 9,671 L .01313 Ac- cord- ingto order of • Con- fine- ment " Twins Mothers Corres- ponding L Batio 1,811 220,807 .0082C 1,386 169,851 .00816 1,337 126,377 .01058 1,096 92,083 .01190 I 831 65,099 .01277 590 45,683 .01292 467 31,253 .01494 302 21,467 .01407 21S 14,28' .0152 i 122 I 9,254 5 .01318 lUtio Triplets Smoothed . . .000056 .000062 .000074 .000088 .000106 .000130 .000158 .000193 .00023 > .000286 Numbers. 10 11 12 13 14 15 16 17 18 19 20 21 22 Cases ot at least 2 children . . Gases of at least 3 children . . HotheiB ot at least 1 chUd Batio of twins to mothers 79 6,694 .01387 39 3,181 .01226 21 1,665 .01261 9 814 6 388 ( 14' ! 1 ) 1 L 59 .0 25 6 3 1 1 1 1 0.01388 Ac-- cord- ing to order . of Con- fine- ment r Twins Mothers Corres- ponding '- Batio 74 5,378 .01378 37 2,964 .01248 19 1,530 .01242 8 740 .01081 5 340 .01471 12 .0157 1 1 r 52 > .01923 21 5 2 1 1 1 1 Since the correction system affects the number of twins and the mothers in the same way, it obviously cannot produce any appreciable difference in the ratios, though it may alter the numbers. This is seen in the results given in the table above. If the number of triplets be smoothed, the result shewn in the final line is obtained. But the numbers are too small to lead to any reliance upon their value, though they con- firm in a general way the dictum that multiple fecundity increases with the issue, thus also with age and duration of marriage. 316 APPENDIX A. TABLE ex.— Shewing Secular Variation in the Frequency of Twins and Triplets. Australia, 1881-1915. No. of Satio of Twinst Katio of Triplets Conflne- Cases of Cases of Cases of to Confinements. to Twins, etc.t Year. raenta 2 or more 3 or more 4 or more dotal).* CluldTen. Children. Cbildien. Crude. Smootbed. Crude. Smoothed. "oT (ii.) (iii.) (iv.) (V.) (vi.) (vii.) (vlu.) (ix.) 1831 63,818 645 7 .00864 .0080 2 64,069 496 3 .00774 .0082 3 68.135 675 ,2 .00843 .0084 i 72,832 629 8 .00863 .0086 5 76,026 661 3 .00869 .0087 .0063 .0063 6 79,009 682 2 .00863 .0088 .0066 .0066 7 83,085 704 4 .00847 .0090 .0064 .0066 8 86,393 875 6 1 .01012 .0096 .0066 .0062 9 87,195 859 3 .00985 .0099 .0068 .0070 1890 91,030 910 5 .00999 .0102 .0072 .0076 1 91,734 941 4 .01026 .0103 .0083 .0081 2 91,980 784 12 .01023 .0102 .0082 .0082 3 90,379 899 11 1 .00994 .0100 .0080 .0081 4 86,384 797 7 .00922 .0096 .0081 .0081 5 91,225 907 12 1 .00994 .0094 .0085 .0085 6 86,526 775 4 .00896 .0094 .0089 .0088 7 90,614 960 6 1 .01069 .0099 .0085 .0089 8 88,993 883 4 .00992 .0104 .0086 .0088 9 90,244 971 9 .01076 .0107 .0088 .0087 1900 92,057 985 7 .01069 .0108 .0084 .0086 1 92,826 1,005 11 .01082 .0107 .0089 .0088 2 92,852 972 12 .01046 .0104 .0088 .0092 3 89,060 877 10 1 .00984 .0102 .0098 .0095 4 93,973 1,005 9 .01069 .0104 .0093 .0097 5 95,060 1,012 11 .01064 .0107 .0099 .0099 6 97,867 1,083 5 .01106 .0107 .0100 .0100 7 100,161 961 13 .00949 .0102 .0099 .0099 8 110,491 1,066 6 .00963 .0098 .0100 .0097 9 112,921 1,142 14 .01011 .0100 .0096 .0096 1910 116,609 1,189 13 .01028 .0102 .0092 .0093 1 120,967 1,236 14 .01021 .0102 .0093 .0089 2 131,726 1,360 16 .01024 .0101 3 134,343 1,369 8 .01019 .0101 4 136,676 1,406 11 .01029 .0102 6 133,444 1,417 10 .01061 .0104 Totls 3,221,694 32,917 281 6§ .010217 .00863 * That is, nuptial and ex-nuptial. t Including triplets and auadiuplets. t That is, the ratio of 9-year groups of triplets including quadruplets to O-year groups of twins, including triplets. % Batio of quadruplets to triplete = 0.018. 16. Secular fluctuations in multiple-biiths. — ^The ratio of multiple births to confinements would appear a priori to be independent of time, but it win be seen from Pig. 86 Secular Fluctuation in Relative Frequency of Births and Twins and Triplets. « 3 .005 ..000 <1.010 is 4 t .005 .000 c V '^' J ~ ' Irac b. '■^V' ^■^ -^1 .036 .030 C .025 C .020 C .015 C 1880 Fig. 1900 86. 10 Curve A denotes the smoothed secular fluctuation of the ratio of births of two or more to the number of con- finements. Curve B denotes the ratio of 11 -year groups of births of three or more to the number of births of two or more. Curve C denotes the crude birth rate and number of births per unit of the general population. that there are indications of a definite secular fluctuation, see also Table CX. above. The number of confinements which constitutes the basis of the experi- ence is more than doubled in the 35 years under review (see column ii.), and the number of twins (which includes triplets and quad- ruplets) is large. The aggregate experience includes 3,221,594 con- finements, in which there were a total of 32,917 births of two or more children, a total of 281 births of 3 or more children, and 5 quadruplets. These give the ratios shewn in the table. In Pig. 85, curve A is the smoothed secular fluctuation-curve of the twins ; curve B that of the triplets (which COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY. 317 were grouped in nines) ; while curve C shews the fluctuations of the crude birth-rates for the same years. The individual values are shewn by dots. It will be observed that on the whole the frequency of twins and triplets rises as the frequency of births diminishes. 17. Comparison of nuptial and ex-nuptial fertility. — ^In columns (x.) and (xvi.) of Table LXXIII., p. 242 hereinbefore, the crude and smoothed ratios for ex-nuptial fertility, attributed wholly to the " never married," were given. The crude results are repeated in column (ii.) of Table XCI. hereunder. If attributed to the " unmarried," which includes the widowed and divorced, the results in column (iii.) are obtained, and the corresponding smoothed results are shewn in column (iv.). Reference to the table shews that the maximum fertility is nuptially attained at about the year of age 18.3 to 19.3, and is about 0.484. The maximum fertiUty is ex-nuptially attained, however, only at about age 21.5 to 22.5, and is about 0.0182 ; that is to say, the maximum is about 3.2 years later, and the proportion at the maximum is only 0.0376, or say 3/80ths. For all ages from 12 to 57 we have for nuptial-fertility ratio 0.1704, and for the ex-nuptial ratio 0.00993. Hence Nuptial and Ex-nuptial FertiUty-ratios. the proportion of the averages is 0.05828. It is obvious that the initial parts of the curves representing the nuptial and ex- nuptial fertility-ratios are not Ukely to be identical, because the nuptial denominator for early ages will be small, and the ex-nuptial denominator will be large. Curves A and C, Fig. 87, denote respectively the nuptial and ex-nuptial curves. By the process indicated in § 2, p. 298, the results in columns (vi.) and (vii.) of Table CXI. are obtained ; these are shewn in Fig. 87 by curve C ; hence the curves are not in planar correspondence. If, however, the curve A be corrected for the effect of previous births, the two curves come into closer correspondence^ ; that is, ex-nuptial fertility has, in general, nearly the same characteristics as nuptial fertility, excepting that the greater measure of restraint operates to make the maximum occur later, and to enormously reduce the ratio. Age 10 Curve ratio. Curve curve A. Curve ratio. Fig. A denotes the nuptial fertility B is the oblique projection of C is the ex-nuptial fertility 1 It is obvious that the ex-nuptial curve does not need the same correction, since oft-repeated ex -nuptial maternity is not likely to occur. 318 APPENDIX A. TABLE CXI. — Comparison of Nuptial and Ez-nuptial Fertility-iatios according to Age. Australia 1907 to 1914. Ratio of Ex-nuptial Births to — Batio of Nuptial Births to Ex-nuptial Bate Computed by Age of Mother. the "Never the " Unmarried." the Oblique Projection." Married." Married. Crude. Crude. Smoothed. Smoothed. Rate. Age. (i-) (ii.) (iii.) (iv.) (V.) (vi.) (vU.) 12 .0000 .0000 .0000 13 .0001 .0001 .0001 U .0004 .0004 .0004 .207 .0077 V5.4 15 .0016 .0016 .0016 .227 .0085 16.5 16 .0043 .0043 .0043 .301 .0113 18.0 17 .0085 .0085 .0085 .458 .0171 20.1 18 .0131 .0131 .0131 .483 .0181 21.3 19 .0162 .0162 .0158 .479 .0179 22.3 20 .0172 .0172 .0174 .464 .0174 23.2 21 .0181 .0181 .0181 .443 .0166 24.0 22 .0173 .0171 .0181 .416 .0156 24.9 23 .0183 .0182 .0177 .381 .0142 25.7 24 .0176 .0174 .0172 .352 .0132 26.4 25 .0163 .0161 .0163 .333 .0124 27.3 26 .0157 .0154 .0154 .319 .0119 28.1 27 .0147 .0143 .0149 .307 .0115 29.0 28 .0157 .01.52 .0145 .293 .0110 30.0 29 .0145 .0139 .0141 .274 .0102 30.9 30 .0157 .0150 .0136 .256 .0096 31.7 31 .0111 .0104 .0131 .241 .0090 32.6 32 .0138 .0128 .0127 .225 .0084 33.5 33 .0131 .0119 .0123 .210 .0079 34.4 34 .0135 .0121 .0119 .197 .0079 35.3 35 .0129 .0113 .0114 .185 .0069 36.3 36 .0127 .0109 .0108 .174 .0065 37.2 37 .0116 .0097 .0101 .164 .0061 38.1 38 .0125 .0101 .0093 .149 .0056 39.0 39 .0103 .0082 .0083 .130 .0049 39.9 40 .0097 .0074 .0070 .108 .0040 40.7 41 .0055 .0041 .0054 .087 .0033 41.6 42 .0060 .0043 .0042 .067 .0025 42.4 43 .0049 .0033 .0030 .0.50 .0019 43.3 44 .0029 .0019 .0020 .033 .0012 44.2 45 .0021 .0013 .0013 .020 .0007 4.5.1 46 .0014 .0008 .0008 .010 .0004 46.0 47 .0007 .0004 .0004 .005 .0002 47.0 48 .0005 .0003 .0003 .003 .0001 48 49 .0003 .0002 .0002 .001 .0000 49 50 .0003 .0001 .0001 .001 .0000 50 51 .0001 .0000 .0000 .000 .0000 51 • The oblique projection brings the maximum points into arbitrary agreement, tlie values for the ages indicated also being determined thereby. The rates lor these ages are tound from those of the nuptial curve by using tlie projection-ratio. COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY. 319 The difference between the nuptial and ex-nuptial probabilities of con- finement are more comprehensively indicated by a decennial table. In Table CXII. hereunder these are given as the number of cases respectively occurring per 10,000 married and per 100,000 "never married" women. The rates, based upon the numbers of the " unmarried," are somewhat smaller. TABLE CXn. — Shewing the Probabilities of Nuptial and Ex-naptial Confinement and their Ratio, for Five-Year Age-gioups. Australia 1907-1914. " Batio oi Probab- No. of No. of Probab- No. of No. of Probab- Probab- ility of ex-nuptial to nuptial matRpn- Married Cases o( ility of " Never Un- No. of ility of ility of ity Age Women at Nuptial Confine- Matern- ity* Married" Women married Women Cases of Bx-nup- Matern- ityt Matern- ity Groups. Census ment in during ai at tial Con- during during Based Based 1911. 8 Years. 1 Year. Census 1911. Census 1911. finement 1 Year. 1 Year. upon the Never Married. upon the Un- Married. 11-14 19 34 2,226 168,778 168,778 152 11 11 .0005 .0005 16-19 8,637 33,245 4,791 214,875 214,905 14,889 862 862 .0180 .0180 20-24 65,506 208,667 3,962 152,967 153,514 21,695 1,765 1,759 .0445 .0444 25-29 109,832 267,886 3,036 78,036 79,918 9,696 1,546 1,510 .0509 .0497 30-34 112,532 204,093 2,257 44,341 47,903 4,822 1,353 1,253 .0600 .0555 35-39 104,825 134,481 1,597 29,953 35,888 2,909 1,208 1,009 .0757 .0632 40-44 94,917 53,143 697 21,483 30,325 1,040 602 427 .0865 .0613 45-49 82,263 5,280 80 15,006 27,172 131 108 60 .136 .075 50-54 60,939 60 1.2 9,784 23,463 7 9 3 .73 .025 55-60 38,905 4 .12 5,698 20,063 ? ? * Probability per annum per 10,000 married women of same age-group. t Probability per annum per 100,000 "never married" women of same age-group. 18. Theory of fertility, sterility and fecundity. — The fertility-ratio ov probability of maternity in a unit of time may be defined as the proportion of cases, which, subjected to a given degree of risk for a unit of time, result in maternity ; and similarly, the sterility ratio or probability of maternity is the arithmetical complement of the probability ; or calling these respectively p and q, p-\-q = 1. If instead of " a unit of time," we write " varioics given periods of time," we arrive at the conception of a varying degree of fertiUtyor sterility, which for brevity, we may call the fertility, q, or the sterihty, 0. That is to say, instead of making a sharp qualitative cleavage between the fertile and the infertile or sterile, both are to be regarded as varying quantitatively. Any compilation shewing the frequency of cases of maternity according to duration of marriage reveals the propriety of this mode of envisaging the question. But we have seen that fertility decreases after a certain age, hence age must also be taken into account. Further, the " degree of risk" varies with the age of the husband. Hence, if x denote the age of the wife, y that of the husband, i the duration of the risk, we have : — (558) q =f{x,y,i) ; and s = 1 — q Fertility and sterility in the sense indicated are determined by the question of a single case of maternity. If instead of this we substitute " result in n cases of maternity," or " result in the bearing of n' children," we arrive at the quantitative conception of fecundity. It is not unlikely that the " degree of risk" varies with the number of previous births. If so, we must write (x, y, i, n) in this last equation. 320 APPENDIX A. If the total number of married women of age x be denoted by xM, the duration of their marriage be denoted by a suffix i, the number of nulliparae, primiparae, and multiparse up to « by the suffixes 1,2, . . . n, then we can have compilations of the types (559). ^M =^Mo + ,Mi+....^Jfi (5601. ,M = ^M'o + ;.M'i + ^M'„ that is, compilation according to age and duration of marriage, or according to age and " issue." It is at once evident that an exhaustive compilation according Ui x, y, i and n is out of the question, since the individual numbers in each " parcel" would be too small. Hence, serviceable tables must ignore some of the factors. In some countries fertihty probably varies but slightly with the age of the husband, and in all the distribution according to the age probably does not materially vary. Hence, by ignoring the issue, tables of " fertility and sterility " and of " fecundity" may take the following forms, the partial tables serving all general practical purposes : — Tables of Fertility and Sterility (effect of " Previous Issue" being Ignored). Arguments of complete tables. Argument of partial tables. (i.) Age of wife, with (ii.) age of husband, (iii.) Duration of marriage. (i.) Age of wife only (i.e., with hus- bands of all ages), (ii.) Duration of marriage. The tables themselves should shew, for each combination of age and duration of marriage, the proportion of married women who have borne one child. Tables of Fecundity (effect of " Previous Issue" being Ignored). (i.), (ii.), and (iii.) as above. (i.) and (ii.) as above. The tables themselves should shew, for esich combination of age and duration of marriage, the proportion of married women who have borne n children, where n is successively 0, 1, 2, 3, 4, etc., etc. Such tables wiU need to be for small age-groups (say for single years), and for durations of marriage, which change by smaU amounts (say one year), inasmuch as the age and duration change together, and the effect of age is considerable, COMPLEX ELEMENTS OP FERTILITY AND FECUNDITY. 321 19. Past fecundity ot an existing population. — The past fecundity of any population as at a particular moment is given by a census, both according to " duration of existing marriage" and according to " age." The usual tabulation according to existing marriage ignores the fact that the record is incomplete, and that for deduction purposes a previous marriage may to some extent modify the fecundity. The results in Tables CXIII. and CXIV. hereunder are deduced from the Census tabula- tions by applying the method outlined in § 4, p. 300, to the crude results. The aggregates for the same " issue" are not, of course, in agreement since in the one case the numbers according to the issue from existing marriages are recorded, and in the other, the numbers according to age include all previous issue. Numbers who boie 1, 2..n Children; also Proportion found to be Sterile. Age of Wife. thoua- ands. ( 1 "30- s S 10^ 3 iTTo V- -e d ^ ft/ •|a 30 70 // V •e a Ik > s \ fi « 1 20 60 o A v^ 1« 5000 ^ \ \ o s S 10 50 ;;;f p £- ^ Ss \ \zo -4 4000 'S SC ^ =^> \ ^_ S \.— X ■< ~ 40 -S >- =^ ^'^ P^ 4- •0 3000 g ^ W 7 u 1 30 1 * 0-4 9 1 w 2000 ^ / s ^ \ ^ S 20 .-' ( # \\ \ |2i « / \ < r \ ^ ^y \ fc 10 / -^ ^ »?s 3^ ^ ^ ' ,„___^ — ^ ^ ,A -^ ^ is-. I ■ 2 1 > , i •( ( 5« g x ^ ^* is ^ = ^ !-^ ) ' 1 1 < s 9 • N r f uml I 9 jerof 10 1 Child 88. I L lien 1 1 3 1 « I S 1 e 1 T I e Curves a to i shew numbers who bore to n children during durations of marriage to 4, 5 to 9, 10 to 14, etc., see Table CXIII. Curves a' to j' shew the numbers who bore to n children according to age and without regard to duration of marriage ; curve a' denoting all under 20 ; curve b' all aged 20 to 24 last birthday ; curve c' all aged 25 to 29, etc. ; see Table CXIV. Curva-s 15 tp 20 shew numbers of wives who bore to n. children for ages 15 to 20 last birthday ; see Table CXIV. These curves are valid only for integral values of the abscissa (number of children). Curve A shews the proportion of wives according to age, but of all durations of marriage, who proved sterile. 322 APPENDIX A. Table CXm.- -Shevnng Issne of 1,000,000 Wives according to Duration of Existing Duration Existing NUMBEB OP WiTEB WHO HAD OIVBII BlETH TO ChILDBBK TO THE NUMBEE 01" — Marriage. 0. 1. 2. 3. 4. 5. 6. 7. 8. 9. Under 5 years . . 5-9 years 10-14 „ 15-19 „ 20-24 „ 25-29 „ 30-34 „ 35-39 „ 40-44 „ 45 and over 73,765 23,504 16,031 9,586 7,374 5,082 2,947 1,904 1,055 970 82,436 28,564 15,059 8,821 8,465 3,806 2,038 1,212 800 585 37,904 50,165 22,961 13,150 9,714 5,450 2,869 1,438 778 606 6,874 47,053 27,141 15,427 12,803 7,701 3,586 1,921 948 821 469 24,421 28,897 16,200 13,916 9,413 4,684 2,600 1,206 1,094 23 7,800 22,421 14,542 13,278 10,078 5,581 3,080 1,582 1,513 1,776 13,774 13,072 12,088 10,095 5,977 3,478 2,024 1,883 344 6,325 10,191 10,253 9,162 8,338 3,884 2,330 2,374 '87 2,285 6,814 8,338 8,043 8,223 4,354 2,611 2,859 "7 718 3,767 6,602 6,858 5,734 4,101 2,618 3,067 Totals for existing marriage 142,218 149,584 144,833 128,055 100,900 79,898 64,146 51,179 41,492 33,260 Total per million (or all ages . . 123,995 146,153 145,107 124,239 103,088 82,140 67,029 53,803 44,026 35,392 • This does not include children by previous marriage , or ex-nuptial children ; it shews the relative frequency of issues of a given number according to " duration of marriage." t The actual total number of wives was 733,773, of which 3747 gave no information either as to durationof marriage or as to number of children ; 12.073 gave no information as to number of children, but stated their age ' and 21,151 gave no information as to age, but stated the number of children. The 3747 were distributed proportionately to the partially specified totals, the two parts being 1362 Table CXIV .—Shewing Issue of 1,000,000 Wives according to Age, at NffMBEE OF Wives to whom had been Bobit Childeen to THE HVKBER OF — Age of wives. 1. 2. 3. 4. 6. 6. 7. 8. 9. 10. 13 1 14 18 7 , , . , . , , , , ^ 16 92 34 , , , , , . 16 249 207 14 17 879 701 61 6 .. 18 1,445 1,723 298 19 1 1 19 2,088 3,002 747 123 8 20 2,987 4,751 1,765 320 22 3 21-24 19,474 29,892 19,136 7,882 2,215 439 108 18 4 25-29 25,137 38,640 38,232 24,981 14,161 6,362 2,357 749 215 S3 7 30-34 18,429 26,026 30,571 27,374 21,084 14,291 8,831 4,645 2,082 838 284 35-39 14,383 15,169 20,990 21,917 19,799 16,043 12,429 8,728 5,785 3,568 1,945 40-44 12,037 10,458 14,208 16,019 18,525 14,877 12,835 10,073 7,847 5,986 4,208 46-49 9,516 7,165 9,619 11,466 12,822 11,945 11,484 9,762 8,827 8,749 6,228 50-54 6,888 4,240 6,378 8,848 7,85(S 8,276 8,145 7,926 7,351 8,484 5,843 55-59 4,171 2,377 2,755 3,338 3,991 4,408 4,599 4,811 4,882 4,427 4,083 60-64 2,938 1,408 1,601 1,803 2,340 2,583 2,965 3,113 3,290 8,106 2,953 65-69 1,913 881 928 1,107 1,307 1,808 1,941 2,101 2,285 2,118 2,175 70-74 1,057 603 503 851 723 891 1,005 1,169 1,168 1,294 1,239 75-79 474 214 296 269 314 424 512 484 562 584 587 80-84 164 61 86 110 113 149 182 181 171 167 155 85-89 48 12 12 22 27 49 39 36 49 28 38 90-94 10 12 9 7 6 8 9 9 7 4 6 95-99 1 1 1 100-104 1 Totals 123,995 146,153 145,107 124,239 103,088 82,140 87,029 63,803 44,026 35,392 28,248 Totals million for existing mar- riaiges 142,218 149,5841 144,833 123,066 100,900 79,896 64,146 51,179 41,492 33,280 26,328 The actual total number of wives was 733,773, of which 343 gave no information as to age, or as to number of children ; 15,477 gave no information as to number of children, but stated their age ; 6432 gave no information as to age, but stated the number of children. The 343 were divided into two groups, viz., 254 and 89, these being distributed proportionately among the partially specified totals. The total additions thiu become for the several ages and age-groups : 0, 0, 0, 6, 11, 28, 70, 167, 1228 COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY. 323 Marriage* at Census of 3rd April, 1911, Australia (Based upon 733,773 Wives.)t Ntimbbr of Wives who had given Bikth to Childkeit to the NnMBER of— 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21 and over. Totals. "l 175 1,741 4,530 5,172 4,845 3,722 2,675 3,467 "63 726 2,364 3,455 3,313 2,789 2,077 2,676 "io 277 1,295 2,145 2,223 2,211 1,560 2,819 "6 99 606 1,199 1,301 1,237 978 1,433 '29 250 592 668 672 514 788 "9 115 282 317 319 287 421 "l 39 157 145 171 141 202 'io 50 67 87 59 88 "6 18 29 29 23 44 "3 6 23 4 23 "l 6 4 9 1 1 "6 6 14 10 22 200,471 183,722 153,846 114,452 109,821 88,571 58,677 39,213 23,981 27,246 26,328 17,463 12,040 6,859 3,513 1,750 856 341 149 59 22 58 1,000,000 28,246 18,826 13,035 7,488 3,834 1,927 941 379 182 68 36 66 1,000,000 and 2385. The luoreased numbers thus become : — For Age-groupa as indicated in table, 3358, 2903, 2074, 1371, 1210, 943, 591, 439, 230, 316 ; in all 13,435. For numbers of children as indicated in table, 2796, 2569, 2816, 2614, 2373, 2142, 1828, 1521, 1298, 1076, 876, 604 493, 288, 125, 62, 24, 17, 6, 2, 1, and 5 ; in all 23,536. These aggregates o£ unspecified and partially specified were then dis- tributed proportionately to the original numbers, see Vol. III., p. 1140-1, Census Eeport. Census of 3rd April, 1911, Australia. (Based upon 733,773 Wives.) NUMBEIl OP Wives to whom had been Born Children to the Number or- 21 Totals. 11. 12. is. 14. 15. 16. 17. 18. 19. 20. and over. •• *■ •• 1 25 126 470 1,446 3,485 5,966 . 9,848 78,943 5 3 148,792 97 31 7 3 3 1 153,375 907 380 160 67 - 16 3 1 142,270 2,437 1,443 766 328 178 78 20 13 1 3 129,939 3,458 2,241 1,287 685 327 156 73- 26 8 8 5 112,052 3,699 2,611 1,482 754 356 196 67 33 8 8 13 83,756 3,021 2,166 1,313 669 335 171 74 36 21 7 8 51,656 2,206 1,592 904 1,759 1,000 548 298 144 70 33 6 7 12 34,172 1,305 762 410 236 113 51 21 11 4 8 22,855 656 463 238 114 58 13 13 8 11 12,681 357 347 194 105 53 14 4 7 6 1 1 5,809 116 76 47 22 10 4 6 3 1,823 24 16 6 4 1 3 1 413 4 1 1 1 92 1 4 •• 1 18,826 13,035 7,488 3,834 1,927 941 379 182 68 36 66 1,000,000 17,463 12,040 6,859 3,513 1,750 856 341 149 59 22 58 1,000,000 2252 2203, 2071, 1993, 1677, 1431, 888, 648, 476, 316, 157, 81, 23, 4, 0, 2 ; in aU. 15,731 : and for the numters oi children as indicated in the table, 591, 717, 730, 693, 602 490, 415,^09, 277, 218, 181, 128, 76 51 18 12,7,2,3,0,0,1; in all 5521. These aggregates for the unspecified, together with the partiaily-Bpecifled, were then distributed proportionately to the original numbers ; see Vol. Vnj. Census Report, pp. 1366-7, 324 APPENDIX A. The results given in Table CXIII. are shewn by curves {a) to (*) in Fig. 88 ; and those in Table CXIV. are shewn by the curves (a' ) to (/ ) in the same figure, the single year results of the latter table being marked 15, 16, ... . 20. Interpolated curves would give the results for any other 5-year age or duration ranges.^ The curves of frequency of cases, according to number of issue, for the 5-year, or for the single-year age-groups, are of the same type, and are essentially dimorphic : strictly they give values only for integral values of the variable.^ Thus they could no doubt be fairly well represented by curves of the type : — (561) y = Aer^" + Bx^+o" in which x has the values 0, 1, 2, 3, . . . . etc. 20. Fecundity during a given year. — A different type of compilation is necessary to reveal what may be called the " existing fecundity." The existing nuptial fecundity is shewn by the number of married women in each age-group, the number who failed to bear a child during the year, and the number who bore the wth child where w = 1, 2, 3, .... etc. This is deduced from two sources, viz., (i.) from the Census record for the numbers of married women ; and (ii.) from the records of one year or for a series of years (1908-1914). The grand total of those who bore a child during the whole period of 7 years, if divided by 7.0666, gave a result substantially identical with that for the year 1911, which may be regarded as satisfactory.* This is seen from the close agreement of the numbers in the two upper portions of Table CXV. It is evident, therefore, that the vital statistics results for the Census year represent fairly satis- factorily the general case, and a 3 or 5-year result with the Census year as middle year would ordinarily be quite satisfactory. ' It is clearly desirable that Census results should be compiled for single years, as soon as public appreciation of the value of a correct statement of age leads to accuracy. 2 Statistical results furnish a number of examples of this character : for example the numbers of families living in houses with 1, 2, 3, ... n rooms, etc. ' If the rate of change of the proportion married be supposed linear, the married female population at the Census is to the aggregate of married females a^s 1 : 7.1272. The ratio of the number of brides is 1 : 6.9473. Theratioof females is 1 : 7.1077, and of population 1 : 7.1160. It is obvious, therefore, that the ratio 7.0666 is very nearly correct. COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY. 325 TABLE CXV. — Shewing foi various Age-groups and for all Durations of Marriage the Number who, during the year, bore the nth Child, where n = to 10 ; and the Total of those who bore a Child later than the 10th. Australia, 1911 and 1908-1914. Age of Xotal Married Women * No. who Bore a Child during the Year.t No. who Bore no Child during the Year. Number for which the Chh.p Born was the — Order Mothers 1st 2nd 3rd 4th 5th Later than 5th. 6th 7th 8th 9th 10th Later than 10th. Speci- fied. -19 20-24 26-29 80-84 85-40 40-44 45- 8,716 65,959 110,591 113,810 105,550 95,578 82,988 4,146 25,957 88,817 25,682 16,839 6,768 718 114,570 40,002 76,774 87,628 88,711 88,810 82,220 8,456 18,089 9,271 3,632 1,279 303 20 619 7,717 8,672 4,327 1,539 816 24 53 3,642 7,109 4,522 1,997 405 29 4 1,085 4,727 4,328 2,277 531 36 246 2,419 3,501 2,243 722 40 62 1,554 5,342 7,476 4,479 561 50 1,093 2,527 2,143 740 48 8 336 1,565 1,848 777 64 4 86 745 1,388 771 70 29 317 970 706 72 8 181 591 607 86 2 57 541 878 221 14 166 65 30 28 7 3 Totals 582,632 113,917 468,715 31,000 28,214 17,757 12,988 9,171 19,474 6,601 4,598 8,059 2,094 1,423 1,699 313 Numbers Gorrebfondinq to the Above Babes upon the Totals for the Period 1908-1914. -19 •8,716 §4,156 114,560 3,410 676 66 4 20-24 65,959 26,277 89,682 13,248 8,043 3,578 1,102 246 60 48 10 2 U 25-29 110,591 33,831 76,760 9,317 8,703 7,065 4,748 2,468 1,530 1,041 348 101 29 8 3 30-34 113,310 25,639 87,671 3,592 4,817 4,624 4,281 8,504 5,321 2,523 1,529 757 326 123 63 35-40 105,550 16,742 88,808 1,259 1,490 1,963 2,274 2,293 7,463 4378 2,130 1,865 1,416 968 580 504 40-44 95,573 6,609 88,964 288 312 418 547 666 717 746 749 677 579 910 45- 82,933 663 82,270 22 19 21 38 36 532 48 60 65 76 74 , 209 Totals 582,682 113,917 X 468,715 31,136 23,560 17,735 12,989 9,213 19,284 6,507 4,558 8,090 2,076 1,864 ■ 1,689 ~ FBOFORTIONS 10 lOIALB OF SAME AOE ; 1911 BEBULTS. -19 100,000 47,668 52,432 39,651 7,102 618 46 ^ ^0 161 20-24 100,000 39,353 60,647 19,768 11,700 5,522 1,645 373 94 '^76 • 12 6 1*6 E«o 6 2B1 25-29 100,000 80,579 69,421 8,384 7,842 6,428 4,274 2,187 1,405 15988 S304 78 26 "7 2 59 30-34 100,000 22,665 77,885 8,205 8,819 3,991 3,820 3,090 4,714 2,230 1,381 657 280 116 50 26 35-40 100,000| 15,958 84,047 1,212 1,458 1,892 2,157 2,125 7,083 2,030 1,751 1,310 919 560 513 26 40-44 100,000 7,076 92,924 317 381 424 556 755 4,686 774 813 80V 789 635 918 7 45- 100,000 860 99,140 24 29 35 44 48 676 58 77 84 87 104 266 4 Froportiokb to Totals of same Aqe ; Based upon the Totals for tee Period 1908-1914. -19 100,000 47,682 52,818 39,128 7,766 757 46 20-24 100,000 89,838 60,162 20,085 12,194 5,425 1,670 373 91 V3 16 3 u U 26-29 100,000 30,591 69,409 8,425 7,870 6,388 4,298 2,232 1,888 941 315 91 26 7 3 30-84 100,000 22,627 77,873 3,170 3,810 4,081 3,778 3,092 4,696 2,226 1,349 668 288 109 66 85-39 100,000 15,862 84,189 1,193 1,412 1,860 2,164 2,172 7.071 2,018 1,767 1,342 917 550 477 40-44 100,000 6,916 93,085 302 326 487 672 697 4,581 750 781 784 708 606 952 4&- 100,000 799 99,160 27 23 25 40 43 641 58 72 78 92 89 252 • Adjusted numbers, see Census Report, Vol. II., p. 19, and also Vol. III., pp. 1136-7. The numbers given are the Census numbers adjusted and multiplied by a factor to make them agree with the mean female population of the year. t In cases where a woman bore twice in the same year, she has been counted twice. The results in this column are obtained from the vital statistics of the year 1911. t The actual figures throughout have been multiplied by a factor (viz., 0.141509 = l-f- 7.0666), so as to make this total, 113,917, to agree with the total above : hence, if the distribution for 1911 were identical with that of the seven-year period 1908-1914, the figures in the several columns would be identical. They are approximately so. § The whole of the numbers in the column are those for 1908-1914, multiphed by 0.141509. II These numbers are obtained by subtracting the totals of those who bore children from the total number of married women. 3^6 APPENDIX A. TABLE CXVI. — Shewing the Number of Married Women at each Age, the Number oi Cases of Maternity, and the Number for all Durations of Marriage, who had not given Birth to a Child. Australia 1907-1914. Wives at 1911 who ■en Birth hUdren.t Bange Proportion of ^s^*: Proportion of Age last No. of Married Women in Years of Dura- Married Women who had not given Age last No. of Mauled Women No. of Wives a Census 1911 wli had given Birtl to no Children Married Women who had not given Birth- at No. Cas Maten 1907 tions of Birth to a Birth- at Birth to a day. Census 1911.t No. of Census hadgiv tono Mar- riage, (up to) Child. day. Census 1911. ChUd. Crude. Smooth ed. Crude. Smooth- ed. (i.) (ii.) iu. (iv.) (V.) (vi.) (vii.) (i.) (ii.) (iv.) (vl.) (vii.) 13 1 0.5 1 1* 1.0000 1.0000 14 18 19 3.7 4.2 13 14 2 .7222 .7388 .8140 ■• •• 15 93 21.2 67 "s .7204 .6530 '56 9,468 769 .08i7 16 349 141.9 183 4 .5244 .5330 56 8,557 678 .0815 17 1,145 494.7 498 5 .4349 .4450 57 7,675 581 .0814 18 2,551 1,219 1,061 6 .4159 .3820 58 6,912 .531 .0813 19 4,499 2,261 1,531 7 .3403 .3403 59 6,293 501 .0814 8,637 4,137.8 3,340 .3867 38,905 3.060 .0786 20 6,933 3,150 2,192 "8 .3162 .3075 '60 5,746 479 .0815 21 10,100 4,423 2,772 9 .2744 .2815 61 5,277 458 , , .0816 22 13,047 5,428 3,422 10 .2622 .2580 62 4,871 435 .0820 23 16,521 6,306 3,973 11 .2405 .2365 63 4,505 412 , , .0823 24 18,905 6,669 4,123 12 .2181 .2165 64 4,161 382 .0827 66,606 25,976 16,482 .2516 24,660 2,166 .0882 25 20,683 6,811 4,123 ■i3* .1993 .1990 '65 3,829 353 .0837 26 21,620 6,903 3,958 14 .1831 .1825 66 3,502 319 .0842 27 22,180 6,751 3,678 15 .1658 .1670 67 3,194 283 .0848 28 22,584 6,691 3,448 16 .1527 .1524 68 2,880 247 .0854 29 22,765 6,192 3,238 17 .1422 .1424 69 2,621 211 .0861 109,832 33,348 18,446 .1679 16,026 1.413 .0882 30 22,784 6,042 3,034 ■i8* .1332 .1339 ■70 2,365 190 .0868 31 22,726 5,065 2,849 19 .1264 .1266 71 2,099 168 .0876 32 22,542 5,240 2,684 20 .1191 .1203 72 1,867 146 .0885 33 22,421 4,722 2,540 21 .1133 .1147 73 1,652 129 .0896 34 22,059 4,338 2,416 22 .1095 .1101 74 1,444 115 .0908 41S,632 25,407 13,523 .1202 , 9,427 748 .0793 35 21,700 3,958 2,299 '23 .1059 .1062 '75 1,224 96 .0921 36 21,350 3,721 2,195 24 .1028 .1029 -76 1,004 82 .0934 37 21,000 3,315 2,101 25 .1000 .1000 77 818 70 38 20,560 3,118 2,017 26 .0981 .0979 78 650 59 39 20,215 2,629 1,942 27 .0961 .0959 79 510 48 104,835 16,741 10,564 ,1007 4.206 355 .0844 40 19,851 2,148 1,880 '28 .0947 .0942 'so 397 38 41 19,457 1,548 1,823 29 .0936 .0927 81 317 30 42 19,026 1,386 1,766 30 .0928 .0913 82 241 23 43 18,543 939 1,710 31 .0922 .0900 83 184 17 Pi 44 18,040 595 1,653 32 .0916 .0888 84 140 13 es 94,917 6,616 8.832 .0930 1.279 121 .0946 -ts 45 17,554 346 1,577 ■33 .0898 .0877 ■ '85 105 10 M 46 17,064 174.2 1,494 34 .0876 .0868 86 80 8 V 47 16,554 85.6 1,403 35 .0847 .0860 87 56 6 u 48 15,975 38.7 1,306 36 .0817 .0852 88 35 5 a 49 15,216 13.2 1,203 37 .0791 .0845 89 24 4 p 82,363 657.7 6,983 .0848 300 33 i.i6o 50 14,303 4.2 1,116 '38 .0780 .0837 '90 20 3 51 13,162 1.5 1,049 39 .0797 .0832 91 16 2 52 12,088 0.9 981 40 .0812 .0827 92 12 1 53 11,100 0.6 914 41 .0823 .0823 93 9 54 10,286 0.1 847 42 .0823 .0819 94 7 60,939 7.8 4,907 .0806 64 6 .0937 95-100 21 • Actually extends to about 1 year greater than shewn. t Graduated. 21. Number of married women without children, all durations oi marriage. — ^The relative numbers of married women of each age, and for all durations of marriage, who are without children, are readily determin- able by means of a Census. That for 1911 gave the results shewn in Table CXVI. above. The smoothed results in column (vii.) of the table are shewn by curve A on Eig. 88. The ratio very rapidly falls to the value of about one-fourth, which is attained during age 22 ; one-eighth COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY. 327 is reached during age 31 ; one-tenth during age 37 ; and the minimum during age 58, which age is, of course, somewhat uncertain. After the age of that minimum the results are very uncertain. Apparently the curve will require several terms of the type Ae""* to empirically represent it, thus the ratio being denoted by a, and the age ^ being reckoned from say 12 or 13, the ratio will be of the form : — (562). .0- = J. + 5e-»f G+e-of +....+ ZP 22. Sterility-ratios according to age and duration of marriage.— The effect of the age of the husband being ignored, the number of cases of sterihty, (or more strictly of childlessness,)^ according to duration of marriage, for women of different ages in Australia was found from the Census of 1911 to be as shewn in the following table : — ' Physiological sterility is the condition, not merely of childlessness, but of childlessness due either to failure to conceive, or to retain the fertilised ovum the full time. The data of ordinary statistics cannot conclusively establish the frequency of physiological sterility, since what is given are merely measures of childlessness. A number of instances are given in the " Handbuch der Medizinischen Statistik," by Friedrich Prinzing, Dr. Med., 1906, Cap. III. ; " Die sterilen und kinderloaen Ehen," pp. 30-40. The following estimations of sterility may be mentioned * No. of Elapsed Period No. of Marriages after Sterile Ratio. Authority. under Observation. Marriage. Cases. Dresden Returns 27,911 5 years 672 0.02407 Dresden Returns 27,911 10 years & more 134 0.00480 AusterUtz, Prag, 1891- 1900 3,920 Not stated 295 0.0753 Hofmeier 2,220 Not stated ? 0.147 Lier and Ascher 2,500 Not stated ? 0.090 Huizinga (Groningen) . . 1,180 Not stated ? 0.115 Verrijn Stuart, Nether- lands 9,443 16 to 21 years ? 0.131 Do., poorer classes 1 Not stated ? t0.141t0.110 Do., middle classes 1 Not stated ? • 0.162 t0.109 Do., well-to-do classes ? Not stated ? t0.160t0.126 "I" Town. { Country. *Other results are: — Spencer, Wells & Sims (Great Britain), 0.125; Duncan (Glasgow and Edinburgh), 0.163; Ansell, 1919 cases. Married Women, 0.079 ; A Swedish County, 0.100 ; Massachusetts, 1885, 0.176 ; Women over 50, 0.119. The whole of the above statements are, of course, defective, inasmuch as sterility is a function both of duration of marriage as well as of age, etc. 328 APPENDIX A. Table CXVn. — Sterility according to Age and Duration of Existing Marriage. Australia, 8rd April, 1911 (Censns). DTJaATION OF BXISTING MAKMAGB. AQE OF TTiTDEE 5 Tears. 5 TO 10 Tears. 10 TO 15 TeAES. 15 TC 20 Tears. 20 TO 25 Tears. AT Time OP Cbssds II •si §3 11 Ss fl ll fl jJ H£ fl %3 Hfi fl Sa s 03 oS g "' oy 00 3S s m Under •14 1 1 1.000 •14 13 18 .722 •15 67 92 .728 •1« 179 338 .530 •17 490 1,044 .469 •18 1,042 2,512 .415 •19 1,496 4,270 .350 •20 2,114 6,69S .316 7 261 .027 21-24 13,378 43,424 .308 474 11,926 .040 25-29 14,724 45,67S .322 3,004 47,785 .063 346 10,587 .»33 1 21 .048 30-34 7,398 19,735 .375 3,998 38,675 .103 2,07fi 40,121 .052 262 8,594 .030 2 21 .095 35-39 4,099 8,11S .505 2,885 16,992 .170 2,693 32,715 .082 1,348 31,792 .042 279 9,324 .030 40-44 2,597 3,575 .726 2,120 6,731 .315 2,096 14,568 .144 1,723 24,408 .071 1,250 32,477 .038 45-49 1,753 1,865 .938 1,712 2,74S .623 1,517 5,280 .287 1,199 9,253 .130 1,365 22,758 .060 50-54 893 894 .999 1,10S 1,187 .933 1,08S 1,865 .584 803 3,230 .244 803 7,780 .103 55-59 431 431 1.000 6'C 531 .998 701 779 .900 S6« 1,088 .520 530 2,265 .234 60-64 247 247 1.000 255 255 1.000 332 332 1.000 420 447 .940 411 791 .520 65-69 140 140 1.000 117 117 1.000 173 17^ 1.000 198 199 .995 305 337 .905 70-74 64 64 l.OOO 74 74 1.000 91 91 1.000 100 100 1.000 127 128 .992 75-79 20 20 1.000 28 28 1.000 37 37 1.000 38 38 1.000 63 53 1.000 80-84 3 a 1.000 7 7 1.000 11 11 1.000 4 4 1.000 17 17 1.000 85-89 1 1 1.000 2 2 1.000 3 3 1.000 2 2 1.000 4 4 1.000 DUEATION OF EXISTING MAKEIAGB. Age 25 TO 30 T ''EAES. 30 1 35 1 fEABS. 35 1 40 1 ''EARS. 40 TC 45 T EARS. Otes 45 Tears. WiTES AT TIME OP Cmavt oS ll &4 1^ Is i| ^1 1^ ll i| m il ^1 1i to oS li 35-39 1 8 .125 I 40-44 221 8,075 .027 45-4S 1,005 28,66« .035 165 6,55e .025 50-54 955 17,237 .055 663 20,004 i)33 121 5,081 .024 55-59 456 4.762 .096 511 9,574 .053 432 12,606 .034 91 3,087 .029 60-64 346 1,590 .218 284 2,946 .096 388 6,437 .060 250 7,789 .032 59 2,209 .027 65-69 292 522 .559 213 978 .218 ' 189 2,120 .08S 22« 4,083 .055 214 6,630 .032 70-74 188 206 .913 125 295 .424 • 106 569 .186 101 1,150 .088 221 5,660 .039 75-75 72 72 1.000 68 73 .932 59 121 .488 4« 305 .151 116 2,994 .039 80-84 15 15 1.000 23 23 1.000 25 26 .962 18 57 .316 46 945 .049 85-8S 2 2 1.000 5 5 1.000 e 9 1.000 5 8 .625 14 200 .070 90-94 1 1 1.000 1 1 1.000 e 41 .146 95-99 1 1 1.000 •• 1 ■■ • The results are from Cenaug Eeport HI., p. 1136. The general resnlts are obUined from an nnpablished series of compilations according to age-groups, and duration-of-marriage groups. In neitiier case were the " unspecified" dislaributed ; such distribution, however, can affect Vbe results only very slightly. An examination of the results given in the table shews that initially the sterility-ratio decreases ; it attains a minimum, and then increases ; see particularly the duration of marriage to 4 years (i.e., under 5 years). The initial fall may be regarded as the normal decrease of cluldlessn^s with increase of the duration of the risk. From the minimum onward, however, the cur re shews the true measure of steriJity for a given duration of marriage, and for any age terminating the given duration of marriage. COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY. 329 The curves on Fig. 89 are the sterility-ratios according to age, each curve denoting a separate range of duration of marriage. By projection^ Fig. 90, shewing the curves of equal sterility, is derived. From these, the correlative durations of marriage and ages, corresponding to any degree of sterihty, can be at once seen. The dots give the positions as determined from the data,^ the curves throughout are smoothed. Fig. 89. t6 ' Sr~" BT Age of Wives Fig. 90. In Fig. 89 the ordinates to the evirvea denote the degrees of sterility : the abscisses denote the age corresponding to the duration of marriage shewn on any curve in question. In Fig. 90, the intersections of the curves with the lines of equal sterility on Fig. 89, are projected, to the ordinates -line corresponding to the mean of the range of durations, viz., 2.5, 7.5, 12.5, etc. years. Smoothed curves have then been drawn shewing the probable position of the curves of equal sterility. Curve A in Fig. 90 denotes the sterility -ratio according to age at marriage where the duration of marriage is 20 years. On Fig. 90 they represent the projected results, and the lines drawn among them, the smoothed general results deduced therefrom. Thus the 1 It has been assvimed that the group-results for the ranges 0-5, 6-10, 10-15, etc., are sensibly correct for the durations 2.5, 7.5, 12.5, etc., as is evident from Fig. 90. This is not quite exact ; the error is not large, however, and the inherent limitations of the determination of the ratio render the measure of uncertainty of but little moment. 2 The three broken lines crossing from Fig. 89 to Fig. 90, indicate the scheme of projection. Thus, the point b, viz., the intersection of the curve assumed to repre- sent a sterility of 0.3 for a duration of marriage of 12.5 years, is found in the graph (plan), Fig. 90, as the point b', viz., on the line parallel to the axis of age at the distance (ordinate) therefrom 12.5, and similarly for point a and c and a' and c'. 330 APPENDIX A. new curves so obtained represent completely the steriUty-ratios according to age taken in conjunction with past duration of maniage.'^ It is obvious that tables shewing average sterility can be constructed (i.) according to ajge at marriage and time since elapsed ; and (ii.) according to age attained after the given interval between it and marriage. As, however, the one differs from the other merely by the whole amount of the duration, it is immaterial in which form they are set out. In the following table (CXVIII.) the former method is adopted ; Figs. 89 and 90, however, give the age attained after a given duration of marriage.* TABLE CXVm.- -Shewing for varions Ages and Durations of Marriage the Degree of Sterility experienced. Aoslralia, 1911 . COBBESPONDING DUBATIONS OF MabEIAGB (iN Yeabs). Sterility-Ratio. 5 10 15 20 25 30 35 40 45 When Tim Age at Mabbiaoe is:—* .025 13.8 15.5 16.6 17.1 17.1 16.7t 16.0t 15.lt .050 19.3 21.3 22.9 23.7 24.3 24.3 24.4 27.6 .075 , , 23.1 24.9 26.3 27.2 27.8 28.0 28.1 31.9 .100 25.8 27.6 28.8 29.6 30.1 30.3 30.9 34.1 .150 29.3 30.7 31.8 32.4 32.6 32.5 33.4 37.9 .200 31.6 32.7 33.8 34.1 34.3 34.5 35.4 .250 33.4 34.3 35.1 35.4 35.7 35.9 36.8 .300 34.9 35.5 36.2 36.6 37.0 37.2 37.9 .350 34.0 35.8 36.5 37.1 37.5 37.9 38.0 38.6 .400 35.1 36.7 37.2 37.8 38.3 38.7 38.8 39.6 .450 36.1 37.5 38.1 38.7 39.2 39.7 40.0 40.8 .500 37.1 38.7 38.8 39.5 39.8 40.4 40.9 41.7 .600 39.0 39.8 40.1 40.7 41.0 41.4 41.7 42.6 .700 40.8 41.4 41.6 42.1 42.2 42.5 42.8 43.7 .800 42.5 43.0 43.1 43.3 43.5 43.7 43.7 44.6 .900 44.5 44.5 44.6 44.7 44.0 44.0 45.0 46.0 .950 45.9 46.1 46.1 46.2 46.3 46.3 46.3 47.0 .975 47.6 47.7 47.7 47.8 47.7 47.6t 47.6t 48.0t 1.000 51.6t 51.5t 51.4t 51.3t 51.2t 51.lt 50.9t 50.7t • The table is thus Interpieted : — Heading horizontally, it the age at maniage was say 16.6 years, and the duration of marriage was 20;year8, 0.025 would be the proportion without children. Similarly it tt\e age at marriage was 17.1 years, and the duration of marriage was either 25 years or 30 years, or reading yertically, for the duration of marriage of 15 years, if the age at marriage were 15.5, tlien 0.025 woiud be sterile ; if the age were 21.3, tlien 0.050 would be sterile ; and so on. t The apparent anomaly in these results may possibly be explained by tlie more fertile not living sufficiently long to be included in the category of those whose duration of marriage attained the numlter of years indicated. The steriUty-ratios given in the table for durations of marriage 0-5, do not accord very closely with those deduced by the method of Part 1 Strictly these curves represent the mean of 5-year groups, both as regarda duration of marriage and age. The corrections to make them instantaneous results, however, are small. ' Data have not been compiled which would enable these resvdts to be worked out with very great precision. For this it would of course be necessary to compile according to single years both as regards age and duration of marriage ; and give results according to " age at marriage" and " duration of marriage" instead of existing age. GOMPLEX ELEMENTS OF FERTILITY AND FECUNDITY. 331 XIII., §§ 11-13, pp. 245 to 250. The probabiUty of a birth, and that of childlessness should together equal unity : For 0-6 years the agreement, however, is closer; see Fig. 71, p. 249, or the values given in Table LXXV., p. 247. As, however, the results for the shorter durations are necessarily somewhat uncertain, these differences are not remarkable. It may be pointed out the results indicated in Table LXXI., p. 238, shew that for the age 51 the probabihty of a birth is 1.17 per thousand, hence the final value should probably be 0.999, rather than 1.000. But tables of this kind are, of course, probably never reUable to this order of precision. 23. Curves of sterility according to duration o£ marriage. — ^The steriUty-ratios determined from the age of the married woman only, are based upon the assurryption that fertility is independent of the age of the husband : this is shewn hereafter not to be the case. Or we may regard the results as true for the average condition {i.e., the condition including husbands of all ages ) . Continuing this assumption and taking the curve for a duration of marriage of 20 years, it is found that the proportion sterile who are married at the ages 11, 12, . . . 51 respectively are as shewn in Fig. 90, Curve A. The ordinate at age 11 is not necessarily zero, but owing to the fact that marriages at that age usually arise from special circumstances, the value of the sterility-ratio is practically zero.^ The curve has a point of inflexion, for marriages at ahoatehge40,{i.e.,d^yjdx^=0 for X = 40), and the sterility-ratio changes most rapidly at about age 28 (i.e., d\/ dx^ = for a; = 28). The curves of steriUty can be obtained by plotting the ages in the vertical columns in Table CXVIII., as abscissae, and the value of the observed steriUty as an ordinate. For every given duration of marriage there will be a different curve. 24. Fecundity according to age and duration of marriage : various distributions and ratios. — ^As already pointed out, fecundity is a function of the age of the husband and of the wife, as well, of course, as of the dura- tion of marriage. It has been shewn herein also, for various durations of marriage, that on the average (i.g., the results being for husbands of all ages combined), and for those only who come under observation in cases of birth, the number of children borne, according to duration of marriage (i), is about I + TT * ; see formula (523) of Part XIII., §§ 34, 35, and Table XC, pp. 279-283. The surface of representation of this is, for the most part, sensibly a plane. It defines the polygenesic^ distribution, see p. 285 ; and thus may be called the polygenesic surface. In the case of this distribution differences of age have much less influence, if any, than differences in duration of marriage. It is important to bear in mind, however, that this distribution, as above stated, applies only to a limited ^ That is the marriages are what have been (somewhat ill-advisedly) called prejudiced" — and do not represent the average liability of becoming fertile. » The word " polygenesis" has been used to indicate the origination of a race arising from several independent ancestors or germs. The above use will, however, lead to no confusion, and is consistent with the general mode of word construction. The word polyphoroua (from ro\v^6pos = bearing many) is used hereinafter for a different function. 332 APPENDIX A. number of married women, viz., those whose total fecundity ha/ppe7is to come under review through repeated child-bearing. In Part XIII., § 36, p. 285, the total number of children borne by married women of given limits of age and duration of marriage has been called the " general genesic," or " fecundity" distribution. For many purposes, however, it is desirable to know the number of mothers (a) instead of the number of children (say, z'= kz, k = Q, \, 2 . . . n) being the number borne by each woman) . It is also preferable to relate the number of married women to the exact number, k, of children borne by each. Let, therefore, ^m, im, zin . . . ^m denote the number of married women who bore 0, 1, 2 . . . n children respectively, the range of whose ages are between Xq and xi, xi and xz, etc., and the range of whose durations-of -marriage are t^ and & m c3 ^ o >; 1 j \<^^ hJ&v ^^^■^ " N ^ S T Y ^ g s ^ n V Ss^ c^- ^ ^ -§%^ j,iss| n SSS ^^5^ ^Si , ^) ^ :^:s_- :^^5 \ \^^ J^f ^k- ^^ \^ V 5 n :^5V '> -■a- /I-»5 . / SI ■ , III 1 16 Fig. 92 shews the characterisics of the age-polyphorous surface, the age being that at the time of the Census. If compiled according to the " age at marriage" is, of, course, materially changed. Number of Children borne. Fig. 92. the form of the contours 336 APPENDIX A. TABLE CXZn. — Shening. for all Durations of Marriage combined, the Relative Numbers of Married Women of given Age-gronps who bore 0, 2, 3 . . . to n Children. Australia, Census of 3rd April, 1911. Age-polyphorous Distribution. Ages No. of Wives Batio of the Number who bore the kth Child to the total Married Women of the Age-groups indicated, where k = Wives. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Over 20 Total. 13 1 1.000 1.0 14 18 7200 2800 .. 1.0 15 92 7280 2720 •• 1.0 16 345 5300 4410 0290 1.C 17 1061 4690 •4850 0420 0040 1.0 18 2557 4149 •4939 0849 0055 0004 0004 .. 1.0 19 4376 3497 •5034 1252 0203 0014 .. 1.0 20 7,224 3034 •4828 1789 0324 0022 0003 1.0 21-24 57,896 2495 •3772 2408 0980 0275 0055 0013 0002 00003 1.0 25-29 109138 1730 •2610 2428 1660 0933 0419 0153 0049 0014 00003 00005 00004 00001 1.0 30-34 112523 1281 1678 •1967 1764 1346 0912 0550 0294 0130 0052 0018 00055 00018 00003 00002 00001 00001 1.0 35-39 104619 95,392 1147 1126 1100 0853 1478 1093 •1522 1215 1359 1095 1095 0842 0934 0587 0743 0400 0574 0240 0440 0129 0309 0060 0177 0026 0104 0010 0055 0004 0024 0001 0013 0005 0001 0001 1.0 40-44 •1238 1.0 45-49 82,237 61,447 1142 1140 0680 0543 0849 0652 1010 0806 •1091 0927 1030 •0956 0971 0829 0896 0700 0825 0567 0730 0438 0589 0292 0407 0188 0290 0106 0164 0056 0083 0028 0039 0013 0021 0006 0008 0002 0004 0001 0001 0001 0000 0001 1.0 50-54 0918 1.0 55-59 37,900 1222 0495 0540 0653 0743 0823 0847 •0878 0878 0799 0734 0539 0384 0236 0118 0058 0031 0012 0005 0003 0001 0001 1.0 60-64 25,065 1312 0437 0479 0542 0677 0742 0828 0845 •0884 0834 0805 0588 0475 0257 0149 0078 0037 0020 0007 0001 0001 0002 1.0 65-69 16,640 9,297 1353 1408 0415 0417 0417 0419 0508 0498 0581 0559 0617 0682 0788 0713 0837 0869 •0928 0858 •0930 0883 0878 0647 0644 0516 0474 0312 0327 0165 0161 0094 0044 0022 0008 0008 0005 0005 0000 0000 0002 0008 1.0 70-74 0869 0079 0043 0009 1.0 75-79 4,254 1425 0416 0524 0487 0531 0712 0825 0790 0872 0881 •0893 0555 0527 0287 0157 0073 0024 0007 0009 0005 1.0 80-105 1,691 1532 0373 0473 0597 0609 0875 0905 •0958 0934 0798 0710 0538 0385 0160 0095 0035 0012 0012 1.0 13-105 Nos. 733773 104761 109720 106195 • 90218 73962 58482 47045 37540 30537 24399 19317 12805 8841 5023 2575 1280 625 245 107 42 16 36 1.0 Batio 100000 14277 14953 14472 12295 10080 07970 06411 05116 04162 03325 02633 01745 01205 00685 00351 00174 00085 00033 00015 00006 00002 00005 1.0 Besnlt as by cxnr.t 14222 14958 14483 12306 10090 07990 06415 05118 04149 03326 02632 01746 01204 00686 00351 00175 00086 00034 00015 00006 00002 00006 1.0 jq'Qte — ^Tbe figures marked with an asterisk are the maxima in the horizontal lines, and those underlined are the maxima in the vertical columns excei>ting in the case of column 0, where .1126 is the minimum. ' t The figures though very approximate to the line above are given by a wholly different distribution of unspecified and partially specified cases. The figures in the Ixidy of the table are, of course, decunals. They are not deduced from those given in Table OXTV., pp. 322-3, but from the results of a more detailed distribution of the unspecified quantities for various age and duration-of-marriage groups, see Table CXXm., pp. 338-9 later. 30. The duiational polyphorous distribution. — ^The data from which the durational fecundity is derived furnish also the numbers required for the computation of the durational polyphorous distribution, viz., that which shews for given durations of marriage, or between given limits of duration of marriage, the relative frequency with which given numbers of children are borne. The ignored elements are the ages of the wives and of their husbands. Thisftable hasjnot been computed, but the necessary data are given ia Table CXXIII. hereinafter. COMPLEX ELEMENTS OP FERTILITY AND FECUNDITY. 337 31. Fecundity distributions according to age, duration of marriage and number of children borne. — The fecundity distribution tables, so far, are of the type z =f(x,y), but if age, duration of marriage and number of children borne, be simultaneously taken into account, then the distribu- tion-frequency is of the type z = f (w, x, y), and cannot be represented by a single three-dimensional graph, for example, height contours upon a plane. It is necessary in fact to have a graph for each value of w adopted in the tabulations. The exigencies of tabulation, of course, also require that a separate table of the values of z shall be given for each value of one co-ordinate (say w), for the values given by double entry of the other two (say x and y). In Table CXXIII., hereunder, the results are tabulated for single years of age from 13 to 20, (last birthday), for the ages 21 to 24, and then for every five year age-group onward. The table gives, for existing marriage, the number of wives, of various ages and durations of marriage, who failed to give birth to children, or who gave birth to 1, 2, 3, etc. In the tables as originally compiled, there was a considerable number of unspeciiied cases, viz., the following : — Class (i.), the larger class, in which the ages were specified. Class (ii.), a relatively small class, in which the ages were not specified. In each of these were three sub-classes as follow, viz. : — (a) in which the duration of marriage was not specified; (6) in which the number of children was not specified; (c) in which neither the duration of marriage nor the number of children was specified. It was consequently necessary to efEect a distribution in order to get anything like the most probable results.^ The method of distribution was that outlined in § 4, Table XC VII., and formulae (543) to (547). That is to say, sub-class (c) was first dis- tributed proportionately among sub-classes (o) and (6), and sub-classes (a) and (6) of Class (i.) were distributed proportionately among the fully specified cases. In Class (ii.) the corrected sub-classes (a) and (6) were then proportionately distributed among the fully specified corrected groups of Class (i.). The details of the distribution shewed that the result was very satisfactory judged by the regularity of the ratios (see § 5 hereinbefore). 1 The method of adopting the fully specified cases as characteristic of the whole, involves merely multiplying each by the ratio of the totals. An examina- tion of actual results shewed that recourse to this procedure was unsatisfactory. It rejects part of the evidence available. To distribute the partially specified oases is, therefore, much to be preferred. 338 APPENDIX A. TABLE CXXm. — Shewing, for Varions Durations of existing Marriage, the Number of Wives in Various Age-groups who bore k Children, where k = Q, 1, 2, etc. Australian Census, 3rd April, 1911. Nnmlier of Wives to whom had been bom Children to the Number of :- - \A Age 13 Age 14. Age 15. Age 16. Age 17. Age 18. a g 1 Total. 1 Total 1 2 Total 1 2 3 Total 1 2 3 4 5 Total. 0-5 1 13 5 18 67 25 92 183 152 10 345 498 514 45 4 1,061 1,061 1,263 216 14 1 1 2.556 5-10 •• 1 1 Totals 1 13 5 18 67 25 92 183 152 10 345 498 514 45! 4 1,061 1,061 1,263 217I 14 1 1 2,557 Age 19. Age 20. Age 21-24. 1 2 3 4 Total. 1 2 3 4 5 Total. 1 2 3 4 5 1 6 7 8 Total. 0-5 1,530 2,203 548 84 3 4,368 2,185 3,445 1,178 140 5 6,953 13,947 20,116 9,708 1,499 104 4 0, 45,378 5-lC 5 3 8 7 43 114 94 11 2 271 493 1,725 4,232 4,175 1,482 3141 73 lOj ..! 12,506 10-15 1 51 1 4| 1 2| 12 Totals 1,530 2,203 548 89 6 4,3761 2,192 3,488 1,292 234 16l 2 7,224l 14,440 21,841 13,94o! 5,675' 1,591^319' 771 n' 2' 57,898 •25-29 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 30 Total. 0-5 15,392 20,412 10,234 1,632 144 5 47,819 5-l( 3,127 7,205 14,778 14,40C 7,689 2,358 500 78 17 1 50,153 10-15 36( 866 1,48C 2,082 2,35C 2,212 1,171 453 133 31 4 4 11,145 15-2C 1 1 5 1 4 2 4 1 1 1 21 Totals 18,880 28,482 26,493 18,119 10,183 4,576 1,675 533 154 33 5 4 1 109.138 •30-34 i 0-5 7,788 8,17^ 4,099 675 55 1 20,792 5-11 4,185 6,678 11,08S 10,308 5,759 1,988 466 97 17 2 40,587 10-lE 2,16( 3,456 6,051 7,827 8,093 7,022 4,372 2,087 727 204 377 49 13 5 2 42,077 15-21 27i 566 906 1,037 1,241 1,244 1,348 1,122 710 155 52 15 a 2 1 1 9,051 20-2E 2 C 1 2 1 2 3 5 2 2 2 • • -. 22 Totals 14,414 18,87e 22,140 19,849 15,149 10,257 6,189 3,306 1,459 585 206 67 20 4 2 1 1 112,525 •35-35 0-5 4,39E 2,76- 1,307 20i 2'. 5 8,701 5-11 3,061 3,06( 4,50S 4,06f 2,22J 78e 203 6C 24 ] 1 17,994 10-lE 2,835 3,21( 5,45< 6,288 6,087 4,94E 3,240 1,511 585 207 57 24 3 2 34,443 15-21 1,425 2,03. 3,458 4,424 4,782 4,596 4,297 3,54« 2,59t l,39i 66« 281 106 39 8 1 33,649 20-2E 295 43E 739 944 1,092 1,128 1,062 1,019 985 89i 62e 324 161 68 37 11 2 1 9,824 25-3( ) ] C 1 1 2 1 1 1 .. 8 Totals 12,002 ii,5oe 15,462 15,927 14,213 11,458 8,803 6,138 4,190 2,500 1,350 630 270 110 45 12 2 1 104,619 •40-44 0-5 2,811 787 244 2! 4 .. 3,869 5-lC 2,30( 1,52; 1,53( l,06i 571 232 53 7 ; ] 7,288 10-16 2,23( 1,97( 2,607 2,626 2,435 1,821 l,04e 48e 177 71 25 5 15,506 15-2( 1,836 1,99! 3,06S 3,654 3,859 3,359 2,931 2,227 1,442 81! 387 177 65 27 10 4 25,865 20-25 1,32J 1,570 2,587 3,634 4,156 4,165 3,984 3,531 3,03< 2,541 1,814 957 55S 264 110 56 17 3 3 34,308 20-35 23E 287 385 585 782 864 890 835 81! 775 721 55C 363 232 109 6C 33 7 A r 1 ?, 8,539 30-35 2 4 1 1 4 1 1 1 15 Totals 10,739 8,140 10,422 11,590 11,809 10,441 8,908 7,087 5,476 4,207 2,948 1,689 988 524 230 120 50 10 9 1 2 95,390 •45-49 0-5 1,901 112 11 1 1 2,026 3,029 5,738 9,914 24,368 30,339 6,816 7 5-10 1,882 624 309 139 46 21 5 1 2 10-15 1,646 1,105 1,001 854 553 305 168 63 2! ! i 2 15-20 1,281 1,055 1,479 1,567 1,516 1,195 826 406 277 132 59, le 12 3 1 17 7 » 1 (1 1 20-25 1,444 1,478 2,312 2,973 3,364 3,130 2,870 2,297 1,666 1,16! 749' 38S 195 88 31 122 68 22 A V S <^ 25-30 1,061 1,053 1,631 2,441 3,050 3,276 3,515 3,250 3,018 2,652 2,100 1,450 91J 50S 263 88 32 21 7 3 1 1 30-35 175 170 242 339 437 • 545 601 738 772 70< 690, 542 422 268 165 ( 35-40 1 1 2 1 1 1 .. Totals 9,390 5,598 6,985 8,314 8,968, 8,472 7,985 6,817 5,764 4,663 3,601 2,402 1,543 868 460 227 , 107 46 14 6 6 3 82,237 •50-54 0-5 969 1 , , 970 1,308 2,099 3,520 8,363 18,424 21,319 5,435 9 5-10 1,221 68 13 4 2 ' 10-15 1,224 451 208 126 45 25 11 4 4 1 15-20 877 600 619 565 406 219 119 51 3( 20 71 2 5 * • •• 20-25 862 748 1,055 1,289 1,291 1,098 782 552 352 16t 83, 44 23 11 4 2 1 25-30 1,019 769 1,237 1,742 2,144 2,329 2,244 2,145 1,761 1,264 847! 454 26! 110 49 22 13 e 1 30-35 704 590 749 1,076 1,580 1,893 2,156 2,275 2,38C 2,414 2,049, 1,439 977 646 265 11? 64 26 14 1 1 ^ 35-40 40-45 129 111 128 151 2 229 304 331 477 1 540 624 634 560 509 2 339 1 189 1 98 49 1 le 17 6 1 3 1 Totals 7,005 3,338 4,009, 4,955 5,637, 5,868 5,643 5,505 5,067 4,489 3,620 2,499 1,784 1,007 508 240 j 128 47 22 e 3 7 61,447 • Ages at date of Census. COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY. 339 TABLE CXXm. — Shewing, for Various Durations oi existing Marriage, the Number of Wives in various Age-groups who bore h Children, where ^==0, 1, 2, etc. Australian Census, of 3rd April, 1911. — Oont. s^i Number of Wives to whom had been born Children to the Number oJ :- - tl 1 2 3 4 5 6 7- 8 9 10 11 12 13 14 15 18 17 18 19 20 o 20 Total. •55-59 0-5 466 466 5-10 578 1 579 10-15 770 60 21 3 1 1 856 15-20 630 285 174 75 37 8 4 1 1 1,215 20-25 576 388 387 390 292 184 112 73 32 26 5 5 2 1 2,472 25-30 493 398 488 625 721 762 661 433 264 143 93 48 16 11 9 1 1 5,167 30-35 544 369 510 751 889 1138 1,176 1,286 1,174 927 707 402 215 118 53 23 10 2 10,294 35-40 463 318 398 558 784 898 1,069 1,311 1,583 1,600 1,515 1,161 873 509 261 111 64 26 9 10 4 4 13,529 40-45 113 57 69, 71 92 128 188 222 275 331 462 425 348 254 126 83 44 19 10 2 2 3,321 45- •• 1 1 1 Totals 4,632 1,876 2,047| 2,473 2,816 3,118 3,211 3,326 3,328 3,028 2,783 2,041 1,454 893 449 218 119 47 19 12 4 6 37,900 •60-64 0-15 905 905 15-20 477 22 499 20-25 459 190 i30 74 22 5 4 1 885 25-30 380 231 265 270 248 168 97 45 29 9 5 4 1,751 30-35 312 2C3 267 320 428 445 417 330 211 142 88 35 11 5 3 1 3.216 35-40 422 235 302 397 551 690 829 802 888 687 524 304 220 56 38 17 12 8 4 1 1 2 6,988 40-45 270 170 207 253 381 468 620 796 887 1,019 1,088 832 642 401 207 105 45 21 7 1 1 3 8,424 45- 64 44 29 45 67 84 108 145 201 234 315 297 319 183 125 72 3b 18 8 2 1 1 2,397 Totals 3,289 1,095 1,200 1,359 1,697 1,860 2,075 2,119 2,216 2,091 2,018 1,472 1,192 645 373 195 92 45 19 4 3 6 25,065 •65-69 0-15 456 _^ 456 15-20 213 1 214 •20-25 340 30 4 1 1 376 25-30 328 115 73 50 19 4 4 1 594 30-35 234 139 180 157 139 102 55 39 26 13 6 2 3 1,095 35-40 204 154 161 233 278 339 335 241 178 103 56 15 16 3 2 4 2,322 40-45 245 117 159 214 278 272 485 554 591 509 399 246 146 69 43 22 11 3 4,363 45- 231 134 116 191 253 308 433 557 749 803 1,009 813 694 447 230 131 63 33 13 8 4 7,220 Total8 2,251 690 693 846 967 1,026 1,312 1,392 1,544 1,428 1,470 1,076 859 519 275 157 74 36 13 8 4 16,640 •70-74 O-20 351 351 20-25 13£ ] 140 25-3C 205 18 1 1 225 30-35 145 75 74 24 19 8 3 2 1 1 347 35-40 119 84 84 116 105 72 29 32 14 3 1 1 660 40-45 110 71 9S 126 138 191 187 141 100 73 31 IS 6 6 1 1 1,293 45- 240 139 133 196 263 367 444 633 693 788 784 579 435 298 150 72 39 8 7 5 8 8,281 TotalB 1,309 388 390 463 520 633 663 808 808 865 816 598 441 304 150 73 40 8 7 5 8 9,297 •75-79 0-SC 27S , 278 30-35 7£ i C C 1 84 35-40 70 32 26 5 ( 2 ( ] C 2 144 40-45 50 38 68 62 SS 35 37 2C i 6 C 1 2 C 1 361 45- 129 103 134 140 181 266 314 315 363 367 380 235 224 120 67 31 9 3 4 2 3,387 Totals 606 177 223 207 226 303 351 336 371 375 380 236 224 122 67 31 10 3 4 2 4,254 80-105 0-3S 117 117 35-4( 3( ] 39 40-4G 27 17 11 1] 6 2 ] ] 1 76 45- 77 45 69 90 98 1 146 152 161 157 iss 120 91 65 27 16 6 2 2 1,459 Totals 25S 62 80 101 103 145 153 162 15E 13E 120 91 65 27 16 6 2 2 1,691 = 1 104,76] L 109,72( ) 106,19E 9C,21S i 73,962 58,48! ! 47,04E > 37,54( ) 30,535 24,399 19,31' 12,80. ) 8,84] 5,02C 2,57! 1,28C ) 62. ) 24£ 10 1 i. : i( 5 3 3 733,773 ^^1 • Ages at date ot Census, 340 APPENDIX A. From the data furnished, distributions (vii.) to (x.) can readily be computed. 32. The duration and age-fecundity distributions. — ^For a series of duration-of-marriage-groups these distributions are obtained by com- puting, for successive age-groups and for each number of children borne, the relative frequency of the mothers within the indicated age-limits who bore a given number of children tathe total mothers of all ages (which are included) bearing the same number of children. These results may be obtained by a re-arrangement of the data in Table CXXIII., pp. 338-9. The distribution is (yii.) of Table CXX., p. 333. The ignored element is only the age of husbands. 33. The duration and age-polyphorous distributions. — ^These, for a series of duration-of -marriage groups, are obtained by computing for a series of age-groups the relative frequency of the mothers within the age- group who bore a given number of children to the total of all mothers in the same age-group {i.e.-, who bore to « children). The results may be obtained by the same re-arrangement as is required for the distribution referred to in § 32, the present distribution being (ix.) in Table CXX., p. 333. The ignored element is, again, the age of the husbands. 34. The age and durational fecundity distributions. — By dividing in each age-group the number of mothers who bore any given number of children, and whose duration of marriage was between given Umits, by the total number of mothers who bore the same number of children (i.e., for all durations of marriage in the age-group in question), the ratios in Table CXXIV. hereinafter are obtained. Each series of ratios is the age and durational fecundity distribution for the fundamental age-group. This case is (viii.) in Table CXX., p. 333. The only ignored element is the age of the husbands. 35. The age and durational polyphorous distributions.— As in the case of the distributions immediately preceding Table CXXIII., pp. 338-9 furnishes the required data. The series of divisors in each age-group are the totals for the indicated Umits of duration of marriage. Thus for married women of a given age and a given duration of marriage, the relative frequency of giving birth to 0, 1, 2 ... w children are obtained, and these are shewn in Table CXXV. below. This case is (x.) in Table CXX., p. 333, and the only ignored element is again the age of the husbands. COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY. 341 TABLE CXXIV. — Shewing, for a Series of Limits of Duration of Existing Marriage, and according to the Age groups given in the Table, the Ratios of Married Mothers who bore k (where /c = 0, 1, 2 .... 20, and " over 20") Cliildren, to the Total Number who, for all Durations of Marriage, Bore that Number. Census 3rd April, 1911. Australia. Duration and Age Fecundity Distribution. Dura- tion Proportion oJ the Number ot Women who, within the Indicated Limit o( Duration of Marriage, Bore k Children to the Total Number o£ Married Women who Bore le Children, where k = of Mar- riage. 1 2 3 4 6 6 7 8 9 10 11 12 13 14 16 16 17 18 19 20 over 20 0-5 6-10 1.000 1.000 9988' 9533 .. 1 0012 0467 5714! 1.000 4286| Nos. •13-19 3,363 1.00 4,162 1.00 820] 107 1.00 1.00 7i 1 1.00, 1.00 0-5 6-10 10-15 9699 0301 9308 0692 7147 2853 2774 7225 0001 0678 9291 0031 0125 9844 0031 9480 0520 9091 0909 1.00 N03. *ao-24 16,632 1.00 25,329 1.00 15,232 1.00 5,909 1.00 1,607 1.00 321 1.00 77 1.00 11 1.00 2 1.00 0-5 5-10 10-15 15-20 S153 1656 0191 7166 2530 0304 3863 5578 0559 0901 7947 1149 0003 0141 7651 2308 0000 0011 5153 4834 0002 2985 6991 0024 1463 8500 0037 1104 8637 0259 0303 9394 0303 .9000 .1000 1.00 1.00 Nob. •25-29 18,880 1.00 28,482 1.00 26,493 1.00 18,119 1.00 10,183 1.00 4,576 1.00 1,675 1.00 533 1.00 154 1.00 33 1.00 5 1.00 4 1.00 1 1.00 0-5 5-10 10-15 15-20 20-25 6404 2903 1503 0189 0001 4330 3538 1833 0299 0000 1862 6006 2733 0409 0000 0341 6193 3943 0522 0001 0036 3802 5342 0820 0000 0001 1938 6846 1213 0002 0753 7064 2178 0005 0293 6313 3394 0000 0117 4983 4866 0034 0034 3488 6444 0034 2379 7524 0097 0194 0776 0030 2500 7500 0000 5000 6000 1.00 1.00 1.00 Nos. •30-34 14,414 1.00 18,876 i.oo 22,140 1.00 19,849 1.00 15,149 1.00 10,257 1.00 6,189 1.00 3,306 1.00 1,459 1.00 585 1.00 206 1.00 67 1.00 20 1.00 4 1.00 2 1.00 1 1.00 1 1.00 0-5 6-10 10-15 15-20 20-25 25-30 3662 2550 2360 1185 0243 2405 2659 2790 1768 0378 0845 2913 3528 2236 0478 0127 2554 3948 2778 0593 0016 1568 4283 3365 0768 0004 0686 4314 4012 0984 0231 3682 4881 1206 0098 2462 5777 1660 0003 0057 1396 6196 2351 0000 0004 0808 5572 3592 0004 0007 0423 4933 4637 0000 0381 4460 5143 0016 0111 3926 5963 0000 0182 3546 6182 0090 1777 8223 0834 9166 1.00 1.00 s Noa. •35-39 12,002 1.00 11,506 1.00 15,462 1.00 15,927 1.00 14,213 1.00 11,458 1.00 8803, 1.00 6,138 1.00 4,190 1.00 2,500 1.00 1,350 1.00 630 1.00 270 1.00 110 1.00 45 1.00 12 1.00 2 1.00 1 1.00 0-5 5-10 10-15 15-20 20-25 25-30 30-35 2617 2142 2082 1710 • 1232 0217 0966 1871 2425 2456 1929 0353 0234 1468 2502 2945 2482 0369 0020 0921 2266 3163 3135 0506 0003 0484 2062 3268 3519 0662 0002 0222 1744 3217 3989 0828 0000 0060 1174 3291 4472 0999 0004 0010 0686 3143 4982 1178 0001 0005 0323 2633 5541 1496 0002 0002 0169 1947 6040 1842 0000 0076 1313 6152 2446 0014 0030 1048 5666 3256 0000 0658 5658 3674 0010 0515 5038 4428 0019 0435 4783 4739 0043 0333 4667 5000 • 3400 6600 3000 7000 3333 6667 0000 1.00 1.00 Nos. •40-44 10,739 1.00 8,140 1.00 10,422 1.00 11,590 1.00 11,809 1.00 10,441 1.00 8,908 1.00 7,087 1.00 5,476 1.00 4,207 1.00 2,948 1.00 1,689 1.00 988 1.00 524 1.00 230 1.00 120 1.00 50 1.00 10 1.00 9 1.00 1 1.00 2 1.00 0-5 5-10 10-15 15-20 20-25 25-30 30-35 35-40 2025 2004 1753 1364 1538 1130 0186 0200 1115 1974 1886 2640 1881 0304 0001 0016 0442 1434 2117 3310 2335 0346 0000 0001 0167 1027 1885 3576 2936 0408 0000 0001 0052 0617 1690 3751 3401 0487 0001 0025 0360 1410 3695 3867 0433 0000 0006 0210 1034 3594 4398 0758 0000 0001 0092 0684 3370 4767 1083 0003 0003 0050 0482 2890 5236 1339 0000 0019 0283 2494 5688 1514 0002 0008 0164 2080 5832 1916 0000 0008 0067 1595 6074 2256 0000 0078 1264 5917 3735 0006 0034 1014 5853 3088 0011 0022 0674 5717 3587 OOCO 0749 5374 3877 0000 0654 6355 2991 0652 4733 4565 0714 4286 5000 0000 4000 6000 2000 6000 2000 6667 3333 N03. •45-49 9,390 ■ 1.00 5,598 1.00 6,985 1.00 8,314 1.00 8,968 1.00 8,472 1.00 7,985 1.00 6,817 1.00 5,764 1.00 4,663 1.00 3,601 1.00 2,402 1.00 1,543 1.00 868 1.00 460 1.00 227 1.00 107 1.00 46 1.00 14 1.00 5 1.00 6 1.00 3 1.00 0-5 5-10 10-15 15-20 20-25 25-30 30-35 35-40 40-45 1383 1743 1747 1252 1231 1455 1005 0184 0000 0003 0204 1351 1797 2241 2304 1768 0332 0000 0032 0519 1544 2632 3086 1868 0319 0000 0008 0254 1140 2601 3516 2172 0305 0004 0004 0079 0713 2266 3763 2773 0402 0000 0043 0373 1871 3969 3226 0518 0000 0019 0211 1386 3977 3820 0587 0000 0007 0093 1003 3896 4133 0866 0002 0008 0059 0695 3475 4697 1066 0000 0002 0044 0370 2810 5378 1390 0000 0019 0339 2340 5660 1752 0000 0008 0176 1817 5758 2241 0000 0038 0129 1503 5476 3853 0011 0110 1092 5433 3366 0010 0079 0964 5217 3720 0030 0083 0917 4875 4125 0000 0078 1016 5000 3828 0078 1276 5320 3404i 0000 0455 6363 3183 0000 1667 8333 0000 6667 3333 0000 4286 4286 143U Nos. •50-54 7,005 1.00 3,338 1.00 4,CC9 1.00 4,955 1.00 6,697 1.00 5,868 1.00 5,643 1.00 5,505 l.OU 5,067 1.00 4,48Q i.m 3,620 1.00 3,499 l.OfI 1,784 1.00 1.007 1.00 508 1.00 240 1.00 128 l.OC 47 1.00 22 6 1.00 1.00 3 1.00 7 l.OU 342 Al-PENDIX A. TABLE CXXI7. — Shewing, for a Series of Limits of Duration of Existing Marriage, and according to the Age groups given in the Table, the Ratios ot Married Mothers who bore fc (where i = 0, 1, 2 20, and " over 20") Children to the Total Number who, for all Durations of Marriage, bore that Number. Census, 3rd April, 1911. Australia. Duration and Age Fecundity Distribution — contimied. Dura- tion Proportion ot the Number of Women who, within tlie Indicated Limit of Duration ot Marriage, Bore ft Children to the Total Number ot Married Women who Bore ft Children, where ft = o£ Mar- riage. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 16 17 18 19 20 over 20 0-5 1006 1- 5-10 1248 0005 0000 10-15 1662 0320 0103 0012 0004 0000 0003 15-20 1360 1519 0850 0303 0131 0026 00] 2 0003 0000 0003 20-25 1241 2068 1891 1577 1037 0590 0349 0219 0096 0086 0018 0024 0014 0011 25-3C 1064 2122 2384 2527 2560 2444 2059 ].sn2 0793 0472 0334 0235 0110 0123 0200 0046 0084 30-35 1174 1967 2491 3037 3157 3650 3662 3867 3528 3062 2540 19711 1479 1322 1181 105t 0841 0426 35-4C lOOC 1695 1944 2256 2784 2280 S329 3942 4757 5284 5444 5688 6(104 5700 5813 5092 5378 5532 4V3V 0833 1.00 6667 40-45 0244 03C4 0337 0288 0327 0410 0586 0667 0826 10,13 1660 2082 2393 2844 2806 3807 3697 4042 5263 0167 0000 3333 t45 ■■ 0004 OCOO 0000 0000 Nos. 4,632 1,876 2,047 2,473 2,816 3,U8 3,211 3,326 3,328 3,028 2,783 2,041 1,454 893 449 218 119 47 19 12 4 6 *55-59 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 l.UO 1.00 l.CO 1.00 0-15 2752 15-1! 145( 0200 20-25 139( 1735 1083 0544 01 3C 0027 001 5 00fi6 25-3( 1156 21 IC 2208 1987 1461 0903 0467 0212 0131 0043 0(125 0027 30-33 094! 1854 2225 2355 2522 23J2 20] f 1.5.S7 09C2 0679 0426 0238 0092 0078 0080 0051 35-4C 128S 2146 2517 2921 3247 371C 3996 3785 4007 SWfi 2597 2065 1846 0868 1019 0872 1304 1333 2105 2500 3333 3333 40-46 082( lEsa 172C 1862 2245 2516 29R8 3656 4003 4873 5391 5652 ,f,386 6217 5550 .5.S85 4892 4667 3684 2500 3333 5000 t45 C195 0402 0242 0331 0395 0452 0520 0684 0907 1119 1561 2018 2676 2837 3351 3692 3804 4000 421 J 500C 8334 1667 Nos. 3,28f 1,095 1,200 1,359 1,697 1,860 2,C75 2,119 2,216 2,091 2,018 1,472 1,192 645 373 195 92 45 19 4 3 6 •50-64 l.OC 1.00 1.00 1.00 1.00 1.00 1.00 l.UO l.CO 1.00 1.00 1.00 1.00 l.CC 1.00 l.UO 1.00 1.00 1.00 1.00 1.0c 1.00 &-l£ 2026 15-2S G94I 0014 2C-2I 151( 0436 005f 0C12 OOOC OOIC 25-31 -145'- 1667 105; 0596 019( 003; 0036 0007 30-3E 104: 2014 259S 1856 143f 0994 0426 0286 0168 0091 0041 0019 0035 35-41 0901 2232 232i 2754 2875 3304 255E 1731 1163 0721 0381 0139 0186 0058 0073 0255 40 -4£ 108S 1696 2294 253C 2875 2651 3697 3986 3828 .'(565 2714 2286 170(1 1323 1564 1401 1486 0833 1.0*; l.CO 0000 1.00 1-45 1026 1942 1674 2258 2616 3002 3300 4002 4851 5623 6864 7556 8079 8613 8363 8344 8514 9167 Nos. 2,251 690 693 846 967 l,826l 1,312 1,392 1,544 1,428 1,470 1,C76 8.59 519 275 157 74 36 13 8 4 •6.5-6! l.OC 1.00 1.00 l.CC l.CO l.CO l.OC i.oo 1.00 1.00 l.OC l.OC 1.00 1.00 l.CO 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0-2C ) 268] 20-2E 1065 002f 25-31 1561 0464 0026 0022 30-3E 1101 193; 1897 051S 0365 0047 0045 0025 0012 0011 35 -4( 0911 2165 2154 2505 201! 1137 0437 0396 0173 0035 0C12 0017 40-4E 0S4( 183C 251J 2722 2558 3017 2821 1745 1238 0844 0386 0301 0136 0197 00061 137(1 0250 t45 183S 3582 3410 4233 5058 5799 6697 7834 8577 9110 9608 9682 9864 9803 1.00 8630 9750 1.00 1.00 1.00 0000 1.00 Nos. 1,30{ 388 390 463 520 633 663 808 808 865 816 598 441 304 1.50 7S 40 8 7 5 8 •70-74 1.0C 1.00 - 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 l.OC 1.00 l.CO l.CO 1.00 1.00 1.00 1.00 1.00 1.00 l.CO 1.00 C-3t 458? 30-3E 130' 022« OOOC OOOC 0044 35-4C 1156 180£ 1166 0241 0265 0066 noor 0030 0000 00.53 40-46 0826 2147 2825 2996 168] 1156 1054 0.595 0216 O160 0006 0043 000(1 0164 0000 flflon 1000 t45 2129 5819 6009 6763 8010 8779 8J46 9375 9784 9787 1.00 9957 1.00 9936 1.00 1.00 9000 1.00 1.00 1.00 Nos. 606 177 223 207 226 303 351 336 371 875 380 236 224 122 67 31 10 » 4 2 •75-79 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 l.OC 1.00 1.00 1.00 l.OC 1.00 1.00 0-36 4517 35-4( 1467 0016 40-45 104S 0270 1375 1089 0486 0132 0065 006? 0063 t45 2973 0714 8625 8911 9514 9968 9935 9338 9937 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 Nos. * 259 63 80 101 103 148 153 162 158 135 120 91 65 27 16 6 2 2 80-106 1.00 l.OC 1.00 l.CO 1.00 l.CO 1.00 1.00 l.OC l.OC 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 Grand Total 104761 109720 106195 90,218, 73,962 68,482 47045 37540 30535 24399 19317 12805 8,841 5,023 2,575 1,280 625 245 107 42I 16 36 Ages. t 45 and over. COMPLEX ELEMENTS OP FERTILITY AND FECUNDITY. 343 TABLE CXXV. — Shewing, for Various Durations of existing Marriage, the Proportion of Women of Various Groups of Ages, who bore 1, 2, 3 ... «. Children, the Total for each Age-group between the Limits of Duration of Marriage being Unity. Australia, Census of 3rd April, 1911. Duration and Age-polyphorous Distribution. Duia tion of Mar- Proportion of tlie Total ot Women within the Indicated Limit oJ Duration o£ existing Marriage who bore Children to the Number of *, in which i = Total No. for the riage and Age. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 " 19 20 OTer $ 20 H Dura- tion. 0-5 5-10 3975 .4931 L097( 111 3 012 1555( LOOO! i333; )000] i .. .. ■* ■•• •• 1.00 8,441 1.00 9 •13-lS 396J 492: )097( )012f iOOOS iOOQ] .. ITOO 8,450 0-5 5-10 10-15 308C 039) 450S 1384 ,208( 1340] )03i; 334] 083C 0025 116£ 4168 0247 0833 065- 3333 0006 0833 066s ■■ •• •• •• •• 1.00 52,331 1.00 12.777 1.00 12 ♦20-24 2554 389C 233£ 0906 0247 0049 0012 0001 oooc .. l.OC 65,120 0-5 6-10 10-15 15-20 3218 0624 0323 0476 426S 1437 0776 OOOfl 214C 2947 1328 0476 0341 2871 1868 2380 0030 1533 2109 0000 0001 0470 1985 0476 oioo 1051 1906 o6i5 0406 0952 0663 0119 1906 0660 0027 0476 0004 0476 0004 OOOO 0476 1.00 1.00 1.00 1.00 47,819 50,153 11,145 21 * 25-29 1730 2610 2428 1660 0933 0419 0153 0049 0014 0003 00005 00004 00001 1.00 109,138 0-5 5-10 10-15 16-20 20-25 3745 1031 0515 0302 0909 3931 1645 0822 0620 0000 1972 2732 1438 1000 0454 0325 2540 1860 1146 0909 0025 1419 1924 1370 0454 0001 0490 1670 1374 0909 oii5 1039 1489 1365 0024 0496 1239 OOOO 0004 0172 0784 2273 0660 0049 0416 0909 ooio 0171 0909 0663 0057 0909 0661 0016 0661 0002 0662 0661 0661 •• 1.00 1.00 1.00 1.00 1.00 20,792 40,587 42,077 9,051 22 •30-35 1281 1678 1967 1764 1346 0912 0650 0294 0130 0052 0018 00055 00018 00003 00002 00001 00001 1.00 112,526 0-5 5-10 10-15 15-20 20-25 25-30 5051 1701 0822 0423 0297 1250 3180 1700 0932 0605 0443 0000 1502 2502 1583 1027 0751 1260 0233 2262 1826 1315 0961 0000 0028 1238 1767 1420 1112 0000 0006 0437 1435 1366 1148 0000 oii3 094] 1307 1081 1250 0033 0349 1054 1037 2500 0013 0170 0772 1003 OOOO 0661 0060 0414 0914 1250 0660 0016 0198 0636 OOOO 0007 0084 0329 1250 0662 0032 0164 OOOO 0660 0010 0069 1250 0662 0038 0661 0011 0004 0062 1.00 1,00 1.00 1.00 1.00 1.00 8,701 17,994 34,443 33,649 9,824 8 *35-40 1147 1100 1478 1522 1359 1095 0842 0587 0400 0240 0129 0060 0026 0010 0004 0001 OOOO OOOO 1.00 104,619 0-5 5-10 10-15 15-20 20-25 25-30 30-35 7265 3156 1443 0710 0386 0273 0000 2034 2090 1274 0773 0458 0336 0000 0632 2099 1682 1187 0754 0451 0000 0059 1466 1692 1413 1058 0685 0000 0010 0783 1570 1492 1212 0916 1334 03i8 1174 1299 1214 1012 0000 0073 0675 1133 1161 1042 3667 loio 0313 0861 1030 0978 0667 0004 0114 0557 0884 0959 0667 0661 0046 0317 0740 0908 OOOO o6i4 0150 0529 0844 2667 0003 0068 0279 0644 OOOO 0025 0163 0426 0667 ooio 0077 0271 0667 0004 0032 0128 0667 0661 0016 0070 0005 0039 0661 0009 0661 0007 0660 0661 0662 1.00 1.00 1.00 1.00 1.00 1.00 1.00 3,869 7,288 15,506 25,865 34,308 8,539 15 .40-44 1126 0853 1093 1215 1238 1095 0934 0743 0574 0440 0309 0177 0104 0055 0024 0013 0005 0001 0001 OOOO 3625 3556 1.00 95,390 0-5 6-10 10-15 15-20 20-25 25-30 30,-35 35-40 9383 6213 2869 1292 0593 0350 0257 0000 0563 2060 1926 1064 3606 0347 0249 1429 0054 1021 1745 1492 0943 0538 0355 OOOO 0005 0459 1487 1582 1220 08O5 0497 OOOO 0005 0152 0964 1529 1380 1005 0641 1428 0069 0632 1205 1285 1080 08OO OOOO 0016 3293 0833 1178 1157 0888 OOOO 0663 0110 0470 0943 1071 1083 2867 0667 0O5O 0279 0684 3996 1133 OOOO ooie 0133 0477 0874 1036 1429 0005 0060 0307 0692 1012 OOOO 0003 0016 0157 0482 0795 OOOO o6i2 0080 0301 0619 1429 0003 0036 0167 0393 1428 0661 0013 0087 0242 o6i7 0050 0029 0007 0028 0010 3003 3009 3007 0661 0002 0002 0660 0001 0001 3661 3001 ( 3000 ( 3661 3000 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2,026 3,029 5,738 9,914 24,368 30,389 6,816 7 *45-49 1142 068O 0849 1010 1091 1030 0971 0829 0700 3567 0438 0292 0188 0106 0056 0028 0013 3006 0002 0001 3001 3000 1.00 82,237 0-5 5-10 10-15 15-20 20-25 25-30 30-35 35-40 40-45 9890 9335 5831 2491 1031 0553 0330 0237 OOIO 3520 2149 1705 0894 3417 3277 3204 0099 0991 1759 1261 0672 0351 3236 0031 0600 1605 1452 0946 0505 0278 2222 o6i5 0214 1153 1544 1164 0741 0421 oiig 0622 1313 1264 3888 0559 3052 3338 3935 1218 1012 3610 o6i9 0145 066O 1164 1067 0878 nil 3020 3085 3421 3956 1117 0994 0665 0O57 0198 0686 1133 1148 0020 0099 0460 0960 1166 0006 0053 0246 0675 1030 06i4 0028 0145 0458 0937 2223 o6i3 0060 0256 0624 1111 0665 0027 0124 0348 1111 0662 0012 0056 0182 •• ■ 0661 0007 0030 0090 1111 0003 0012 0029 00005 0007 0013 00664 0009 0661 0002 0661 0005 1111 1.00 1.60 1.00 1.00 1.00 1.00 1.00 1.00 970 1,308 2,009 S,520 8,363 18,424 21,319 5,435 9 •60-59 1140 ( 3543 3652 08O6 0927 1 0956 3918 0896 0825 073O 0589 0407 0290 0164 0083 0039 0021 0008 0004 0001 OOOO 0001 1.00 61,447 344 APPENDIX A. TABLE CXXV. — Shewing for Various Durations of Existing Marriage the Proportion of Women of Various Groups of Ages, who Bore 1, 2, 3 ... n Children, the Total for each Age-group between the Limits of Duration of Marriage being Unity. Australia, Census of 3rd April, 1911. Duration and Age-polyphorous Distribution — continued. Duia- tionol Mar- Proportion of tlie Total of Women witliin tlie Indicated Limit of Duration of Existing Marriage who Bore Children to the Number of Ir, in which ]c = Total No. riage, and Age. 1 2 3 4 5 6 . 7 8 9 10 11 12 13 14 15 16 17 18 19 20 over 20 for the Dura- tion. 0-5 5-10 10-15 15-20 20-25 25-30 30-35 35-40 40-45 45 and over 1000 9983 8995 5185 2327 0954 0528 0342 0340 o6i7 0701 2346 1570 0770 0358 0236 0172 0245 1432 1565 0944 0495 0294 0208 0035 0617 1578 1210 0730 0412 0214 o6i2 0305 1182 1395 0864 0579 0277 0000 0066 0744 1475 1105 0664 0385 o6i2 0033 0453 1279 1143 0791 0566 0008 0295 0838 1250 0979 0668 0000 0129 0512 1140 1157 0828 0008 0105 0277 0900 1183 0997 0000 0020 0180 0687 1120 1391 1.000 0020 0093 0391 0858 1280 0008 0031 0209 0646 1048 0004 0021 0115 0376 0765 o6i7 0051 0193 0380 0002 0022 0082 0250 0002 0010 0048 0132 0002 0020 0057 0007 0030 0007 0006 0003 0000 0003 0006 1.00 1.00 1.00 1.00 1.00 1.00 1,00 1.00 1.00 1.00 466 579 856 1,215 2,472 5,167 10,294 13,529 3,321 1 •54^59 1222 0495 0540 0653 0743 0823 0847 0878 0878 0799 0734 0539 0384 0236 0118 0058 0031 0012 0005 0003 0001 0001 1.00 37,9b0 0-15 15-20 20-25 25-30 30-35 35-40 40-^5 45 and over 1.00 9559 5186 2170 0970 0604 0320 0267 0441 2147 1319 0631 0336 0202 0184 1469 1513 0830 0432 0246 0121 0836 1542 0995 0568 0300 0188 0249 1416 1331 0789 0452 0280 0057 0959 1384 0988 0556 0350 0045 0554 1297 1186 0736 0451 ooii 0257 1026 1148 0945 0605 0i66 0656 1271 1053 0839 0052 0442 0983 1209 0976 0029 0267 0750 1292 1314 0023 0109 0435 0988 1239 0034 0315 0762 1331 ooio 0080 0476 0764 0009 0054 0246 0521 0003 0024 0125 0300 o6i7 0053 0146 0009 0025 0075 0006 0008 0033 0001 0001 0008 0001 0001 0004 0003 0004 0004 1.00 1.00 1.00 1.00 1.00 1.00 1.00 100 905 499 885 1,751 3,216 6,898 8,424 2,397 •60-64 1312 0437 0479 0542 0677 0742 0828 0845 0884 0834 0805 0588 0475 0257 0149 0078 0037 0020 0007 0001 0001 0002 1.00 25,065 0-15 15-20 20-25 25-30 30-35 35-40 40-45 45 and over 1.00 9953 9043 5522 2137 0379 0562 0320 0047 0798 1936 1269 0663 0268 0185 oioe 1229 1644 0693 0364 0161 0027 0842 1434 1003 0490 0265 OOOG 0320 1269 1197 0637 0350 0026 0067 0932 1460 0623 0427 0067 0502 1443 1112 0800 o6i7 0356 1038 1270 0772 0238 0767 1354 1037 oiio 0443 1167 1112 0055 0241 0915 1397 o6i8 0065 0564 1126 0027 0069 0335 0961 o6i3 0152 0619 0009 0099 0319 o6i7 0050 0181 0025 0087 0007 0046 0018 0011 0000 0006 1;00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 456 214 376 594 1,095 2,322 4,363 7,220 •65-69 1353 0415 0417 0508 0581 0617 0788 0837 0928 0858 0883 0647 0516 0312 0165 0094 0044 0022 0008 0005 0000 0002 1.00 16,640 0-20 20-25 25-30 30-35 35-40 40-45 45 and over 1.00 9929 9112 4179 1803 0852 0382 0071 0800 2161 1273 0549 0221 0044 2133 1273 0758 0212 0044 0692 1758 0974 0312 0547 1591 0129 0419 0087 1091 1427 0587 0086 0433 1446 0707 0058 0485 1090 1008 0029 0212 0773 1103 0028 0045 0565 1255 0015 0240 1248 06i5 0139 0922 0046 0692 0047 0474 0000 0239 0008 0115 0007 0062 0013 0011 0008 0000 0013 1.00 1.00 1.00 1.00 1.00 1.00 1.00 351 140 225 347 660 1,293 6,281 •70-74 1408 0417 0419 0498 0559 0682 0713 0869 0869 0930 0878 0644 0474 0327 0161 0079 0043 0009 0008 0005 0000 0008 1.00 9,297 0-30 30 35 35-40 40^5 45 and over 1.00 9405 4861 1385 0381 0476 2222 1053 0304 0000 1806 1745 0396 0000 0347 1717 0413 oiig 0417 1052 0534 0i39 0975 0785 0000 1025 0927 0069 0554 0930 0000 0222 1072 0i39 0166 1084 0000 1122 0028 0694 0000 0661 0055 0354 0000 0198 0000 0092 0028 0027 0009 0012 0006 •• 1.00 1.00 1.00 1.00 1.00 278 84 144 361 3,387 •75-79 1425 0416 0524 0487 0531 0712 0825 0790 0872 0881 0893 0555 0527 0287 0157 0073 0024 0007 0009 0005 .. 1.00 4,254 0-35 35-40 40-45 45 and over 1.00 9744 3553 0528 0256 2237 0308 1447 0473 1447 0617 0658 0672 0263 1001 0i32 1042 0i32 1103 oisi 1076 0925 0822 0624 0446 0185 0110 0041 0014 0012 •• •■ 1.00 1.00 1.00 1.00 117 39 76 1,459 80-105 1532 D373l0473 0597 0609 0875 0905 0958 0934 0798 0710 0538 0384 0160 0095 0036 0012 0012 1.00 1,691 • Totals for ages indicated. Ages at the time of the Census. COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY. 345 36. Fecundity-distributions according to age at marriage. — By sub- division, according to duration of marriage, of the numbers in Table CXXIII., pp. 338-9, and subsequent rearrangement, tables can be prepared giving very approximately the distributions corresponding to the ages at marriage ^- As this involves the relative numbers marrying at successive ages, it is essential to know the frequency of marriage at given agesi This is furnished by Table LIV., p. 190 2. The results are as follow : — TABLE CXXVI. — Shewing the Relative Number of Marriages according to Ages of Brides. Australia, 1907-1914, ' and the Average Age for each Year Group. Alleged age (last birthday) 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Mean age Ko. of marriages per 1,000,000 12.66 3 13.66 13 14.67 242 15.67 1,620 16.61 7,992 15.57 22,885 18.54 43,889 19.52 64,027 20.52 81,033 21.49 90,337 22.49 92,609 23.49 87,491 24.49 79,199 25.49 68,610 Alleged age (last birthday. 26 27 28 29 30 31 32 33 34 35 36 37 38 39 Mean age No. of Marriages per 1,000,000 26.49 58,749 27.48 48,897 28.48 40,286 29.48 30.48 33,259 26,627 31.49 21,480 32.49 16,927 33.49 14,553 34.49 11,548 35.49 10,451 36.49 9,415 37.49 8,444 38.49 t 7,540 39.49 6,702 Alleged age (last (birthday) 40 41 42 43 44 45 46 47 48 49 50 51 52 53 to 95 Mean age No. of marriages per 1,000,000 - 40 49 5,931 41.49 5,225 42.49 4,584 43.49 4,003 44.49 t 3,481 45.49 3,014 46.49 2,598 47.49 2,230 48.49 1,906 49.49 1,623 50.49 1,375 51.49 1,160 52.49 t 975 7,064 ' * Smoothed for misstatement of age. reciprocals of 1.105, 1.110, 1.115, etc. t Smoothed to a curve by a multiplier changing regularly, viz., the The preceding table shews that, from the age 18 onwards, the average age is, sensibly, the age last birthday plus one half-year, and no serious error wiU result if it be so taken even for the ages earlier than 18. Hence a correction can be readily made for the effect of mortality, and asynthetic table prepared in the following way : — Let a, h, etc., denote the marriages at ages (last birthday) Xi, xz, etc., reduced for a half-years' mortality ; a', b', etc., these reduced for one and a half years' mortality ; a", b", etc., the same reduced for two and 1 Original compilation according to age at marriage is, of course, the best method of obtaining the proper numbers. ^ This gives 8 years' experience in Australia of the frequency of marriage at different ages, the total oases being 301,918. . » These numbers are deduced from those shewn on pp. 190-191 by distri- buting the 111 unspecified cases. 346 APPENDIX A. a half years' mortality, the mortahty being both of husbands and wives,* and so on. Then, ignoring migration, the numbers according to age, as, at a census, and for a given duration of marriage, will be as shewn in the following table, viz. : — TABLE CXXVn. — Scheme of Compilation of Numbers according to Duration of Marriage. Duiations Aqe at Census. of Marriage. »! «2 ■■is «* Xn Xe X-, etc. 0-1 .. 1-2 .. 2-3 .. 3-4 .. 4-5 .. a b a' c b' a" d d b" a"' e d' c" 6"' a" f e' d" c'" 6'v 9 r e" d'" etc. etc. etc. etc. etc. The total numbers of married women for durations of to 5 years, 5 to^lO years, etc., are consequently : — (564). .0^6 = (a) + (a' + b) + (a" + b' + c) + {a"' + b" + c' + d) + (aiy + 6" + c" + rf' + e) + (6'' +c"' + ...+/) + etc. (565). .bMio = (a^) + (a'' + 6') + (a™ + 6" + C) + etc. (566). .10-^15 = {a-^) + (a"' + 6'^) + (a*" + 6^' + c^) + etc.; etc.; etc. It is obvious that a synthetic table can be prepared by means of which the partition can be effected of a group of married women between given limits of age and duration of marriage : in this way the mean age of any element may also be readily ascertained. Obviously the successive quantities vertically are, with sufficient precision for the purpose in view, respectively — in actuarial notation : — m^i^—^l'x) ; ■rr^x-x (1— k'«-i)-P'*-i •P'x;m^-^ (^-yx-^)-P'x-,'P'x-i'P'x m denoting the number of marriages, according to the age of the woman, ^ For rigorous results the fact must be taken into account that the death of hiisbands also removes the women from the category " married." Hence the correction for mortality includes the probable number of deaths of wives, and of husbands, diminished, however, by the joint deaths, which are counted, of course, once only. COMPLEX ELEMENTS OF FERTILITY AliTD FECUNDITY. 347 +Zu)} If the external factors, therefore, are made unity in (574) and (575) the internal will be, respectively, ^, -g-, -^, and -|-, -§-, and ^. It is evident from these results that the elimination of systematic error involves in aU cases the assignment of a high " weight" to the central value. But it is equally certain that if the central values be considered liable to deviations from the general trend of the surface, which, compared with the systematic errors introduced are small, we may practically reach a better result by emplojdng (571) or even (573).^ Another and more satisfactory method of obtaining values of ^^v is to smooth the series of the values of the type K, P, U, etc. ; i.e., with y constant ; and independently those at right angles thereto, viz., K, L, M, etc., i.e., with x constant. The means of the two results for each point are then adopted as a jQrst smoothing, and the process repeated as often as is found necessary. This leads to more rigorous results, but can be readily employed only when the original results do not deviate largely from the general trend of the surface. 40. Diisogens, their trajectories and tangents. — ^The general nature of surfaces such as are here under consideration has been indicated in Pt. XII., §§ 21 and 22, pp. 201-203, and the fundamental formulsB of orthogonal trajectories have been given. The system of contours upon such surfaces (diisogens) probably do not conform to any simple geo- metrical specification ; the present imperfect data certainly do not point to their representation by any system of curves of a simple character, though the settlement of this question must remain for more extended investigation and more accurate data. At any point {x, y) whatsoever, dy I dx furnishes the relation by means of which the birth-rate equivalence 1 The question of the adjijstment of such values, has been systematically treated by E. Blaechke, Ph. D., see his " Methoden der Ausgleiohung von Wahrsoheinlichkeiten," Wien, 1893. See also Phil. Trans., Vol. 186, 11., pp. 870-5, 1895. See also Part XII. herein, § 39, pp. 230-2. COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY. 353 of pairs of ages may be expressed in the form K^ = a; + OA y = a constant. For we shall have, for the direction of the tangent to a diisogen, dy/dx = tan 6 — 1/6? say. Hence it follows that (576). .X — Ay cot Q = X ^ Ay -5— = G ; that is x-\-OAy = G If k be the recripocal of K then kG will be the constant value of the birth-rate for the diisogen in question. Ordinarily dx/dy is negative. Parallelism of the tangents of diisogens to the a;-axis would imply that the increase of the age of the husband had no influence whatever on the birth-rate, while the parallelism of the tangents to the y-axis would denote that the age of the wife had no influence. If, therefore, the age of the wife has, in general, the preponderating influence, the diisogens must make a smaller angle with the a;-axis than with the y-axis. If the diisogens are inclined 46° to each axis, then the birth-rate is constant when x -\- yis constant.^ 41. Digenesic age-eauivalence in two populations. — As already shewn, the diisogens or their orthogonal trajectories determine the cor- relative changes in the ages of husbands and wives which give equivalence of birth-rate, i.e., diisogeny. The diisogenic factor G in formula (576) for any pair of ages {i.e., of husband and wife) is the coefficient which must be multiplied into the age of the wife so that the product, plus the age of the husband, will be continually proportional to the birth-rate. It holds, of course, only for a moderate range of age-differences about the point for which it is ascertained. Thus the expressions : — (577) j; — • «/ -5- = constant ; x ~ — y = constant, apply only to a limited region. For two populations the differential coefficients are not identical. Hence, for a given difference of age in the wife, the equivalent difference of age in the husband is not the same. The factor to make one equal the other may be called the masculine factor of age-equivalence, E. Similarly the factor to make the difference in the wives' age equal, for a given difference in the age of husband, may be called the feminine factor of equivalence, E'. Suffixes can be used to denote the ages (of. husband and wife) to which these factors exactly apply. 1 Roughly speaking this representa the general character of the relation indicated (on Table 3, facing p. 852, Phil. Trans., Vol. 186, Pt. II.), by KorOsi. Thus, for quite a large range of ages, the birth-rate would appear, according to that authority, to depend merely upon the sum of the ages of husband and wife, and not upon their individual ages. This condition may be called equilateral diisogeny, and is probably not a general condition. 354 APPENDIX A. Let 8 y denote any small difference in the age of wives at the point x, y, common to the populations A and B, the tangents to the diisogens making the angles 6^ and ^j, respectively, with the 9;-axis. Then since 8a;, = hy cot da and Sa;;, = 8y cot 0j, we have (578). . ..E 8a-6 Sycotdi Gft tan 9„ dFa(x)/dx Sxa Syootda Ga tan ^j dFt(x)/dx Similarly — (579).. hy„ hx tan 6^ (?„ 1 hya Sx tan d^ G^ E that is, the masculine and feminine factors of age-equivalence are recip- rocals. 42. Birthrate-sauivalences for given age-differences. — The factors of age-equivalence merely disclose the equivalent differences of age for two populations for a given age-difference in either sex, but not the birth rate equivalence. This latter depends not only upon the direction of the tangents to the graphs of the diisogens in plan {i.e., upon the tan gents to their horizontal projections), but also to the angle of slope i/r of the orthogonal trajectories. The tangent to any point Q, on a trajectory will be required. The angle it makes with the z-axis will be ^ so that 1 + ^ = 90°. The follow- ing procedure will always be abund- antly accurate for determining the age-equivalence and digenesic effi- ciency for any point Q the co-ordin- ates of which in plan are x,, y^. Let P' P P", Q' Q Q", and R' E R" in Fig. 95 be three diisogens (the values of which are known), crossed by the orthogonal trajectory P, Q, R, which in general is, of course, a curve of double curvature (tortuous curve). Let this trajectory be projected orthogonally on to the horizontal plane X Y passing through P : this proj ection is the broken line P q r , the proj ections of short stretches of the diisogens being similarly the broken lines q' q q" and r'rr"; P'PP"is itself in the plane of projection. Let the curved hne Pq be denoted by Xi, and the curved line Pr, of which Pq forms part, by X^, measured along the curve ; and let also the difference of birth-rates for P and Q {i.e., Qq) be denoted by 8i, and the Fig. 95. COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY. 355 difference for P and R be denoted by 83 ; then we may assume that the curved triangle, P Q R r q P in relation to lengths along the curved axis P q r, is, with sufficient precision, given hy S = bx + cy^, and therefore that the tangent at the point Q is dh/dy = 6 + Sc^. Thus we shall have : — When 82 = 281, this becomes 81 y-l — 2X2^1 + 2X-^ and when, in addition, X2 = 2Xi, the expression becomes, of course, (582) tan i^ = Si /^x =82/^2 The direction of this line of slope tangentially passing through Q, and making the angle ^ with the horizontal plane, is shewn by the pro- jection S q T, which is tangential to p q r, passing tangentially through the point q. It, of course, makes the angle d' with the X axis. Con- sequently the angles d' and (/«, or their complements Q and ^, give all the necessary relations required. Since the line, Qq, in the figure =81 = /J, — jSj,, viz., the difference of birth-rates indicated by the diisogens at P and Q, the horizontal equivalent thereof, Sq = s, say, measured in the direction of the tangent to the orthogonal trajectory at Q is : — • (583) s = 81 cot i/* = (i3j — j8p) tan I, since ^ + 1^ = 90°. Thus, in plan, the rate of change of the birth-rate at any point a;, 2/ on a diisogen can be ascertained from the position of the diisogens on either side, and the position of the orthogonal trajectory through the point. Thus the age-equivalence of this difference of birth- rate is to be found by dividing by the sine and cosine of the angle which the orthogonal trajectory makes with the co-ordinate axes, 9 and 9' ; their sum, 9 -{- 9' = 90°. Consequently the masculine birth-rate- equivalence, H say, for wives of the one age, is : — (584) H = (jS, - jSp) tan ^ oosec 9 since 1 /sin 9 = cosec 9, and the feminine birthrate-equivalence H', for husbands of the one age, is (585). . . ■ . , fl' = (;8, - jSp) tan ? sec 6 356 APPENDIX A. We thus have, from these two equations, for two populations, A and B for any common small difference of birth-rate, the ratio : — (586) Hb^taii^t, cosec 6^ _ __, i^ b H„ tan ^„ cosec 6^ ' H\ H'j, tan ^6 sec Bj, tan ^a sec $„ These relations, however, can be determined very readily from appropriate graphs of the populations. 43. Diisogeny in Australia. — Diisogeny is doubtless best exhibited by the maternity rates, not the birth-rates, the ratios to be ascertained being the proportions which the number of cases of maternity bear to the number of women at risk in any age-group with husbands of any age- group. In order to ascertain the nuptial maternity rates of Australia accord- ing to pairs of ages, the nuptial cases of maternity have been taken out for the seven years 1908 to 1914 inclusive, that is, for the Census year 1911, and for the three years before and after that year. In order to relate these cases of maternity in age-groups to the numbers of married couples in the same age-groups at the Census, they have been divided, not by 7, but by a number which gave the true average, viz., 7.13143.^ The results thus obtained are shewn by the uppermost of the figures in Table CXXIX. hereunder. Thus the results used are equivalent to a total 5,232,988 married women, among whom maternity was experienced 814,617 times. This gives an annual maternity rate of 0.15567. But of this number of married women, 7.6368 per cent, were 60 years of age and over, and 12.7667 per cent, were 55 years of age and over, so that about 87 per cent, were of child-bearing age. Hence the birth-rate for married 1 This figure was ascertained in the following way ; — ^The number of females in the years 1908 to 1914 inclusive were multiplied by a linear changing ratio (deter- mined from the intercensal period 1901-1911) in order to obtain the numbers of married women during the years in question, the results being as hereunder : — ■ 1908 2.0187C6 X .33355 = 677,377 1309 2.038512 X .33818 = 696,148 1910 2.103318 X .34081 = 716.832 1911 . 1 2.156781 X .34344 = 740,725 1912 . 1 2.224484 X .34607 = 769,827 1913 2.301011 X .34870 = 802,363 1914 . 1 2.361643 X .35133 = 829,716 Tot.ll N 0. of married women in 7 years 5,232,988 No. at Census date Total birthg in 7 years Births in Census year 738,773 Total population Census population All females, 7 years Census females Total married females, 7 years Census year (wliole) _ 31,697,28.= ~ 4,455,005 _ 15.224,455 2,141,970 = 7.11498 = 7.10769 _ 5.232,988 734,226 = 7.12722 = 7.13162 = 7.12758 This was found to agree with other deductions as to the number of years, viz., 7-|- e where e was a small fraction (as shewn above) varying between 0.10769 to 0.13162. The actual divisibn used was 7.13143, the reciprocal of which is 0.140224. This, multiplied into the births during the 7 years, gave the uppermost figure shewn in the table, Complex elements of pertiliTy and fecundity. 35? women of 13 to 54 years of age inclusive was 0.17845, or for women of 13 to 59 years of age inclusive, 0.16854. Korosi's results were 46,926 children from 71,800 married couples, in 4 years, that is 0.16339 per annum. 1 The numbers of husbands and of wives recorded in the Australian Census of 3rd April, 1911, were not equal. It was deemed probable that the number of wives recorded would be the best basis for determining the distribution according to the age of the married women at the Census : in this way the numbers exposed to risk are ascertained in each age- group. The adjusted distribution^ gives the numbers which constitute the denominators of the ratios. In general there is a considerable number of cases for each pair of age-groups adopted ; the table discloses the number. It is evident, however, that in extreme instances the numbers are small, and the maternity rates consequently ill-determined.* They may be regarded, however, as well ascertained where the number of mothers has been shewn in heavy figures. The age-distribution as at the Census probably differs but little from the average distribution over the 7 years, which yielded the births : hence the ratios ascertained may be accepted as very closely representing the true amounts. The results are shewn in Table CXXIX. hereunder. f^ 1 The average crude birth-rate for Australia for 1908-14 was .02745, and for Hungary for 1908-12, 0.3632. Apparently the Budapest matemity-rate is not larger than that of Australia. 2 The following is a conspectus of the data; — ^ Unspecified as Husbands whose Husbands whose Total regards Wives were Wives were Wife's Age. with them. Absent. Unspecifledas regards 4,108 620,846 11,084 husband's age . . 2,368 506 2,874 1,045 3,919 Wives whose husbands were with them . . 619,106 4,614 623,720 112,129 735,849 Wives whose husbands were absent 108,892 1,161 110,053 Total Wives . . 5,775 733,773 , , The adjustment was effected as follows : — ^The 506 doubly unspecified cases were divided into 185 and 321, that is in the proportion of each to their sum, and those were distributed proportionally among the wives and husbands unspecified. Next the 1161 wives, unspecified as regards age, were distributed proportionally among the 108,892 whose ages were given, thus making up the total 110,053. A like proceeding was followed in the case of the 1045 husbands, unspecified as regards age, so as to make the total 112,129. The individual totals were then reduced by multiplying throughout by 0.981485, so as to form the same aggregate 110,063 as in the case of the wives. One half of each was then distributed proportionally to the individual original numbers, thus making the grand total 733,773. See Table I., pp. 1106-7, Vol. III., Census Report. ' In general, tables prepared in this manner have the advantage that it may at once be seen whether any change of a ratio, necessary to make it conform to a general law, is probable or otherwise. A result like that shewn for ages of husbands 17, and of wives 16, viz., 1.55, is of course not impossible, but it would not be true for a very large number of cases. 358 APPENDIX A i> 2 "J, _ ™ 3« ^m 01.3 ««! a o O M at! ■3 o OSes Pn a> SS ■§■0 ^! o o "SP >> »^ o m g u^ S ■» 53 a " n s ». o o Q >> 3 £ S o gi ffieiH.S S I ■9 ■sa . oa _Oi "-'53 00 S a-a ■sS "» 2 S w .a ® 3 til 9s za o Hi &4 o S 0-- ^1 6= ■3SS . ■gS56 I 'spnvqsnH jo B83v 0000 0020 or^o 0100 0030 eoota aof-ic4 o-^.N S®.*^. S^.^. eCiHO lOOOO a«0 COODO 00000 OCDO O O N . oooo "*00O Or-ltf> d d ooi-ir- oeoo CO'O'i-i OQiHO 00 r*-^ ?7 0Den s . . . d ^8 d iHr-trH d d oqoo "1 ill rH «Sd rH lis rH CO (OrH-« rH d OS d d oi-g Id"-. r-1 «-"g f-i d O.rHO t-ilO §55 rt 5s§ „^2 OD OrHO . . . d d r-tCOCC ar-oo i-i**" (M d "as §a| 3SS m lOiHO m eo §S| d T-1 : : : : 0(M»0 rH d OCOO) OOr-lrH r-t si 3SS - 2 04 d « 1 d CO-*CT d d 000 " 1 5^1 pi 00t>CD d r| 10 1-1 OCQO d : ocoo d d COr-ICO d d OtMO l-J ^ d C^e- I ■^"§ coo t^ d 3SS d rH : : : : d :e- iH d : : d^^- : d d "*CO00 ci d d d 1-t d ;o- CO • : : : dff^ : d d rH d*- : d :*- : : : i-ISw NiSc-i eo^eo mUn COMPLEX ELEMENTS OF FERTILITY AND FECUNDITY. 359 3. «S M I> 00 t*- t> ■^ OS -tt* CO u:i xn t> X ■S :« It-" '. '.00 I ii-T I"^ : :°°" ! OS r- vn "•* J^^ f_l m « "^ CO OS in 10 .s »H ■* TV t* <74 ,—1 pH CO 00 •di in CO • ■ ■ ... 00 i •tH ■CO • cq • in • in ■ •OS ■iH 1-t CO rH tH Oi (M t- t- *co" i-T CO 00c 00c ococ OU3C OOC oeic o-^o o«o OCOO O05C OCOO OfHO 0-n OMC ? »'l si«i «•! m ^09 = d d d'-^d 00 COCl no(^^ t^oom t.ia 1 llg iii S"i d d c d d 00 §S „-«o «"=> ' ^OJO eOE^t- OS coo oocqco v~aaa iHinoa 000 „-fHCO g>oin N^-s gp. S-.S vn"*Q S"° co-g J^S °"S ° S 00 in CO in"° M'^=> 00 rHiH n«(Nl r-iioeo XU9 0D CO CO CO t>l-HC^ Or-lO CO s-.i ^§11 |SS "3 -•^3 d d °"o 00 ^^ .h" inrH Aoocn rtoqw rtO eo«> ! sss s?§s SI sss SSS ^'y r-l d-^g » s c. 00 R^ Stoo Sofd §§S ooo»t~ (Mt^OS CO CO in CO (Mm owe n(M S'Sm S;iHffi S"S xi'^S C^»H« ° S =• 3 ■SiHO S |S OOD^ miog r-liHiH 1> OOC o>3; S ^"^ o'n ^-- - s ,^. gS d S'" •*rteq 00 in to CO cam 000 OOC =!£ 0: R*S t^'-s ^-"3 = 3 ' 3 S^ ^^ (N aeus Oi^t- o-*o iHiHiH '^.2 M ««S ""™ - ^- s^: rH rtlOOO lOiACC GOrHQO T-* «tg S - s H 6 :°- ■*T-I nuato t- tHCOCO "^Sg ° 6 6 :^. ■= 3 rH -*eoeo iHr-lf-* a" in ° 3 i-t in 00 3 ; I ; : : eo . . . ; ; ; rH •saSv 3 s « -f S SS in Svn vn ss i ss sss O"* S5S s 2S 2St S5§ §"§1 1-1 ea 5 ' .2 K 1i sa ■"5 39 53 a* •gB °!1 >> aS 0.3 SS 11 §5 "i P 360 APPENDIX A. Diisogenic Surface. Australia. Ages of Wives = y. S 30 EL9B CSOet'lllSIUIIlUqEO ^^ .§ ^' % J' £^ M- t I ^ « t s'gr^t^'?^!!^!! i 'S^^N^^^-^^^ ^^^ ^S ■"■ ^ ^ ""■ -- ' "" H ^ ^ 'Vsn""^ ^S^^^ J^'^^SS 5 ^^S^ i;^^ S^ ,' ~? [ ^^ \\ V V \ ^ ^^ o \ ^ ^^^ ^^^ ^ ^ ^ t* '^"^ "^ "^ ' " t -RiJEJ \rfcS3S5VvSsl|||||l§vsNv^4 ^t ?^z:S7■-■4Iat^^S^^'SSSs5|5S5 5^'^^X Xti ■jzm^iii'i - \2ux\^^^^^ \\J/ .'///// 1 .\ V h-.'--l- -1 -Yv -vi •■- i- - ' 'zl-MtiiJt^ntutui i--i-ll-li l- y^^M-tutT-' 2ZQE52' JIhLZJQL Attttt '^MtttuJt'Jt^ltlttutut^ J. \s±htTt%im "" ^ WittkujL^xninmjjt'ihtt'iM ^ MtTittfi1fit%tt^ ^t lKlttlt^JU:ttuMttJUtH t H i %it'Zt$^ttumtttt%t4-tt-----l--- t imih'Uitttitrrnitht 1 ? ^titi.itktitjjtt'tit . ^ xttitttttmtiittz^it '^'''iqtiiitittLuut^ " G jtiittiUt il-itt it •a ' \ 1 'i //M/// ' / / 1 « ^j,^ mttl-tltlittMitttt -.- s J. luLtiu-tutt^i-iittuit I 1 ' tU%1-tutttiUMttJmii t ' tizrittltjitr'^ :tlifi " z -\z7tl-ttJ-li^ll21-ttUt 14 a " jszt^ttTittiiiiittmtiti ^ a \%z.7ttTttiittt' diverse, consequently in the aggregate of mortality from all causes the pecuUar incidence of each is to a great extent masked ; and as regards the secular trend of mortality the intervention of epidemics may produce great irregularities. Many diseases have a well-defined annual period, while others have not ; these periods, however, are not identical in phase. The aggregate of the deaths from all causes, therefore, gives a less definite indication of an annual period. Inasmuch as diversity of phase and of ampHtude do not wholly obliterate the periodicity, the general death-rate, viz., 8 = D / P, i.e., the deaths divided by the number of the population, is as follows : — (600) . . 8 =D/P =D I (t) / 1 + ao -f 2„^i a„ sinn (^ + aj } / j PJ^{t) \ in which 6 v:> ei fraction of a unit of time (say of a year), w = 1, 2, 3, etc., and both D^ and Pq are means over a unit of time, as at a particular epoch. Thus the graph of a death-rate, extending over several units of time (years), ir; made up of a non-periodic curve — representing the general trend — ^upon which is superimposed a periodic curve repeating itself during each unit upon a scale varying with the death-rate itself. ^ 2. Secular changes in crude death-rates. — ^The general lowering of the general crude death-rate in the western world has been remarkable, and is best exhibited by deducing the general trend of the rates for each country. The death-rates for Austraha are shewn in Table CXXXIV., from 1881 to 1915, for males, females, and persons ; see columns (ii.) to (iv.). In order to partially eliminate the irregularities of results for single years, quinquennial means were formed, see columns (viii.) to (x.), and the smoothing of these for "persons" gives the values in column (xiv.), the maximum value 0.01570 being that for the year 1884 and the minimum 0.01066 that for the year 1911. This fall to about two-thirds of its earlier value in 27 years is remarkable, and is accounted for not only by a stiU greater decrease in infantile mortaUty, but also in general mortaUty up to 60 or 65 years of age. It is worthy of note that the year 1895 was characterised by a halt in the decrease exhibited by the general trend of the death-rate. 1 So long, of course, as the character of the periodicity is maintained. MORTALITY. 373 The rates of infantile mortality are given in columns (v.) to (vii.), the quinquennial means in columns (xi.) to (xiii.), and the smoothed result or general trend in column (xv.). Here again the fall has not been continu- ous, see values for 1894-5. The character of the lowering of the rates doea not therefore fall under any law susceptible of simple mathematical expression. TABLE GXXXIV.— Shewing Secular Changes of the Death-rates, and of the Infantile MortaUty-rates in Australia, from 1881 to 1915. Year. Death Bates X 100,000. Infantile Mortality Kates X 10,000. (Quinquennial Mean Death Rates X 100,000. Quinquennial Mean Kates of Infantile Mortality x 10,000. Cieneral Trend of Death Kates (Smoothed) X 100,000. 1 Trend of ! Morflity lothed) 1 .0,000. 1 General Infntile (Smc X 1 Males. re- males. Per- sons. Males Fe- males. Per- sons. Males. Fe- males Per- sons. Males. Fe- males. Per- sons. (i.) (ii.) (iii.) (iv.) (V.) (vi.) (vii.) (viii.) (ix.) (X.) (xi.) (xii.) (xiii.) (xiv.) ,(xv.) 1881 1,589 1,328 1,469 1,232 1,095 1,165 1,636 1,348 1,504 1,372 1,203 1,293 1,528 1,293 1882 1,746 1,419 1,596 1,446 1,265 1,357 1,675 1,380 1,540 1,363 1,195 1,284 1,552 1,284 1883 1,654 1,381 1,529 1,302 1,138 1,222 1,708 1,404 1,569 1,353 1,186 1,274 1,569 1,272 1884 1,804 1,460 1,646 1,348 1,168 1,260 1,722 1,417 1,582 1,342 1,176 1,263 1,570 1,261 1885 1,747 1,434 1,604 1,360 1,221 1,292 1,689 1,397 1,555 1,330 1,166 1,251 1,562. 1,251 1886 1,659 1,392 1,537 1,348 1,189 1,271 1,676 1,381 1,541 1,316 1,155 1,238 1,546 1,238 1887 1,583 1,317 1,461 1,235 1,091 1,164 1,651 1,386 1,520 1,300 1,140 1,222 1,526 1,221 1888 1,589 1,300 1,456 1,251 1,072 1,164 1,611 1,336 1,485 1,281 1,122 1,203 1,500 1,203 1889 1,672 1,385 1,540 1,400 1,234 1,319 1,603 1,323 1,474 1,260 1,101 1,182 1,474 1,185 1890 1,554 1,287 1,431 1,152 1,009 1,082 1,570 1,289 1,440 1,237 1,078 1,159 1,444 1,163 1891 1,618 1,328 1,484 1,232 1,074 1,155 1,553 1,274 1,424 1,212 1,053 1,135 1,410 1,138 1892 1,419 1,144 1,291 1,142 970 1,058 1,496 1,223 1,369 1,188 1,024 1,108 1,368 1,112 1893 1,502 1,227 1,374 1,240 1,072 1,149 1,459 1,186 1,332 1,167 998 1,086 1,326 1,086 1894 1,386 1,128 1,266 1,107 952 1,031 1,419 1,147 1,292 1,158 993 1,076 1,292 1,074 1895 1,372 1,102 1,245 1,099 921 1,012 1,403 1,131 1,276 1,161 997 1,079 1,280 1,078 1896 1,414 1,135 1,283 1,202 1,045 1,126 1,411 1,139 1,285 1,177 1,012 1,096 1,282 1,096 1897 1,342 1,065 1,212 1,126 967 1,048 1,416 1,145 1,289 1,196 1,031 1,115 1,284 1,114 1898 1,540 1,267 1,412 1,364 1,175 1,272 1,404 1,130 1,275 1,204 1,038 1,125 1,283 1,125 1899 1,411 1,156 1,291' 1,246 1,085 1,167 1,395 1,157 1,263 1,198 1,034 1,117 1,273 1,117 1900 1,314 1,026 1,178! 1,086 915 1,002 1,403 1,123 1,270 1,181 1,019 1,097 1,255 1,097 1901 1,366 -1,064 1,222, 1,122 947 1,037 1,362 1,086 1,231 1,145 993 1,062 1,231 1,062 1902 1,383 1,102 1,249 1,142 997 1,071 1,322 1,052 1,194 1,103 946 1,019 1,201 1,020 1903 1,837 1,080 1,215 1,183 1,025 1,105 1,302 1,037 1,176 1,053 892 970 1,172 971 1904 1,212 988 1,105 891 756 825 1,269 1,019 1,150 1005 852 930 1,144 920 1905 1,214 950 1,088 906 724 818 1,235 994 1,120 952 800 878 1,118 872 1906 1,201 973 l,092i 901 760 833 1,212 974 1,098 887 734 813 1,098 827 1907 1,211 977 1,099' 884 734 811 1,200 957 1,084 867 711 791 1,085 792 1908 1,224 981 1,107, 855 697 778 1,188 952 1,075 849 702 777 1,076 770 1909 1,151 906 1,033 787 642 716 1,184 946 1,070 820 671 748 1,070 751 1910 1,154 924 1,043 817 675 748 1,192 947 1,074 804 650 729 1,087 733 1911 1,182 940 1,086 759 ■ 607 685 1,186 941 1,069 790 641 718 1,066 718 1912 1,251 984 1,123 801 630 717 1,189 946 1,072 791 640 717 1,070 708 1913 1,193 953 1,078 788 653 722 1,200 944 .1,079 776 626 703 1,079 702 1914 1,167 927 1,051 791 635 715 .. • . .. 1915 1,208 916 1,066 743 605 675 ■• •• •• •• The results in the above Table are shewn in Curves A and B of Fig. 98, the dots shewing the quinquennial means and the continuous line the general trend. The correlation between the two curves is fairly well indicated, because, although the ratio of the annual number of cases of deaths of children under 1 year of age, to the annual number of deaths of all ages is somewhat variable, there is some degree of general correspond- ence when a mean is taken over a number of years. See Fig. 98, p. 377. 374 APPENDIX A. The following example sufficiently illustrates the variable character of the ratio of infantile to total deaths, shewn in lines (a) and (6) here- under : — TABLE CXXXIV.A.— *Ratio x 10,000, of Infantile to Total Deaths, accoiding to Sex. Australia. Year .. . .j 1902.| 1903.i 1904 Males (a) ..'2,155:2,206 1,890 Females (6) . . 2,4771 2,469J 2,039 Females (c) . . 2,410| 2,295; 2,202 Males (d) (d) H- (c) = (e) 2,138 .887 2,078 2,024 .905 .919 1905. 1906. 1907. 1908. 1909. 1910. 1911. 1912. 1913. 1914. 1,930 l,97l{ 1,925 1,834 1,804 1,880 1,720 1,798 1,832 1,871 2,030 2,099 2,041 1,915 1,907 1,966 1,787 1,874 1,973 1,955 2,123 2,056 2,003 1,960 1,930 1,905 1,886 1,873 1,864 1,858 1,976 1,923! 1,893 1,863 1,839 1,819 1,807 1,798 1,792 1,788 .931 .935 .945 .951 .953 .955 .958 .960 .961 .962 1915. 1,672 1,798 1,855 1,786 .963 ♦ The figures on lines (a) and (b) are the ratios of the annual nunjhers of male and of female infantile deaths to the annual number of total male and of total female deaths respectively. The figures on lines (c) and (d) are the smoothed ratios for females and males respectively. The figures on line (e) are the ratios of male to the female ratios as determined from the smoothed ratios (c) and (d). Although the ratio oscillates between somewhat wide limits, the female ratio is invariably higher than the male-ratio : the general death- rate of females, however, is lower than that for males. These results indicate that the proportion of infantile deaths to total deaths for both sexes is rapidly decreasing ; the decrease for females being more rapid than for males. This is best seen by forming quinquennial means from which the general trend can be readily ascertained. The magnitude and general trend of the ratios of infantile to total mortality in the case of females and also in the case of males, are shewn respectively by curves M and N in Fig. 98, p. 377. 3. Secular changes in mortality according to age. — The death-rate for any age-group is the ratio of the number of deaths per unit of time (per annum) therein to the average number of persons in the group during that unit, i.e., to the number at risk. ^^ This ratio is markedly different for the two sexes. The following table, viz., CXXXV., based upon the censal results and intercensal experience since 1881 ^ shews that for nearly all ages a remarkable diminution in the death rates has taken place. That this must be so is obvious from the results given in Table CXXXIV. In a later Table, viz., CXXXVI., the average, also according to age, of the ratios between the death-rates of the sexes is given. These average ratios are the ratios of the sum of the four ratios given in each age-group for females to those given for males, and may be referred to the epoch 1900.0 for all comparisons as to any possible change with time. ^ 1 Actuarially, the ratio of the number of deaths experienced by persons be- tween given limits of age to the total number of units of time (years of life) lived within those age -limits by the population considered. 2 The results for 1911 are reaUy based upon the deaths occurring during the nineyears 1907to 1915inclusive. The actual populations for these years are assumed to be distributed according to age as at the Census of the middle year, viz., 1911 which must be substantially correct. MORTALITY. 375 It is obvious from the table that estimations of the frequency of death based upon tables compiled on the experience of past years are erroneous, if applied at the present time. * We shall investigate hereinafter the law of change. TABLE CXXXV. — Shewing the Mean Death-rates in Age-groups deduced for Various Epochs, and niusirating their Secular Changes. Australia, 1881 to 1915. Average ratio ol MALES. RATE X 100,000. Females. Bate x 100,000. Female to Male Age Death Rate. or Age- 1881- .1891- 1901- 1907- 1881- 1891- 1901- 1907- Data. group 1891, say 1901, say 1911, say 1915, say 1891, say 1901, say 1911. say 1906.0 1915, say Age. Sm'thed 1886.0 1896.0 1906.0 1911.0 1886.0 1896.0 1911.0 result. 0-0* 25,439 23,473 19,341 16,360 21,340 19,333 15,562 12,867 0.0 t.l866 14,366 12,738 10,112 8,540 12,414 10,786 8,349 6,862 0.5 .8395 1 3,576 2,685 1,804 1,559 3,427 2,519 1,684 1,389 1.5 .9371 2 1,379 982 677 642 1,336 963 631 575 2.5 .9524 3 891 628 441 409 834 617 412 382 3.5 .9477 4 692 497 350 301 648 488 325 300 4.5 .9571 0-4t 4,549 3,777 2,801 2,455 4,035 3,276 2,365 2,023 2.5 .8614 S-9 384 310 222 222 355 293 201 202 7.5 .9236 10-14 253 219 192 173 235 192 171 163 12.5 .8973 15-19 528 366 300 256 406 315 272 221 17.5 .8372 20-24 793 541 410 364 597 447 370 341 22.5 .8326 25-29 870 651 473 431 781 586 468 432 27.5 .9349 30-34 890 737 552 508 813 703 539 475 32.5 .9416 35-39 1,007 902 714 666 976 847 674 586 37.5 .9374 40-44 1,236 1,029 918 841 1,090 836 746 641 42.5 .8233 45-49t 1,591 1,311 1,222 1,120 1,262 1,000 890 794 47.5 .7525 50-54 2,085 1,737 1,522 1,511 1,568 1,273 1,044 1,050 52.5 .7199 55-59 2,803 2,454 2,091 2,153 2,037 1,793 1,497 1,473 57.5 .7157 60-64 3,717 3,624 3,095 3,174 2,694 2,677 2,293 2,177 62.5 .7231 65-69 5,528 5,207 4,708 4,678 4,423 3,753 3,619 3,471 67.5 .7587 70-74 7,488 7,104 7,584 6,972 6,218 5,704 6,074 5,523 72.5 .8069 75-79 11,778 11,686 11,845 10,900 10,076 9,967 9,378 9,162 77.5 .8350 80-84 15,275 16,210 16,450 16,815 14,490 13,984 13,306 14,575 82.5 .8704 85-89 27,169 26,041 27,372 26,783 24,227 21,960 22,836 21,701 87.6 .8427 90-94 24,661 26,917 30,677 30,896 28,455 26,497 29,433 28,960 92.5 1.0026 95-99 45,050 37,500 36,974 39,111 32,207 45,941 41,188 38,319 97.5 .9938 100-4 24,188 39,844 33,724 113,043 18,621 47,312 39,224 107,229 102.5 1.0075 • Nominally at the instant of birth, but not really so. Foi: the first week after birth the curve is quite distinct from the general death-rate curve after that period. The values given are deduced from the results for the five age-groups, to 4 inclusive, by formula (197), p.68 herein. If computed on the basis of A* = A + Ox+ Bo* , see C. H. Wicliens' Journ. Austr. Assoc. Adv. Sel. XIV., 1913, p. 535. The values for will be .27640, .26330, .22790, .19460 and .22740, .21470, .17840 and .15090. But true values of ix, are really much greater than these. t Between these limits (inclusive) the ratio is 0.8593. t The ratio of death-rates using .27640 -h etc., to .22740 + etc., is 0.8017. 4. The changes in the ratio of female to male mortality according to time and age. — ^The ratio of female to male mortaUty, according to time, may be deduceci from the rates given in Table CXXXIV., and those according to age from the rates given in Table CXXXV. To avoid the irregularities of individual years the former ratio is obtained by dividing the results in column (ix.) by those in column (viii.), Table CXXXIV. The quotients are given in Table CXXXVI., and are shewn by the dots on curve C, Fig. 98. This is the ratio for general mortality. For infantile mortality the results in column (xii.) of Table CXXXIV. are divided by those of column (xi.), and these are shewn by ' Thus the actuarial tables used by insurance societies err on the side of con- servatism ; they are based upon death-rates which are nove excessively high. 376 APPENDIX A. small crosses on curve D, Kg. 98. The firm lines denote the general trend of these results. They give some indication of correlation with the general and infantile death-rates, see Curves A and B, and the difference between the two curves is less marked ; see Mg. 98, p. 377. TABLE CXXXVI.— Shewing Ratios of Female to Male Death-rates, and Female to Male Rates o£ Infantile Mortality. Based upon Quinquennial Means. Aus- traUa, 1881-1913. Year Ratios of Female to Male Death- | Ratios of Female to Male Rates of of Rates (Curve C) 1 Infantile Mortality (Curve D). De- i cade. ' 1 1880. 1890. 1900. 1910. ; 1880. 1890. 1900. 1910. .821 .800 .795 1 .872 .863 .809' 1 .824 .820 .797 .793 .877 ! .869 .867 .811 2 .824 .818 .796 .796 .877 .862 .858 .809 3 .822 .81:! .796 .788 .876 .855 .847 .807 4 .823 .808 .803 .876 .858 .848 5 .827 .806 .805 .877 .859 .840 6 .824 .807 .804 .878 .860 .828 7 .840 .809 .797 .877 .862 .820 8 1 .829 .805 .801 . . ! .876 .862 .8.29 9 .825 .829 .799 .874 .863 .818 *2.0 .7925 .8089 *2.5 ' .8160 .7986 .8766 .8630 .8565 *3.0 .8232 ; *7.5 ; .8290 .8112 .8013 1 .8762 .8611 .8270 * These are means of five quinquennial means, except in two instances where they are means of four quinquennial means. That the ratio of female to male mortahty varies with time, having changed from 0.824 in 1881 to 0.788 in 1913, shews that life-tables for males and females, based on experience dating many years back, can no longer represent the facts with sufficient exactitude. The curve, shewing the ratio of female to male mortaUty according to age, may be deduced from Table CXXXV., and in view of the overlap or the partial overlap of the 1907-15 results on those of 1901-11, the epoch to which the ratio may be referred is 1900. ^ These ratios are based upon the sums of the four ratios for each sex, given in the table. The result is shewn in Kg. 98, curve K. There are two maxima and two minima in the curve, at the ages indicated in the table ; see p. 377. The dotted curve L, from which the curve K departs during the reproductive period of life, is symmetrical about an axis passing through the age 47. It is not unlikely that this departure from the curve L is due to the vicissitudes of reproduction ; see the reference hereinafter to the gestate force ofmartality. ^ Though not strictly exact, this assumption is sensibly correct. MORTALITY. 377 IVIortality Cuives and theii Relations. Australia. Curves A, B, C, D. Year 1880 3 4 6 8 1890 S 4 6 8 1900 2 4 6 8 1910 S 4 6 Ages 10 Cuives H, I, and J. 90 Curves H, I, J. Curve A shews the trend of the quinquennial means of the annual death-rates for " persons" from 1880 to 1913 for the Commonwealth of Australia : the dots shew the quinquennial means themselves: see Table CXXXIV., p. 373. Curve B similarly shews the trend of the quinquennial means of the infantile mortality rates : the dots shewing, as before, the quinquennial means ; see Table CXXXIV., p. 373. Curve C. — The dots shew the ratios of the quinquennial means of the death-rates for females to the quinquennial means of the death-rates for males, and the continuous line shews the general trend of these results ; see Columns viii. and ix. of Table CXXXIV., p. 373. Curve D. — The minute crosses shew the ratios of the quinquennial means of the rates of female infantile mortality to the quinquennial means of the rates of male infantile mortality^ and the continu- ous line shews the general trend of these results ; see columns xi. and xii. of Table CXXXIV., p. 373. Curves E. — The firm lines are the graphs for males and the broken line the graphs for females, of the results given in the vertical columns of the lower part of Table CXXXVIII., p. 379. Curves E. shew the changes in the ratios of decrease of mortality for ages to i, the firm line indicating the results for males and the broken line those for females ; see pp. 378-380. Curve a shews the mean of the results for ages to i, so reduced that the mean agrees with curve J ; see pp. 379-380. Curves H and I are drawn through the terminals of ordinates representing the means of the factors of decrease and increase. They shew the effect of age ; see pp. 379-380. Curve J may be regarded as the corresponding line for " persons." The scale needs modification. The line denoting unity may be taken at 0.9547: thus 0.9 ' and 1 '.0 are the correct places for 0.9 and 1.0 in relation to the curve ; see p. 380. Curve K. — The ratios of female to male mortality according to age, are shewn by curve K, the data being indicated by the dots, and the smoothed result by the continuous ciurve. The smoothed results are given in Table CXXXVII. ; see p. 377. Curve L. — This curve is symmetrical about age 47, and is coincident with curve K from age 62 years onwards ; for its significance see p. 376. Curve M. — The broken lines joining the points shew the ratio of female infantile to total female deaths for successive years. The dots shew the quinquennial means of these, and the firm line shews their general trend ; see p. 374. Curve N. — Similarly the fine zigzag lines are the lines joining the points defining the ratios of male infantile deaths to total male deaths for successive years. The dots shew the quinquennial means of these, and the firm line their general trend. The ratios of the ordinates to curve K to the ordinates to curve M, are given in line (c) in Table CXXXIV. A ; see p. 374. 378 APPENDIX A. TABLE CXXXVn.— Shewing for the Period of 1881 to 1915 the Average Ratio of Female to Male Moitality, accoiding to Age. Australia. DATA. SMOOTHED RESULT. Age- Ratio X Age- Batio j^gg. Ratio Atesaqe Ratios of Fbmales to Male Sbath-bates. 1000 ;group. ! 1000^ ^""P- j lOOOil Age. Ratio xlOOO Age. Ratio X 1000 Age. Ratio xlOOO 0.0 817 15-20 837 65-70 1 759 0.0 817 ^.«A 35.0 944 (») 7 10 70.0 773 %. 0-1 840 20-25 833 70-75 807 5.0 894 127 40.0 882 Of > 75.0 810 f 10* 1-2 937 25-30 935 75-80 835 7.0 903t • 01 45.0 787 eis 80.0 850 ISO* 2-3 952 30-35 942 80-85 870 10.0 883* «8a« 47.0 753 •'"t 85.0 892 • f s* 3-4 948 35-40 937 85-90 843 15.0 836* !>•• 50.0 730 ttt 90.0 935 • ss« 4-5 957 40-45 823 90-95 1.003 17.5 834t tit 55.0 713 Sflfi 95.0 980 • «0* 5-year means 45-50 753 95-lOC 994 20.0 839 .w.o 710 100.0 10!',6 10S8* 0-5 861 50-55 720 1100-105 1.007 25.0 894 797 60.0 716 1 1t 10!^ 1044 104«« 5-lC 924 55-60 716 1 30.0 944 7S8 620 724* 7t4* 10-15 897 60-65 723 32.5 950t 7tt 65.0 741* 7< 1* 15-20 837 65-70 759 35.0 944 7 10 70.0 773* SIO* * Curve of ratios identical witli curve L in Fig. 98, shewn by broken lines. t MaYimiim values, t Minimnm values. (a) Columns (a) are the values to curve L shewn by broken lines in Fig. 98. This curve is symmetrically situated about an axis, passing through the axis of abscissae at age 47.0, For the significance of curve L reference should be made to the text. 5. Secular changes in mortality vary with age. — ^For any age or group of ages, let /xq denote the mortality at a particular date, adopted as time origin ; and let p denote its rate of change — ^the sign being negative if it be decreasing — so that (601). ■n't = /-tpe^' = jtioe*^*^'- The last form is necessary only if p be Twt constant. It will be found probably in aU cases that p is a function of time, and it is also a function of age. The results for small age-groups are of course irregular, so that it is only in extended age-groups that the laws of the secular changes according to age and time are rendered obvious and unequivocal. This can be seen by an analysis of the results given in Table CXXXV.,^ and it is important to know whether for any given age p is sensibly constant for any sensible period. The analysis is effected by forming a series of sums of age-group- results from Table CXXXV., and calculating the coefficients which, multiplied into the results of any period will give those of a later period. 1 For example the sum of the rates to 49 gave the following indication : Year (o) 1886.0 p 1896.0 P 1906.0 Males m Males .(c) FemalesW FemaleB(e) .12101 X .12118 X .10550 X .10636 X .8314 = .09843 x .8050 = .09755 x .8052 = .08495 x .7909 = .08412 x .7928 = .07804 .8050 = .07853 .7882 = .06696 .7909 = .06659 P 1911.0 V(.8219) = .07036 V(.8050) = .07046 V(.7679) = .05868 V(.7909) = .05922 Data Computed Data Computed The constant ratios .8050 and ,7909 therefore reproduce the results fairly well, for males and females respectively, though with a decennium as unit for the ratio-value, we find the value of the ratio is i„/J = 0,8052 — 0,000127t — 0,0001573{' for females, t being expressed in years reckoned from 1886,0, The results are computed by takine the square root of the quantities .8314 and .7928 : allowing each the weight 2 and 9016 = V( ^19) the weight 1. This gives 0,89723, the square of which is .8050. The factors to divide into 12101 .09843, etc., are respectively 1, .8050, .64802 and .58142 ; the division gives .12101 12227 12043 and .12101 the mean of all being .12118 from which by inverting the procedure the above values for malpa are deduced ; similar resiilts give ,10636, etc., for females. i"»ics The values, foimd as shewn, suggest that, for the purpose of obtaining values for successive dates multiplication by a factor and its powers, or say an annual guingwnnial or decennial coefficient oi I'ortofion, has advantages over the employment of differences. MOBTALITY. 379 The quantities in columns ii. to iv., and vii. to ix., of this table, for males and females respectively, are deduced for the corresponding series of age- groups shewn ; the ratios are assumed to be true for the centres of the ranges of ages, an assumption which is sufficiently exact for the purpose in view.^ TABLE CXXXVin.— Shewing the Changing Ratios for different age-groups as between different dates. Australia, 1881 to 1915. MALES. FEMALES. AGE GROUPS. 1886 1896 1906 Ratio 1886 1896 1906 Ratio to to to Means to to to to Means to 1896 1906. 1911. Total. 1896. 1906. 1911. Total. (i.) ("•), (iii.) (iv.) .8459 (V.) (vi.) (vu.) (vlii.) (ix.) (X.) (xi.) 00 . . Moe' .90772 .9165 1.0528 .95182 .89722 .8268 .9050 1.0474 .. .9416 « .83102 .8445 .9019 1.0361 .93212 .87982 .8219 .8891 1.0291 1 .. .8665" .8197* .8642 .8473 .9733 .85732 .8175 2 .8248 .8349 .9654 2 .. .8439 i" .81822 .9480 .8544 .9815 .84902 .8094 2 .9112 .8456 .9777 3 .. .8395 2 .83802 .9274 .8565 .9838 .86012 .81712 .9272 .8563 .9901 4 .. .8469 2 .83922 .8600 .8464 .9723 .86782 .81612 .9231 .8582 .9923 Means • .8832 .8523 .8817 .8705 1.0000 .8864 .8395 .8725 .8649 1.0000 0-4 .9112* .86122 .8765 .8843 .9251 .90112 .84962 .8554 .8714 .9138 [5-14] . . [14-24] .91132 .88472 .9541 [.9092 [.8600 .90672 .87582 .9543 [.9038 [.8908] .8286' .88482 .8732 .87162 .31792 .8754 5-24 .85642 .88472 .9030 .8770 .9175 .88582 .90182 .9043 .8955 .9392 25-49 .90982 .91532 .9193 .9139 .9561 .8983 2 .91382 .8827 .9014 .9453 50-64 .95302 .92652 1.0194 .9557 .9998 .95482 .91742 .9723 .9433 .9893 65-79 .98382 1.0029' .9343 .9815 1.0268 .9683.2 .99092 .9520 .9741 1.0216 80-104 . . 1.03662 1.00542 2.5304 1.1229 1.1747 1.14812 .96802 1.4448 1.1354 1.1908 Means \ i .9418 „ .9327 1.0305 .95588 =1.0000 .9592 .9236 1.0019 .95351 = 1.0000 .94362 .93432 .96352 .92472 The quantities shewn in the table for the 10-year intervals are the square roots of the quantities originally given. In the totals these are counted twice. In the means 1 denotes the arithmetical mean, 2 the mean of the squares. The irregularities of the results are doubtless due in part to actual irregularities in the death-rates themsdves, and ia part to errors in the data. They shew unmistakeably that the death-rate up to age say 60 decreases with time, and that, at any rate above age 80, the rate for males increases with time. The results exhibiting this are illustrated by curves E, F, G, H and I, Fig. 98, E shewing the six results given in Table CXXXVIII., for males by firm lines, and the six results for females by broken lines. The thick line divides the values under unity, viz., those 1 Let a series of quantities, a and A, be respectively the numerators and denominators vrhich give the ratio for any range of the variable. Then it is assumed that I W ( ^2(i)/(-is4)= o„/ A,„ = a,/A, where a„ and A„ are the values for the middle range, the suffix notation being — k, — 1, 0, 1 .... k. Obviously in general such an assumption is invalid ; the true range is that which gives a value of a' / A' equal to a„, / A„. Later the assumption will also be made that the mean of a series of ratios may also be ascribed to mid-point of the entire range. The error of such an assumption is best illustrated by setting out the two results thus : — (^)..p,„ = (p, +P, +..+ P„)/« = (^^ + ^4-....+ -Jl-)/n (7) ■P m = tti -f- fflg -F . , ^1 -I- ^2 -I-..-I- An Although in general />',„ is not equal to p,n, if the successive ratios are in arithmetical progression, they are in agreement, and /j„, = a,,,/ Am above. If these successions of ratios are sensibly linear in their changes, the error will be negligible. 380 APPENDIX A. which represent a decrease, from those which represent an increase (on the upper side). It would appear from this figure that the change is some- where between 70 and 80, and that the rate of decrease of mortaKty unmistakably diminishes as age increases. Curve G shews the mean of the results multiplied by a factor so as to make the average agree with curve J. Curves F shew the changes in the ratios of decrease for ages to 4, the firm hne denoting the results for males and the broken Hne those for females. Curves H and I are drawn through the ratios, to the total, of the means of the factors of decrease (or increase) : they illustrate the general correspondence in the male and female cases of the effect of age, the curve J being the probable general indication, i.e., for persons. The line denoting unity may be taken as at 0.9547 : thus the broken Kne at 1.0474 will be reaUy unity in relation to the curve. ^ It is obvious that advances in hygiene, therapeutics, and social condition will be marked by diminished mortality. Whether that wiU extend over all ages or wiU characterise all but the older ages, depends upon whether the term of life is virtually sharply fixed or not. We shall consider the matter further in a later section. 6. Fluent life-tables. — For many purposes (much of insurance business for example) the ordinary tables of rates of mortality (fi^ or m^), of probabilities of living or dying within a year (p or g ), or of expecta- tions of fife (gj.), of the population survivors (Ix), at age x, etc., are satis- factory because they represent not only a considerable body of past experience, but also are 'on the safe side' for the major part of the uses to which they are applied (determination of insurance premiums, etc.). For the accurate prediction of Ufe, however, existing tables are not at all satisfactory, because, representing past experience, they take no account of the fact that the rates of mortality for the major part of fife are rapidly diminishing, that is the probability of lite is increasing for every age, say up to 60 for both sexes in many and probably in all, civilised countries. Hence for estimations of the true probabihty of Ufe, for the evaluation of payments for annuities, etc., existing lite-tables are seriously defective. To avoid this difficulty it is necessary to constrwct fluent life-tables, extrapolated for as many years as may seem safe. Such tables are, to the extent they are extrapolated, prediction tables. In these, past experience is brought under review in two ways : that is (o) as to the values of the various functions as they existed at a given moment, and (6) their trend, or variation with time. As the variation with time is not linear probably an annual coefficient of variation would best attain the object in view, and could be readily appUed.^ 1 Any resulting "error of scale" may almost be ignored. 2 Thus if this were 0.993 for example, the values of the factors for successive years would be — to three decimals — 0.993, 0.986, 0.979, 0.972 . . .0.9454, the last being for the 8th year forward. A linear diminution of .007 would have given .9444. MORTALITY. 381 It is only by means of fluent life-tables that accurate predictions of survivors for any given age can be ascertained. In Fig. 99, shewing the change of death-rates with time, the dots denote the values according to the data : the system of curved hnes shews what may be regarded as the general trend of the mortaUty-rate for. the various age-groups. The results for individual age-groups are irregular, but they unmistakably point to a diminution of the type e""**, where t denotes the period elapsed, m however having a different value for each age and sex. This index factor (m)'has no simple relation to age or to the magnitude of the mortal- ity-rate itseK, but is probably related to the two combined ; that is, it is a function of fx, and x. We shall first deal with the method of evaluat- ing it, and it wiU simphfy the matter if m be not treated as a function of time as in the final form of (601). Change of Rates of Mortality aecoiding to Age and Time. Males Females \ 'J^A^ let - X - ■0*2 -.^ . I flJ «4 ■040-^4----^ - 5-79 12-^r ^^-^ ^.A~- ^f- •038 . -»lrj ~'-- — ,i!ta| '■^ ^ds I ^'t. — — N 70-74 ^^ \ •034 - ^ ^7 ^.7,1 1 -T .-65;:i6r ^.^ V JH v — ^^^ 02 i- -:U.H .J tt 7^ §5^;^;= ::== ==^^i J^-^ i.= •028 -Y^^~"V != = = ===. 00^\- \ -02C \. V P "^ N \ (^ ff ^ \ " rt i.. \ '^l 1 -022 t %. V \l \ 1 A ^ )> Aa \ \ S ~%. N - -ozo \^ \ ^, \ \ \> ^ ' \ s L. ^ N ^ - \ N .MR \ \. S \ _ \~A N \-. ^ \ S N. .riM S» ^ ^^ s ■014^ Ng ^ ^ §1 n s ^^ vV 1:^ \ ''"^S^v "^ -, ^N 5^ : s. \ Vv ^. ^.. -^ ^5a S&^\ s ^-, ^^ ^^^ ^ .— \ i^ ^ ^. v^- ^. i^~^ 4iMs V.^; ^ ^\ ^. l^ ^.^ X'-w^ .\ v; •^ ^^ ■""'^C^^^-s ■^•^ ^ ^ y^ .004 4^ ^:i ^^ "~~- ■-- -fii 1^ ^ ¥^- ""S ^^•> i; = S -^ ==- ^ H~~" Zero •ooo _ r« 10 m 1! )o 1 20 31 4 ) 1 8( 90 l< 10 1 ) n r 3 1) 40 Zero Note. — The scale for the older ages, viz., .00 to .16, is shewn between the graph for the males and that for the females, the zero corresponding to .028 in the graphs for the lower ages. The curves shew the general trend of the improve - ment. Zeio 30 4U Year (to which rate of mortality applies). Fig. 99. The dots shew the rates of mortality according to the data ; the curved lines denote the general trend. The scale of the upper part of the graphs is shewn in the middle, the divisions representing ten times as great a quantity as in the lower part. The extrapolation of the curves to the year 1940 give an indication of the con- tinuation of the improvement. 382 APPENDIX A. 7. Determination of the general trend of the secular changes in mortality. — ^The results given in Table CXXXV., shewing a decrease with time — except for very great ages — ^in the rates of mortality, are best studied in Fig. 99. As this figure, however, gives only the rates of mortaUty as ordinates, and the epochs to which they refer as abscissae ; and does not shew the ratio of the improvement, it is necessary to evaluate this ratio. To do this the mortality at any epoch m^nst be divided by that at some epoch of reference. Thus we may assume — see Fig. 99 — ^that over greater or lesser stretches of time, the curve of variation of the inortaUty is of the form (601) with m constant ; that is : — (602) fit = fto^* > hence log m = log fi,Q + ''* '"? ^ The logarithmic homologue of this relation being a straight line, as shewn, the values of fi^ and r may be found by the " method of least squares." Or, put Bt = fit / fJ-O' thea, reckoning t from the year for which fig is taken, the general trend of the change in mortality can be computed by the following formulae, the derivation of which from (602) is obvious. (603) r'loge= U"^ + '^ + eto) In this expression n is one less than the number of dates for which jj, is known : r' is, of course, the mean value of r. Having found r' log e the mean initial value of the rate of mortaUty is : — (604). .log fi'o=-\log fig+log fii+log iiz + --—r' log e («i+<2+-)} /(n+l) and /x is the mean value to be substituted for the original jtio to compute later rates; that is, the general trend may then be taken as fi't = fi'^ (c*")' the value of e*" being determined according to the unit of t {i.e., for a year, a quinquennium, a decennium, etc.)."^ Within what limits an assumption of the relation expressed by (602) may be supposed to exist is of course to be ascertained by graphing the results on a suitable scale. 8. Modification of the general trend by age. — In order to discover the relation between age and the present secular improvement in mor- taUty, it will suffice to take the terminal values only into account ;^ provided we restrict ourselves to the most consistent results. The improvement for 25 years has therefore been computed, and is as follows, the tabulated results being the values of ^^25 / fJ-o '■ — ■ 1 The following instance will suffice to disclose the significance of the method : Year .. .. 1886 1896 1906 1911 Sum of Squares (a) Date 01379 .00982 .00677 .00642 of residuals. (6) Adopting terminals .01379 .01016 .00748 .00642 (c) By (603) and (604) .01386 .00993 .00711 .00602 (b) - (a) 00000 + .00034 + .00071 .00000 .00000062 (c) - (a) .. + .00007 +.00011 +.00034 -.00040 .00000029 The values of e" for a unit of 5 years, by (6), i.e., adopting terminal values 0.8582 : by (c), i.e., by above method 0.8463, MOBTALITY. 383 TABLE CXXXIX. — Shewing the secular improvement for 25 years in the Bates of MortaUty. AustraUa, 1886-1911. Agel 2 3 4 7.5 12.5 17.5 22.5 42.5 47.5 67.5 Males . . Females Batio* .436 .405 1.076 .466 .430 1.084 .459 .458 .1.002 .435 .463 .940 .544 .533 1.021 .684 .651 1.051 .485 .544 .890 .459 .571 .804 .680 .588 1.157 .704 .629 1.119 .846 .785 1.078 • Ratio of male to female ratio of improvement. The smaller the ratio the greater the diminution of the mortality. These results shew (i.), that in general the diminution of mortality is more marked in young life than in old ; and (ii.), that the diminution is not identical for mules and females. Changes in rates of mortaUty, whether due to causes outside human control or otherwise, may be regarded as due to changes in the relation between the human organism and its environment. Factors known to be operative in various organisms, and which are possibly operative in the human case, are : — (i.) Evolution of the protective reaction between the organism and its environment, (ii.) Changes of the food supply in amount and quality, (iii.) Changes due to the reactions of the organism to economic conditions, in respect of its nutritional and neural apparatus, etc. (iv.) Changes in individual and general hygiene, in therapeutical and surgical knowledge, and in prevailing traditions which affect the vitality of the organism ; etc. For our present purpose it is not material whether the change is what may be called internal — as (i.) above — or external : either or both may be regarded as changes in environment, i.e., provided they are regarded as either actual or virtual changes. In short, the effect upon the death and morbidity rates, of any given change in human environment, necessarily varies with the modifiability or " plasticity" of the human organism. The plasticity, however, is not the only element which iufluences the results. The rate of a general improvement in environment will probably be masked to soine extent by evolutionary disturbances, as, for example, by dentitional and puberal changes and, ia the case of females, by the de- mands made on the organism by the exercise of the reproductive function. Hence, a priori, it is not to be expected that the secular variation of mortaUty according to age wiU reveal any simple progression with age. Moreover, to maintain the same rate of improvement for the ages of least mortaUty, as for those of greater mortaUty, is probably, from the nature of the case, very difi&cult. Let Bx denote the ratio of change in jUj. in a given unit of time ; R being supposed to vary only with age (x). Excepting at the age of minimum mortaUty, a given value of yu is characteristic of two ages, viz., x>ne less and the other greater than^this'minimum age. Since the plasticity 384 APPENDIX A. of the organism} diminishes with age, a given (absolute) change in environ- ment will tend to have less effect on the later than on thg earlier age, other things being equal. It follows, therefore, that, in so far as plasticity alone is concerned, B^ will be greater than Bx+k- If the plasticity degrade continuously with age we may suppose that it could be expected to vary probably either as 1 /(a; + 0)*+"* or else as 1 / c"** , the value of a in the former representing the interval between fertilisation and birth, or say 0.75 year, since the plasticity is initially a maximum, and is greatest in utero. Consequently if it were necessary to take plasticity alone into account the reciprocal of the last quantity should be a factor distinguishing between the equal values of /x. for different ages. The former expression, it is found, does not represent the facts ; the latter possibly would do so but for the other elements influencing the result. For the purpose of analysing these complex relations between age, the change in the rate of mortahty, and the magnitude of that rate, we shall make use of the Census Life Tables for AustraUa for 1881-1890, and 1901-1910, see Census Report, Vol. III., pp. 1209-1218. For exact ages and 1, the ratios of ^^ ^re used, and for the purpose in hand it wiU be abundantly accurate to take Mi = 1 ('"^z-i + "^x) for ^g^s 2 and above 2,^ m being the central death- rate for each age in question. In order to fix upon values of the mortahty with which to associate the ages and ratios of change, the geometric means of the mortahties used in computing the ratios have been adopted, which is consistent with the first form of formula (601). It will also be assumed that the tabular values may be referred to the central point of time of the period from which the data are derived.* As already defined, Bt denotes the ratio of change for the time, that is Bt = nt //-lo a-s before, see formula (603). But there will be some advantage in fixing our attention upon the ratio of improvement rather than upon the ratio of reduction of mortahty. Thus if there be no im- provement (diminution) in the death-rate with the lapse of time, the quantity considered should be 0, and on the other hand the vanishing of death altogether would be denoted by unity. Let B denote this ratio of betterment (or of improvement), then : — (605) oBt = 1 - ^Bt = (fio -/x() //xo 1 The fixation of plastic elements, by means of which the growth and recon- stitution of the cellules of the organism are ensured, or anabolism , and the production of heat and energy by the oxidation of dynamic elements, or kataboUsm, constitute together the metabolism of the organism. The rate of metabolism or of waste and repair may appropriately be said to measure the plasticity of the organism. The plastic and dynamic elements, for example, the albumins, fats, hydrocarbons, etc., require also the presence of mineral salts and vitamines, in order to properly fulfil their nutritive and dynamic functions. The modifiability of the organism may of course be affected by its environment as well as by age : but its potential modifiability may be regarded as the measure of its plasticity. 2 The error of this assumption is, of course, nearly negligible for most purposes for almost any ages, and for the present purpose is quite negligible. The central death-rate is the number of deaths occurring between any age limits divided by the mean population. » That is, the table for the period 1881-1891 can be regarded as referable to the point of time 1886.0, and the table for 1001-1911 to the moment 1906.0. MOKTALITY. 385 with suffixes to denote the age to which the formula refers. As afeeady indicated, the magnitude of B will be influenced by various circumstances. For example, the ratio of improvement will probably be low (and as a matter of fact is low) for those ages which are characterised by the lowest rates of mortaUty^ ; that is for the ages when vitality is greatest a favourable advance in the environment will produce a relatively small effect. To analyse the effect of the value of the death-rate upon the improvement we may divide B by the geometric mean of the death-rates measuring' the change ; that is by : — (606) ixm = -\/(f*o M*)"' and call the ratio of the betterment to this quantity, A, or the relative betterment,^ thus : — (607). . A( = £( / /im = (1 — /^t / A*o) / ViH-oH't Since the Umits of B are and 1, this quantity can attain to considerable magnitude when t is considerable, and is therefore a sensitive measure of any improvement in the rate of mortahty. The following Table gives for males and females the values of fim, B, and A, the values for fig and iit being those given by the analysis of the Census results for thirty years, and the interval being referable therefore to the period between 1886.0 and 1906.0. For values of E, if required, we have simply 1 — B. The values of B are shewn in Fig. 100, curves B and B' ; in which also the mean death-rates \/(/'^o/^t) ^^^ shewn, viz., curve A male, and curve A' female. These exhibit the following characteristics : — Curves o£ Relative Improvement for 30 Years in Death-rates. Initial Point. 1st Maximum Age. Amount. 1st Minimum. Age. Amount. 2nd Maximum. Age. Amount. Remarks. Males Females 0.175 0.215 2.8 yrs. 0.508 2.7 yra. 0.520 12.8 yrs. 0.209 13.2 yrs. 0.224 23.3 yrs. 0.491 24.5 yrs. 0.400? Later values, are irregular Upon plotting the ratio of the betterment, viz., the values of A for males and females, we obtain the results as shewn upon Fig. 100 by curves C and C", representing the ratio of improvement in the case of males, and curves C and C", representing the ratio of improvement in the case of females. These exhibit the foUowing characteristics : — • Ratio of the Relative Improvement to the Death-rate for 20 Years. Initial Value. 1st Maximum. Age. Amount. Minimum. Age. Amount 2nd Maximum. Age. Amount. Remarks. Males Females 0.70 1.07 9.2 yrs. 164.6 9.5 yrs. 176.4 13.8 yrs. 94.8 (13.8 yrs. 108.3) 16.8 yrs. 109.4 (16.8 yfc. 100.5) Results after- wards irregular 1 This corresponds with the age at which the reproductive function commences to unfold, viz., at about age 12. Probably what may be called the age of effloresenoe of the organism is generally its period of highest vitality. ' This is suggested by the word iSeXn^ucris, i.e., betterment ; /3 is already appropriated for birth-rate, etc, 386 APPENDIX A. The values for age cannot be deemed to closely represent the f £icts ; to obtain these a table of deaths occurring on successive days after birth would be needed, and not merely extrapolated results based upon succes- sive years. For all other ages, however, they represent the facts with considerable accuracy. It will be convenient to call the ratio A the mortality improvement ratio. TABLE CXL. — Shewing the Mean Mortality, the Relative Improvement in Mortality in 20 Years and the Ratio of this Relative Improvement to the Mean Mortality for Males and Females. Australia, 1886.0 to 1906.0. 1 MAItBS. Femaies. Exact Maies. Fbmaies. Bxact ilmprovemeBt. Mean ; Death ^ , Ratio - Mean ' Plas- Death ' mprovement.|| ilmprovement. Mean : Death ' Ratio Plas- Mean Death Improvement. Age. !] Ratio Age. ,] Batio ' Bate Ee- to ;„ tieity ; Kate Be- to Rate ! Re- 1 to ticity Bate Re- to 1896.0 ative. i DeathlCurve ' iS96.0l latire.; Death 1896.0 lative. i Death Cnrve 1896.0 lative. Death 1 Rate 1 ' Rate, i! ^ 1 i Bate. Bate. 25100 1751 .70 279.0 20150! 2151 1.07: 45 1 01217! 244 20.1 27.7;; 00963 310 32.2 1 04640 45l' 9.7 265.0 04270 463 10.8 46 01283 239 18.6 26.4! 00988 312 31.6 2 01750 499: 28.5 251.8 01660: 514 31.0:, 47 01353 239 17.7 25.0' 01018 317 31.1 3 00796 507 63.7 239.2 00752! 519 69.0; 48 01426 244 17.0 23.8 01055 321 30.4 4 00559 500 89.5 227.2 00522; 503 : 96.4 49 01503 249 16.6' 22.61 01095 322 29.4 5 00441 488 110.7 215.9 OOlOft 492 120.3i 50 01583 251 15.9 21.5; 01140 320 28.1 6 00354 469 132.5 305.5 003241 470 145.0i 51 01668; 256 15.3 20.4 01190 319 26.8 7 00299 449 150.1 194.8; 002691 432 160.61 52 01758 260 14.8 19.4 01249 317: 25.4 8 00267 423: USA 185.1 0023.=)' 395 168.11 53 01855 263 14.2 18.4 01319 316 23.9 9 00243 387 riCl.3 17.-).8 00215; 377 175.3! 54 01964 267 13.6 17.5 01398 310^ 22.2 10 00222 331 ir,.'!.l 167.0 00201' 354 176.11 55 02081, 268 12.9 16.6 01488 304' 20.4 11 00208 264 126.9 1.58.7 OOinS 305 1.58.0 56 02209' 266 12.0 15.8! 01586 295 18.8 12 00206 219 106.3 150.7 0019.5i 244 125.1 57 02349 259 11.0 15.0l! 01694 284 16.8 13 00216 210 S7.2 143.2 00204 225 110.7 58 02503 246 10.6 14.21 01814 267 14.'7 14 00241 223 95.01 136.0 00221' 238 107.7; 59 02667; 229 8.6 13.511 01948 244 12.5 15 00284 282 on.3 129.2 00243 257 105.8 60 02842 212 7.5 12.9 02081 217 10.4 16 00335 359 100.9 122.8 00273 282 103.3 : 61 03030 192 6.3 12.2 02249 185 8.2 17 00385 421 109.3 116.6 00300 306 9S.;i 62 03234 172 5.3 11.6, 02418 155 6.4 18 00429 450 104.9 110.8 00343 333 97.1 63 03465 156 4.5 11 0: 02605 127 4.9 19 00466 466 100.0 105.3 00375 357 95.2 : 64 03746 149 4.0 10.5 02831 115 4.1 20 00490 476 97.2 100.0 00404 373 92.3 65 04098 155 8.8 9.9 03137 138 4.4 21 00531 483 01.9 95.0 00431 380 88.2 66 04520 166 3.7 9.0 03514 179 5.1 -■>2 00556 487 87.6 90.3 00457 379 82.9^ 67 04971 167 3.4 9.0 03898 201 5.2 23 00577 491 85.1 85.7 00479 371l 77.5 68 05423 151 2.5 8.5 04253 194 4.6 24 00598 490 81.9 81.5 00504 372 73.8|| 69 05862 118 2.0 8.1, 04596 165 3.6 25 00616 485 78.7 77.4 00535 383 71.6: 70 06285 069 1.1 7.7 04945 116 2.3 26 00630 473 74.9 73.5 00565 396 70.1 71 06721 013 2 7.3! 05314 053 1.0 27 00642 458 71.3 69.8;! 00593 403 67.9 72 07240 —030 —.1 6.9; 0574C —005 —0.1 28 00651 440 67.6 66.3 00619 398 64.3 73 07883 —051 —.7 6.6 06260 —042 — .7 29 00660 426 63.5 63.0 00638 392 61.4 j 74 08647 —052 —.6 6.3 06893 —049 — .7 30 00668 . "1 61.5 59.9 00652 382 58.6 , 75 09484 —044 : — .5 6.0i, 0763 —033 — .4 31 00680 392 57.6 56.9 00664 361 54.41 76 1035 —032 —.3 5.7 0843 —006 — .1 32 00696 374 52.6 54.0 00676 338 50.0 77 1126 —021 —.2 1 5.4 092e -1-019 -1-0.2 33 00714 360 50.4 51.3 00692 324 46.8 78 1222 —012 — .1 1 5.11' lOlC 042 0.4 34 00736 352 47.8 48.8ji 00714 315 44.1 ' 79 1321 —008 ; — .1 4.91' llOS [ j, : 064 , 0.6 35 00763 342 44.8 46.31 00737 314| 42.6 80 1422 —Oil — .1 ■ 4.6 121c ) 085 .7 36 00795 330 41.5 44.1- 00762 317, 41.6,1 81 1530 —013 — _■ 4.4' 1321 ' 104 .8 37 00826 318 38.5 41.8 00790 321 40.6 82 1645 —018 — !i 4.2 1 144( ) 121 .8 38 00862 303 35.2 39.7; 00816 3241 39.7 1 83 1775 —02s — -^ 4.0! 156! ) 13J .9 39 00902 289 32.0 37.7| 00838 323; 38.5. 84 1915 —037 '; 3.8| 170! > 141 .8 40 00943 281 29.8 35.9;' 00859 316 36.8: 85 3.6;' .. 41 00987 274 27.8 34.l! 00876 3071 35.0 86 3.4 .. 1 1 .'. 42 01037 266 25^ 32.4 00896 30ll 33.6 87 3.2'' .. 1 :; 43 01094 258 23.6 30.7 00919 30i: 32.7 88 ^ 3.1 44 01154 250 21.7 29.2 , 00941 307 32.6 89 1 i 2.9, . . jl MORTALITY. 387 Moitality Curves and their Variation with Time. Ages ,mfi >. . 5 J) 25 1 -■ — (70 fC / \ \ ' 160 ^ 1 / /• > I / 150 'j 1^ ,/ / i'- ! [^ C / / 1 E HO : 1^ 1 [t l' j 1 • ■ 130 / c ■■. 1 L I j ■ 120 li'J 1 \ pLI |^•■ 1 \ ' 1 ■■ li ■ ,' \ C 1 'A'' /' .^ \ 100 \\ '1 \^ /'- \ ■•. \\ ,} -~ \ ll' 90 c c ^N, N //>\ ■^ 1 w ! s / 1 ' s / \ [/ 1 ... s. L ^ / ■'*' -■ 60 A R j ^ V 1 i - ( f 1 "OJ ' ^J -ij \ B 1 17 '' \\ \- <0 m ' ' V, \ / \fd v- R A / « J , -\ >. / / V / i~~ A V ^ ^-' '., \ / _^ -ST- A \ / t^^ A ^ ('. 7( n fi r A V R IT n'- ... \ ^ N D WO .0 ^ s\ •■ E n' i (rof( r c c c c 1) H K ^ ■ 10 2 :i II 4 [) i J b u 7 )'"- 8 J - 9 10 n Curves A and A' represent the geometric means, according to age. of the rates of mortality lor 1886.0 and 1906.0, for males and femalesrespectively. Curves B and B' are the ratios of the diminution in the ^ rate of mortality in 20 years to a the geometric mean of the S rates, in the case of males and 8 females respectively. ° Curves C and C are the o ratios of the improvement m (last mentioned) divided by the ~ geometric mean rates of mor- « tality. Curves C" and C" are the curves C and C respectively, plotted to an extended horizon- tal scale but with the same vertical scale. Curves B and E' — the plasticity curve — shews, in a roughly approximate way, the general trend of the mortality- improvement ratio : see § 10. Fig. 100. Ages. 9. Significance of the variations in the mortality improvement ratio.- The following relation between the changes in mortality and in the mortaUty-improvement-ratio is important. The variations of the curve of the mortality improvement ratio are reciprocal to those of the mortality itself ; that is, x and 17 being the ordinates to the mortaUty-improvement-ratio curve, and x and y the ordinates to the mortahty curve, we have, practically for aU ages,' : — (608). .tj'/t = Ky /y' ; or s = K /r 17, 17' and y, y' being successive ordinates, and s and r their respective ratios.^ K, however, is not a constant ; nor is it any simply expressed function of x, though generally it is a Httle less or a little greater than unity. * Certainly for all ages for which themortality ratio caa be very accurately evaluated. ' That is ri and y are values for x, and r/' and y' values for a; -|- 1. 388 APPENDIX A. This reciprocal relationship reveals the fact that as the mortaUty at the beginning of life decreases with the successive years, the relative- improvement-ratio increases in very similar proportion. This reciprocal movement of the mortaUty-ratio, as compared with the mortality-im- provement-ratio with increasing age, probably continues throughout life, and certainly continues till at least age 70. The values of the coefficient K in (608) above, are given in Table CXLI., Km denoting those which apply to males and Kf those which apply to females. The ratios s=r]'/rj and 1/r = y/y' are also shewn, viz., by the smaller figures between the values of rj and y respectively. This coefficient K may be called the beUiotic coefficient.^ TABLE CXLI. — Shewing the ratios between the mean mortalities and the mortality- impiovement-ratios for successive ages. Australia, 1886-0 to 1906-0. Ratio Ratio Ratio of Values of K I of Values of K of Values of K. Exact Ratio Mor- Exact Ratio Mor- Exact Ratio Mor- Ages from ol Mean tality Im- Ages from of Mean taUty Im- Ages from of Mean taUty Im- to Mor- prove- Fe- to Mor- prove- Fe- to Mor- prove- Fe- (x). talities ment Males. males. (I). talities ment Males. males. («). talities ment Males. males. a/r) ratios, (s) K,„ ^/ (lA) ratios, (s) ^,„ ^r (1/r) ratios. (») ^^ ^^ 0-1 5.409 13.945 2.578 2.154 28-29 .986 .940 .953 .983 56-57 .941 .918 .976 .952 1-2 2.651 2.934 1.107 1.110 29-30 .988 .968 .980 .973 57-58 .938 .962 1.026 .930 3-3 2.198 2,232 1.016 1.011 30-31 .982 .936 .953 .945 58-69 .939 .809 .862 .914 3-4 1.424 1.401 .984 .970 31-32 .977 .913 .934 .936 59-60 .938 .869 .927 .888 4-5 1.268 1.237 .976 .978 32-33 .975 .958 .983 .959 60-61 .938 .850 .906 .854 5-6 1.246 1.197 .961 .956 33-34 .970 .949 .978 .972 61-62 .937 .837 .893 .837 6-7 1.184 1.133 .957 .919 34-35 .965 .936 .970 .997 62-63 .933 .849 .910 .821 7-8 1.120 1.055 .942 .914 35-36 .960 .927 .965 1.009 63-64 .925 .882 .954 .904 8-9 1.099 1.005 .914 .955 36-37 .963 .927 .964 1.012 64-65 .914 .950 1.039 1.202 9-10 1.095 1.024 .935 .939 1 37-38 .958 .913 .953 1.009 65-66 .907 .970 1.069 1.295 10-11 1.067 .780 .731 .862'! 38-39 .956 .911 .953 .997 66-67 .909 .916 1.008 1.126 11-12 1.010 .838 .830 .800 39-40 .956 .930 .973 .978 67-68 .917 .929 .795 .966 12-13 .954 .914 .958 .926 40-41 .955 .932 .976 .970 68-69 .925 .824 .891 .853 13-14 .896 .977 1.090 1.054 41-42 .952 .924 .971 .981 69-70 .933 ,545 .584 .670 14-15 .849 1.045 1.231 1.079 42-43 .948 .920 .971 .997 70-71 .935 .931 .427 .459 15-16 .848 1.077 1.270 1.090 43-44 .948 .918 .968 1.023 71-72 .928 .926 16-17 .870 1.022 1.175 1.085 44r-45 .948 .927 .978 1.009 72-73 .918 .917 17-18 .897 .959 1.069 1.089 45-46 .949 .928 .978 1.006 73-74 .912 .908 18-19 .921 .953 1.035 1.072 46-47 .948 .948 1.000 1.015 74-75 .912 .903 19-20 .951 .972 1.022 1.045 47-48 .949 .963 1.014 1.013 75-76 .916 .905 20-21 .923 .946 1.025 1.019 48-49 .949 .974 1.026 1.003 76-77 .919 .910 21-22 .955 .953 .997 .998 49-50 .949 .958 1.009 .993 77-78 .921 .914 22-23 .964 .972 1.009 .979 50-51 .949 .967 1.019 .997 78-79 .925 .914 23-24 .965 .963 .998 1.003 51-52 .949 .965 1.017 .993 79-80 .929 .916 24-25 .971 .961 .990 1.030 52-53 .948 .959 1.011 .997 80-81 .929 .916 25-26 .978 .952 .973 1.034 53-54 .945 .958 1,014 .982 81-82 .930 .917 26-27 .981 .951 .969 1.017 54-55 .944 .946 1.002 .982 82-83 .927 .918 27-28 .986 .948 .961 .987 55-56 .942 .935 .993 .980 83-84 .927 .920 The ratios in the Table (1/r) are the values of the mortality at any age divided by the mortality at the age greater by one year ; that is. the tabular values are the quantities M» //*.v+i. The tabular ratios of the mortality-improvement-ratios are the values obtained by dividing the mortality-improvement-ratio for any age by that of the age less by one year ; that is the tabular values are the quantities X . /\ The coefficient K is that quantity which, multiplied into the ratio of the reciprocally the ratio of the mortality-improvement-ratios. mean mortalities, gives ' From /SeXTiwTiKos, bettering or amending. MORTALITY. 389 If the value of the ratio y is required for a single unit of time (1 year), we have, on the assumption of a geometrically progressive decrease in mortality, fi^ — fi^ ; consequently : — (609). .5, = [1 -(/Li//x„)%„ and A,= (1 r- i^.^J/VC/^o ^^^) (610) I* = l^t/f^o' and,i^=/*of The form of the expression for A is independent of the unit of time, though of course its numerical vulue is dependent on that unit. 10. The plasticity curve. — ^If we except the period between exact ages 14 and 17, the. beltiotic coefficient continually decreases in value from age 10. If a curve be drawn representing the general result, it is found (from the 20 years' improvement in the mortahty conditions) that it is fairly well represented by the equation y = 278.95 (0.95)^ This curve, viz., E and E' on Fig. 100, may be called the plasticity curve, and its ordinates are given in Table CXL. The amount by which the beltiotic curve {i.e., the curve of the mortality -improvement-ratio) falls short of the plasticity curve, does not, however, and least of all initially, constitute a measure of the great difficulty of attaining to the limit, which plasticity would admit of, were it not for the great difficulty of initial adjustment to a new environment, and to the exhaustion of energy involved by puberal developments. For the analysis of these questions, however, the available data appear to be inadequate,^ and they will not be further discussed here. No simple relation expresses the variation of the constants 278.95 and 0.95 with the unit of time over which the improvement is measured. 11. Bate of mortality at the beginning of life. — The mortality at the beginning of Ufe is probably considerably affected by local cir- cumstances ; consequently for the first two weeks and perhaps even the first month of life it would be difficult to assign any particular law of change of mortafity with age.^ Statistics for Saxony gave a first minimum rate at 8 days, and a lesser maximum 15 days, and those for Sweden gave 1 It may be noted that for the relative improvement to be unity we must have ;li, = in (605), that is to say, death must vanish. But no diminution of mortality in a geometrical ratio can reach zero, for though ii„ i' may be as small as we please, it cannot become zero with f positive and t finite : moreover, when the death-rate is large the value of \ cannot be great with any practicable change of death-rate. ^ See " The improvement in infantile mortality : its annual fluctuations and frequency according to age in Australia." by G. H. Knibbs, Journ. Australas. Med. Congress., Sept. 1911, pp. 670-679 ; see also " Die Sterblichkeit im ersten Lebens- monat, Zeit. f. Soz. Mediz., Leipzig Bd. v., p. 175, 15th April, 1910. 390 APPENDIX A. a somewhat similar indication^, while Austrahan records do not lend any support to this recrudescence of the rate of mortality. Prussian statistics shew a minimum rate for 9 days and a rise to 14 days. ^ The statistics in Austraha are imperfect, and some distributing was necessary owing to the want of precision in stating the exact interval after birth. The defective statement of _ age does not, however, afEect the deductions hereinafter. In the following table the results for the fractions of the first day are merely computed : the rates, calculated without regard to migration, the effects of which are nearly neghgible, and are not accurately ascertainable, are determined by deducting the deaths from the total births in order to ascertain the numbers of survivors. The rates so found shew that from the end of the first day the law of mortality is expressed by /x^. = [i-^/x, for 5 or even 6 days. The generality of this expression can be extended, if it be put in the following form, viz. : — (611). ,/Xj. = jiti [1 +/(«)]/«, consequently 1 +/(a;) = a;/Xj.//i, and / {x) for the first 5 or 6 days is zero. The shorter expression indicates that after the first 24 hours, and for about the first week of life the probability of death diminishes as the length of time lived, reckoned from the moment of birth. The following rates are computed for " persons" {i.e., males and females) from the records of about 500,000 births and the deaths which resulted from them. TABLE CXLn.— Death-iates per diem at the Beginning of Life. Based upon 499,674 Births, and the Deaths occurring therein. Austialia, 1909, 1910, 1912 and 1913. Age- Death- Death- Death- Death- group rates Age rates Age " rates Age rates or Age per Days. per Days. per Days. per Days. Diem. Diem. Diem. Diem. 0* .015000 4 .001416 40 .0002237 200 .0001233 0- * .014061 5 .001137 50 .0002117 225 .0001142 i- * .012355 6 .000975 60 .0002035 250 .0001063 i-* .010143 7 .000853 70 .0001948 275 .0000986 ^:* .007934 8 .000767 80 .0001875 300 .0000923 j_l* .006330 9 .000703 90 .0001804 325 .0000865 0-1* .009404 10 .000653 100 .0001740 350 .0000821 1 .005743 15 .000497 125 .0001594 365 .0000802 2 .002927 20 .0003961 150 .0001464 3 .001899 30 .0002678 175 .0001337 1095 .00001084 4 .001416 40 .0002237 200 .0001233 • Approximate estimates only. There are no available statistics for the accurate estimation of the frequency of death during each of the first 24 hours of life. ' Op. cit., p. 676. The results are given on graph No. 7 on the page men- tioned. » See Handbuch d. Med. Statistik., Fr. Prinzing, 1906, pp. 281-2 ; also G. Lommatzsch. Zeit. f. saohs. Stat. Bureau, 1897, Bd. xliii., p. 1. MORTALITY. 3dl We may take the mean of \x./x for the first 5 days as the value of the mortahty at the end of the first day ; this gives the rate 0.005729 per diem. Using this to determine 1 + / («), we find that its values are as follow : — TABLE CXLin.— Shewing the Values o£ xfxx /ii-,, that is 1 + / (x) in (611). 1 + / (a;) 1 + / (a;). 1 + / (X). Exact Exact Age, Exact Age, Age, Days. Crude. Smooth- ed. Days. Crude. Smooth- ed. Days. Crude. Smooth- ed. 1-5 1.0000 1.0000 30 1.4023 1.4023 125 3.4779 3.4373 6 1.0211 1.0200 35 1.4442 1.4442 150 3.8331 3.8067 7 1.0422 1.0438 40 1.5619 1.5620 175 4.0841 4.1022 8 1.0710 1.0714 45 1.7038 1.7043 200 4.3044 4.3342 9 1.1044 1.1028 50 1.&476 1.8466 225 4.4851 4.5131 10 1.1398 1.1380 55 1.9894 1.9889 250 4.6387 4.6493 12.5 1.2280 1.2280 60 2.1313 2.1312 275 4.7329 4.7532 15 1.3013 1.3013 70 2.3802 2.3930 300 4.8333 4.8352 17.5 1.3526 1.3526 80 2.6182 2.6330 325 4.9071 4.9057 20 1.3828 1.3828 90 2.8340 2.8500 350 5.0157 4.9749 25 1.3990 1.3990 100 3.0372 3.0450 365 5.1096 30 1.4023 1.4023 125 3.4779 3.4373 From 5 to 10 is a second degree curve, the 1st difif. for a unit being = H- '0200, 2nd di£f. = -l- .0038. From 40 to 60 is a straight line, the common difference for a unit being + .02846. Th» curve from 60 to 120 is a second degree curve, the 1st did. being -1-0.2622 and the 2nd diff. -0.02238. From 125 to 350 is a third degree curve, the first rank of difference being + 0.3694, - 0.0739, and + 0.0104, the last being the common differences. The results in the above table shew that although for the first few (five) days the death-rate diminishes as the duration of hfe, this rapid rale of diminution is not continued, but the rate falls off more slowly — and on the whole'continually — tiU the minimum death-rate occurs. Ages c,»,E. 3.or 1 1 1 1 M 1 1 1 rnrnu. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 nm Mortality Curves. Curve A A shews the values of 1 + Hx) for 90 days, see (611) p. 390. Curve A' A', shews on a smaller scale the values of 1 -1- f (x) for 360 days. Curve B B is the curve of rates of mortality for 360 days ; the dotted line shews what would be the curve if the hyperbolic law held throughout. Curve C is the curve of rates of mortality for males, and curve D B is that for females. Curves C and D* are the same as C and D, except that the vertical scale is increased tenfold. Cmve B B is the curve of mas- culinity of the rates of mortality according to age ; see Table CXLVI. Curve F F is the curve of the ratios of the rates of mortality for males " not married " to those for married males, according to age. Curve G G is the curve of the ratios of the rates of mortality for females " not married" to those for married females, according to age. ays- Fig. 101. 392 APPENDIX A. The characteristics of the dimiaution of the initial death-rate may be summed up as follows :• — (a) For the first 24 hours of life satisfactory data do not exist to determine the characteristics of the death-rate (see below). (6) From the end of the first to the end of the fifth day the rate varies inversely as the duration of life. (c) From the end of the fifth day the rate of diminution rapidly faUs off till about the 20th day, then less rapidly till the 30th day, then the rapidity of the falling off of the rate of diminution approximates to what it was from the 5th to the 20th days, but after that decreases slowly and fairly regularly. {d) No simple function expresses these changes in the variation of the death-rate, and £hey probably differ somewhat in different countries. If the expression (611) is put into the form : — (612) ^. = ^e«(-i)' this can be fitted to a considerable range of the curve, provided that minor fluctuations are ignored. It cannot, however, represent with sufl&cient accuracy a year's results. To fit any two points on the curve besides the origin we have : — (613) log^M: =Mog^^ ^ log 2/ ° X— \ in which 2/ = x^j./fji^ = 1 + f (x). When b is foxmd a can be readily obtained from (612). For the values of (i for fractions of the first day it may be assumed that the curve is /Xo^"'' For this to give .995729 at the end of the first day we must make jno = 0.015573 (per diem), and this would be the mortality for x=0, viz., at the moment of birth, and is equivalent to a death-rate of 5.684 per annum. This may be put ia another way, viz!., it is equivalent to a rate of unity per 64.21 days {i.e., 365 -=- 5.684), and implies that such a rate, if operating uniformly for that period on a group of children for 64.21 days, the group being kept constant, would in that time account for the death of all bom. 12. Composite chaiacter of aggregate mortality according to age. — Before dealing further with the variation with age of the rate of mortality, it is desirable to review the nature of the aggregate rate of mortaUty. The general rate of mortahty for any age, Dj,/Pj=/ij., viz., the aggre- gate number of deaths of persons between given infinitesimal limits of age occurring in a unit of time, divided by the average number of persons of the same age ^ (the average being taken over the unit period in which the ' In practice D and P are taken between limits x and x', say, in which case /t is not given but instead the average over the range. The difference is dealt with later. MORTALITY. 393 deaths occurred) is made up of the rates from each cause, and if regarded from the summation point of view — see (596), p. 370 — ^is compounded of a series of rates, the graphs of which are by no means similar. For example, in " causes of deaths," Nos. 31 and 32, the real number at risk are those shewn in hne 2 below, the variation with age is quite unlike the variation with age of the total mortahty, and is by no means identical in the two cases, as wiU be at once seen from a Table given hereinafter. The results are as follows : — 1. Age-group 10-14 15-19 20-24 25-29 30-34 35-39 40-44 45-49 50-54 Ttl. 2. Oases of Maternity 1907-1915* 211 54,527 262,866 317,815 238,746 155,813 60,970 6,075 79 3. Cases of Puerperal Septic semia 1 96 370 515 459 307 112 12 1? 1872 4. Ratio .00474 .00176 .00141 .00162 .00192 .00197 .00184 .00197 .01266 ? 6 Cases of other Ac- cidents ol Preg- nancy & Labour 4 161 587 816 799 829 397 55 0? 3648 6. Ratio . . .01896 .00295 .00223 .00257 .00335 .00532 .00651 .00905 .00000 ? • Actually births. These are, however, only slightly too great. The correction may be neglected for the present purpose. The results shew that out of a total of 100 deaths at all ages from puerperal septicaemia and other accidents of pregnancy and labour, 34 will arise from the former cause, and .66 from the latter ; and also that the distribution according to age differs considerably for septicaemia ; the proportion dying at different- ages remaining more nearly constant than in the case of deaths from other accidents of pregnancy and labour. The fall to a minimum occurs at about age 23.4, when the ratio is about 0.00219. The minimum in the case of septicaemia is at about age 23.1, and the ratio is about 0.00139, the proportion of the deaths from other accidents of pregnancy, etc., being here 0.61 of the two combined.^ Causes of death may be classified, as regards their relative frequency according to age, as foUow, viz.^ : — (i.) Normal, viz., those in which the relative frequency is similar to the relative frequency of death from all causes combined ; » See formula (292) and (294), p. 92 herein. ' The causes of death given in a Table hereinafter may be classified accord- ing to the scheme indicated, and are as follows, viz. : — Glass (1.) Hoima], — 9. Influenza ; 12. Spidemlc niseases ; 16a. General Diseases ; 18. Cerebral Hsemorrhage, etc. ; 18o. Other Diseases of the Nervous System ; 20. Acute Bronchitis ; 22. Pneumonia ; 23. Other Diseases of the Respiratory System : 24. Diseases of the Stomach; 25. Diarrhosa and Enteritis; 27. Hernia and Intestinal Obstruction ; 28o. Diseases of the Digestive System ; 29. Acute Nephritis, etc. ; 80a. Other Diseases of the Genlto-Urinary System; 32a. Diseases of the SUn and Cellular Tissue ; 322>. Diseases of the Organs of Locomotion ; 35. Violent Death ; 38. Ill-deflned Diseases. Class (il.). — Infantile, Sub-classes (o), (ft) and (c). — 5. Measles (ft); 7. Whooping Cough (a); 8. DiphtheriaandCroup(6); 14. Tubercular Meningitis (6) ; 15. Other Forms of Tuberculosis (c) ; 17. Simple Meningitis (a) ; 33. Congenital Debility and Mal- formations (a) ; 33a. Other Diseases of Infancy (o). Class (ill.). — Senile.— 16. Cancer and other Malignant Tumours ; 19. Organic Diseases of the Heart ; 19a. Other Diseases of the Circulatory System ; 21. Chronic Bronchitis ; 28. Cirrhosis of the Liver ; 34. Senile Debility. Class (iv.). — Median. — 1. Typhoid Fever : 13. Tuberculosis ; 26. Appendicitis, etc. ; 31. Puer- peral Septicaemia ; 32. Accldente of Pregnancy and Labour ; 36. Median ; 30. Non-cancerous tumours of the female genital organs. Organic diseases of the heart and other diseases of the circulatory system are hardly to be included in the " normal" series, because the death-rate in the first year of life is not very great. 394 APPENDIX A, (ii.) Infantile, viz., those which characterise iafancy only ; (iii.) Senile, viz., those which characterise old age only ; (iv.) Median, viz., those which characterise middle age only. The infantile causes of death may be subdivided into three sub- classes, viz. (a) those in which the mortality is greatest in the first year of life ; (6) those in which it is later than the first year ; and (c) those in which the mortaUty is greatest in the first year, but is followed by an irregular mortahty for all ages. It is obvious that, apart from variations in the distribution according to age, and general differences in local salubrity, epidemics will cause differences in mortahty rates according to age, hence to be representative of a country, the deduced mortality rates must be taken over a sufficient period of time. The results in the Table. CXLIV. hereinafter are based upon 9 years' experience, viz., from 1907-1915 in Austraha, and the distribution of the population, according to age and sex ia assumed to be as at the Census of 3rd April, 1911. Before analysing these results it will be necessary to consider the character of curves of organic increase or decrease. 13. The curve of organic increase or decrease. — The curve e" (or e"*) and its variants, may, for obvious reasons, appropriately be called the curve of organic increase or (orgswiic decrease). In considering its appHcation to the increase of population by birth or the reduction of population by death, etc., certain characters of the curve deserve notice, and will now be considered. If to adapt it to a given instance, the ex- pression be put in the more general form hereunder, we may note that : — (614) ^ =^6"^+° = (^e'')e»* = ^'e"* =4'm* in which m = e" Hence the addition of a constant to the index of e affects only the vertical scale of the graph of the curve, while n affects its horizontal scale. If w be constant the final form in the above expression is satisfactory, but it it change with x, then the appropriate expression is — (615) y = ^e«^*W = ^e"^-»*W = ^e"'^ = A^{xf and the form of ^{x) will be determined by the law of change in n'. Geometrically this is equivalent to changing the a;-scale as x increases. In order to ascertain the form of X^^X/,. (v.) The relative increase of the ordinates of distribution G is less rapid than those of distribution H ; then Xg ,q = eo Later, viz., 1839, Ludwig Moser published in Berlin his " Die Gesetze der Lebensdauer. 406 APPENDIX A. Later he discovered that a further modification, viz., the introduction of a term Cx, that is, an arithmetical progression, gave the formula a wider extension. Thus his second modification was the expression : — (6295) .jn^ = A + Gx + Be'. The significance of expressions of this type is seen at once from (630) hereinafter, that is : — r B (629c). . loge y = —jBc'dx =K - ^^— ^ ^ -^ or «/ = kg<^ according to Gompertz ; or (629«i). .log« y= — ](A + B(f)dx= K— Ax— then initially «/,-or (y^) = 1 and I—?/,, will denote the ratio of the aggregate of deaths up to the age x. ' Let [jL=(x) denote the rate per unit of time*^ at which death occurs at the " exact age" x ; then the number dying in a unit of time, whose ages are between x and x + dx, is the number living between those age- limits, multiphed by the rate of dying, that is, yfj. dx. ^ Thus if /x be re- garded as positive (630) — dy= yfidx ; or — = —0 (x) dx y By integration we obtain : — (631) log 2/ = — /^ (a;) da; : or «/ = e"-^''' W<*^ Equations (630) and (631) are the bases of the theory of an " actuarial population." The number of survivors at each age obviously depends on the form of ^{x), and is completely determined when that function is known. Various forms that have been adopted for ^{x), and their integrals have already been given, formulae (629) to (629/). The probabihty at birth, of hving to age x is y^, as given by (631) above. The probabihty of dying before age x (vj,, say), is the arithmetical complement of the probabihty of hving, viz., l—y^ ', that is : — (632) v^ =l-2/^ = l_e 'S{x)ix Similarly the probabihty {p^) of persons of age x hving to age a;+l and {q^) that of dying before that age, are respectively : — (633) 'Px={yx+\)/yx ; a.rAqx= {yx — yx+ii/Vx = l—Px- The average of the death-rates (M) of persons dying between ages Xi and xz IS : — 1 rX^ 1 r^'-i (634) M= \ u,dx= ) ^^^ *^** toward the end of the first year f {x) ia large — about 4 — - compared with unity. Also it is evident from curve A', Fig. 101, that it is approxi- mately a constant at about 320 days to perhaps 400 days, thus /t, = S/i^ /x, and would appear to have become constant at least for some range of x. Such, however, is not the case. If it were we should have x/ix =Jc [1 + f (x)] a, constant. We obtain, however, the following results : — .03808 .02325 .01602 .01428 .01354 — — .01650 .03078 .02070 .01435 .01333 .01350 — — .01500 which shew that 1+J (x) is not expressible by any simple relation. The results for males for 24, 3^, and 4 J years can be expressed by fix = Moe""**. and for females this expression is also fairly approximate. MORTALITY. 413 26. Genesic and Gestate elements in mortality.— If the infantile and juvenile, and the senile elements of the mortality be subtracted from the totals, the residuals will constitute the genesic element in the case of males, and the gestate elements in the case of females. The rate of diminution seemed to be constantly 0.97 per half-year (see Table CXLVII., p. 412) from age 8.5 to 11.5 for both sexes. This is equivalent to 0.73752 for 5 years, and the adoption of this gives the results in columns (iii.) and (viii.) of Table CXLVIII. This may be regarded as the measure of degradation of the power of adjustment to environ- ment. The residuals smoothed as shewn on Fig. 103, are given in columns (iv.) and (ix.). On this figure the heavy curve, M, denotes results for males, and the light one, P, results for females. The computed mortaHty curves and those given by the crude data, are shewn in columns (v.) and (vi.) for males, and columns (x. ) and (xi. ) for females. The agreement in general is fair up to 62.5 years. Afterwards the results diverge somewhat. It has, however, to be remembered that these divergencies are not really large, and do not make large differences as between the computed and actual numbers of deaths. Fig. 103. The curves shew tlie genesic (M) and gestate (F) elements in mor- tality. TABLE CXLVin — ^Illustrating the component-elements of the Force of Mortality. AnstraUa, 1911. '*"- Male Eates ol Mortality, x 100,000 Female Rates of Mortality, x 100,000. SenUe Element. Juvenile Element. (Jenesic Total. Senile Element. Juvenile Element. Gestate Sm'thd. Total. Sm'thd. (Com- Ob- (Com- 1 Ob- puted.) served. puted.) 1 serve d. (i.) (ii.) (iii.) (iv.) (V.) (vi.) (vii.) (viii.) ' (ix.) (X.) (xi.) 2.5 .. 640 640 641 574 574 574 7.5 221 221 220 201 201 200 12.5 . . 163 10 173 173 149 4 153 153 17.5 .. 2 120 133 255 255 110 120 230 220 22.5 . . 7 89 252 348 364 3 81 268 352 341 27.5 . . 24 65 343 432 432 10 60 366 436 433 S2.5 . . 64 48 413 525 508 29 44 435 508 475 37.5 . . 151 36 467 654 666 74 33 467 574 586 42.5 . . 318 26 497 841 841 170 24 462 656 641 47.5 . . 617 19 484 1,120 1,122 354 18 426 798 796 52.5 . . 1,121 14 387 1,522 1,522 686 13 356 1,055 1,057 57.5 . . 1,930 11 220 2,161 2,161 1,254 10 215 1,479 1,479 62.5 . . 3,173 8 3 3,184 3,179 2,177 7 3 2,187 2,181 87.5 .. 5,022 6 5,028 4,693 3,623 5 3,628 2,201 72.5 . . 7,693 4 7,697 7,034 5,814 4 5,818 5,580 77.5 . . 11,455 3 11,458 11,136 9,041 3 9,044 9,379 82.5 . . 16,635 2 16,637 17,387 13,674 2 13,676 15,026 87.5 . . 23,632 a 23,634 27,557 20,188 2 20,190 22,492 92.5 . . 33,926 1 33,927 31,673 29,161 1 29,162 30,007 97.5 45,071 1 45,072 40,475 41,314 1 41,315 39,873 02.5 . . 60,744 1 60,745 1.23393 57,531 1 57,532 1.16876 27. Noim of mortality-rates.- — A study of mortahty rates for the same country at different times, and for various countries, shews that the real nature of the mortality curve will probably be revealed only by 414 APPENDIX A. obtaining a norm of mortality rates on a wide basis. Such a norm would necessitate a compilation for a large series of populations, of the foUowiog data, viz. : — (a) Infantile deaths according to hours for the first week of life ; then according to days for the first month of life : and according to weeks for the balance of the year. (b) Deaths in childhood according to months for the second year ; and according to quarters for the third year and afterwards ; (c) annually — or better semi-annually — ^tiU 15. Afterwards the annual number of deaths. The " number Uving" would preferably be deduced for the first 12 months (making corrections, however, for migration), by subtracting the deaths from the recorded births. Afterwards, or at any rate after the second year, the census data would in most cases be preferable to use. The combination of a large number of results, viz., all deaths in any age-group, and the sum of the populations in the same age-group from which such deaths were drawn, would probably disclose the true laws of the incidence of death. Only in large bodies of figures can it be hoped that the minor chance influences will counteract one another. 28. Number of deaths from particular causes. — ^The actual numbers of deaths according to sex and age, which occurred ki Austraha during the 9 years 1907-1915 from various causes, were as shewn in the following table, viz.. No. CLXIX., their relative frequency from all causes together, but retaining the age-groups, that is their ratios to the totals for the same sex, being shewn on the last two fines, see pp. 416-417 : — 29. Relative frequency of deaths from particular diseases according to age and sex. — ^If for each sex and for each age-group in that sex, the number of deaths from each cause be divided by the total deaths from all causes, the quotients are the relative positions of the disease as rewards their contribution to the totahty of deaths. Thus they measure the gravity of the incidence of any disease in question. This has been done and the' results are shewn in Table CL., on pp. 418-419, MORTALITY. 415 30. Death-rates from particular diseases according to age and sex. — It has already been pointed out that the incidence of death according to sex, has diverse characters as regards its relation to age ; see § 12, p. 393 hereinbefore. If the ratio of the number of deaths which occur in one year from any disease, in any age-group, and for either sex, to the average number of persons of the same sex in the age-group be found, this ratio will be the annual death-rate for the particular disease in question. ^ Thus the ratios are exactly analogous to the values with accents in (628) of § 19, p. 402 ; that is, they are the individual components of the death- rate for the same sex and age-group. They represent the ratio of the number of persons of a particular age-group who will (probably) die of the particular disease in question during the one year. These ratios, multiphed by 1,000,000, are shewn in Table CLI. and are thus the (partial) death-rates for each disease and for the two sexes, see pp. 420-421. The forms of the rate-of-mortahty curves for each disease are shewn on Fig. 104, the heavy hues denoting the curves for males and the lighter line those for females. They illustrate the marked differences in the incidence of death as between the sexes for the same disease, and accord- ing to age as between different diseases. 31. Rates of mortality during the first twelve months of life.— The incidence of death during the first twelve months of life is so varied that the means for the successive years 0, 1, 2, 3, etc., cannot be regarded as giving a satisfactory indication in regard thereto. Even in the first month of hfe, the frequency of deaths greatly varies for the successive weeks therein, so that a month is clearly too large a unit to adopt for rigorous results. Consequently, a tabulation for the first four weeks is necessary as well as for each of the succeeding eleven months. The population on which the ratios were based was 399,823 male births, and 38,027 females, which was reduced by the deaths themselves and increased by the net immigration of the same sex. ^ ■ 1 The sum of the mean populations for each sex and for the 9 years under review were distributed according to the Census of 1911, the middle year. This gave the divisors by means of which the rates were computed. • The immigration is by no means wholly negligible for accurate results : thus it was estimated to be — for each sex — 267 for the eleventh to the twelfth month, while the deaths were : males, 933 ; females, 768. Its neglect does not, however, obviously make a large error, since the deaths are drawn mainly from those born in the country under consideration, 416 APPENDIX A. ^- 1 3S US'* C4t> coa O0» too 00 00 II KO ^O OO i-IO 04 CO ^00 Mr-I 1-4 1-tI-l 04 (N 04 iH iH S3 Y-t 1-Ci-H lO'* OOO eooi OOO o» ss IP OO OO oo OO 00 i-i ^00 5:2 00 -# -*co t-ll-i ^o Tt<0 coco om OO OO 09 ;2^ r--* 04rH rHrH OOO i-lfH -Hrt O4 04 '"' COCO iHiH COCO OO OO 11 S3 OrH S3 OOO ■dto S2 0(M 00-* t-o OOO ^S «g ss s§ 04i-l iHi-l CO 04 S-Sg ss eoo4 iHrH tHiH iHi-C E:S rHi-l «C» COi-1 II r-«H eoo tow S^ OC4 OC4 §s §1 ;^^ O-^i ©Ir^ 1-Ci-l 00 04 om 1-1 CO iHrH (NCO gsg 1-1 CO r-lrH Ni-i •#co OCO c>-o ooo S2 ISS^ 00'^. (M-iji '^ in-*^o r-l ia\a t-<» GOO 1-1 1-1 iHi-l OO eoTjf CO CD CO'^ 04 OrK OO sss CO 9 i-«iH i-t 4r. o-* o OO << l-i.-l OO 00 IM aai x- OCO coco OCO 1-1 o rUrH r-tOO '■i'O OIO 5:5:; SSg OOl ■*r- Ot- COrH X_r7 rHrH eoco o-* iHiH l>CO 0004 CDO r^(M C-1 ^ vnm com ODO ODO coo OOCO g5S OO CO 00 OCO CO 00 r*rH x_o l-H CO ■*oi rHC4 t-O I- CD o^ !::2 ON 0-* ON i- OOD OrH o«> (NCO OOO WGO oeo OCO r-co <3>W coo ^O ot- r-00 r-to r-lTj< sss into car- eoe 01-* □or- l>CO COCi tPOO IN 1-1 coco OOl r-tr-l !MO T* t- lTC^ iHi-I eot-t •HCD oa» OCO r-04 OCO i>o t — < in oc» OtII CJr-H xaoo Oi-< tO'^ b-OO OCO 00 CO OO M ?3 r-l «eo lO W_ cqr-f 00 00 .aid i-IO wr- o c* MO 1-iN XO -^^ F-ii-i t-oo OICO iHrH r-KM I-I01 coco i-Hi-H iHrH oeo ■^^ t-w =00 r-l CO 03D ■*« 00^ CO"* i-irH ot- CO-* 00 V-": OCN OOO ot- o t- COCO »o2® i-ii-i coco eO(N NrH i-liH ^t- ■^U-S r-rH T-tOS i-rio Oi-( TS-(0 OCO COO OOO t-(N 1*00 ED CO Oi-i o o OCO COO 1-f uO"* c-i—< ■* COCO O04 « •*co i-Hi-l OOO r-'M ■Ml^ -(CO COO r^CO i>o t--* ■*o "*0 OO t-co ■HrH l-H OS -**o iHt-l 1-1 Oi rHr-l t~CO (M coco 1-1 04 rH tH t- lOO coo CO 04 '"' eo-M 1-1 rH «^ OOCO r-irH 00 U3 OirH r-co CO"* COO l>t- OI> OO ot- 00 t- OOD TjiN OO «i.n 'i.'^. TtCO oo .HO OO i-fiH OSb- OO t-o OO OO l>I> rHO OO OO xos S3 OO CON OTt. 31^ rHN SE: 1252 r-.H Olio OO lOOS rHN OCO ON OO OO ^CO rHO OO OO ox ^OS OO OrH 9!S CON ■ OO lAO OO com OO rH lO OO OO COOS Nvn OO OO NO OSO Tt<^ t^-* ■*co ^?^ K^ rH coo o rHOS l-lrH tJ(tJ< iHiH iH tHiH 0-* X ' rHrH tHN N rH Ti«CO xco rHrH t>i> OiKO ErE: 53 1^ ino HOJ OOO ^00 ot- OO OO OO NCO t:5e lO rH iHi-l tH NN rHO O rHrH COCM rH NrH x'"co" rHrH t>I> ooo 52^ INO toos OO OO OO NCO C01> rHt- OCO n!S OrH MCO iHrH NrH NN cqo rn" m 1-1 rn" t-N .. t>rH rHrH o"o ©1ft (MOS rHO ^O N OS OCO «cS iftN N OS rH OiO i-H MrH rHiH CON XrH rH TtlX m^ Er92 ODO (NQO iHOO i>o COrH GO'S* lAiO ON oos tn£> OO OO OS CO cox OO OO rHrH OX c>.x es^ ^'^ CMOS mco OCO OtH IH Wr-I rHiH CNiH NN rH^O rH^rH rn" CO tiTjv rH lo"-*' 00 r- -oi-^ OSO t-o Ot-H Nt> NO Its OS ON C3SX OrH OO OO OS OS OO OO tJIO XO ION XtO 00 -# SS; 00^ 00>f3 [^<«t4 X 1ft CO r^ tHf^ rHrH tH iHr-t T-lrH COrH NN OO rH rH rH ■d OSN rHlft i^t-i I>^ 00 CO ON ON ON oin ON rHX rHO OO ■<*N !>■* t-iO OS CO 2IS (M(N I>CO rH »Oi-( T-(r-l »H rHiH N^ NN OSlft T-^ ION 1-1 lO rH NrH 00l> rH lOTjT 92 CO 00 T*.^ OS CO I>CO — op^jj- ON ot- COO NO 5!^ OSO coos QOOJ om C3S rH rH iHiH rH rH rH CO 1-i rHt> rH 92 "i oim Nth NrH oc- Tjllft XX OO OO Oift ^S mco OI> NOQ tH T-( rHN rH rH OSt- CAW to in OlO OI> OO OON THin oco Q^ o-^ OOS OOS !>■* rHt- rHO OO OO Lft ^ ION ON lON 5SS rH T-iT-i coco ^^^ c^o" M-m ti°o NOO CTSlft OO rHO OO rHrH NCO rHO rdS ^C0 OO CON coco rH iHrH rHN ift rn" l>t^ co"-* OS CO D-00 ■^m tHO COX (Mr-f t-OH I>«* OtH xm OO OI> OSlft i>o NO OO OO COOS TJ(0 OS OS IftCO 04 to CON rHrH rHrH OrH ^^.M iHrH rHiH NCQ rH CO Ift NrH INO coco" t^'* OiM ^o N91 oos OrH i>o Olft XN OO OO N'* OOS OSOT xco iH NW rH rHrH lO"-* (pfN- MO 00^ inrH TttTti OrH O^ iHr-t iH i-i T-i rHi-i Nt-I OSN co"n s-::^ Nt^ OrH ox OO OO COrH OO rHI> Ot- T-I(N t-os CO CM (MiH (MtH iHiH iHrH 'rHi-H ■<*■* rH"N OSt- ''^vO Ot- rHO i>o OO 1ft Ift COO NCO OO OO C^OS OO OX 0-* lOO t-l 00 OS iHrH iHiH N-* rHrH rHrH COCO__ rH^rn" ss r^O oo OO OO COCO OSO NrH iH i-MiH iHi-H NN OS_^l>^ rH''r-r OiW OSM 0-* OOS COOS COCO OO OO OS rH OO OSlO COT* rHt> «#IH H-^ i>ir5 00 OS OO OO OO OO iftX Xl> cnira OO OO cor- OO rHrH XN cot- NO ^-* jH rHrH rHrH \n-^ ift(M 1-t ■<*cc NN xt> isTo co"co" OS^ r-O OrH U^N OO Tt tHO cox OO CON OO CDift 03 CO m(M NX ■*N osvo rH 00 «■* CM 01 NrH rH rHrH T*r^ r-l iHi-l NOS rHrH N'rH' ■ss rHrH SfH Sli, a^ Hh aii( I^Ph a^ SfR gh !^N Sh l^fM Sh Sn 1^;^ ah :^f^ S;^ ^^ 1^^ SfR Sph gPH * * 4 q-)CQ o li 1 o go 3 ■3 * ii l| IS li «■ so 1 CO 1- -.1 >, ii ^8 1. li Si r a i M ■s s 1: II ■a r |l « R EH 1> It ll CM O p P < w Q o <1 o PM o p p o o 33 ■^ do Q e ■a o o X ss ii 00-**QO 04^ OO ss (MrH ss is i m 00+^00 §§ OO rH §?s ss O»Q0 coco ss oo rH tJI ss t-co OO § . o ■ o com OOO rHrH OO rHrH COo OtH §s oo oo OO ON S3 oo .s •s D-O ss OO is rHrH lOiO mt« tHfH ss f^O OrH SS ss •«*co oo is oo ii S3 S8 ss CO eg coo 23 s^ ii =3 §§ oo Ota oo oo ss CM rH rHrH com oo ss SS '^CO oo rHrH s§ ss §§ OO oo tHCO OSkO CO 00 00 lO r-r-( CM CO mr- r-rH So S3 s tO-SiO oo mot OOO a oo oo mr- ■^r- oo oo oo oo iHO oo oo rH(M oo oo r-ir-i ss OTtI oo lAO OTti t-(M S55 mrH r-T*l H oo oo oo oo oo rHrH oo oo OrH OO oo « t-^ om mm rHOO ooS fl §55 oo oo oo oo o-<* com o 5 oo oo oo OrH oo oo OO oo O0I> SISg oo oo da 00 oo l-t1-i mcD rHO (MrH § oo oo oo oo oo OO oo oo_ coco co-^ 1-tCO oo t*«o COTji rHO t>-<*l om COOi i>o c--* 00 CO «-»3M (?IH« t^TH t^r^ Oco rHrH mco mco (MCO i oo oo oo oo cq00 coo cDO ^-rH rHCD 0(M rH'* "3 S5§ OOO I>rH IV CO rHrH .g 03 3 oo oo oo oo (Mcq oo oo oo '^ mco 00 00 oo 00 1- fit §ss 1 op oo *=* oo oo r-<(M oo oo oo OO OO oo \a 3SS ana OI> o-<* oco coco too rHffI rH©I a 1 OrH oo rHTH Oi-H y-ta^ rHrH coco •*-* cot- oo oo oo *=* j-i'SZ oo OO OO OO oo oo oo oo tKCD «Oi-l eoo r-I> 1 eoi> (MCq oo m^ji s oo oo oo oo oo oo oo coo •«* coco OOrH !>■* rHCO coco mm 00 CD moo C30 '^m t-OO CDO eoo om oa taS'^ oeo t-co rHTH rHCO ta-^ o oo oo .-KM oo oo OO oo oo oo oo oo ^ «c- ^-(^^ CD(N 00 CO r-(CO -*o tH COrH ■4 CD 00 la tnco eocn ot- CM CO CM CO COO •-I oo Tjieo oo oo (M(M r-lrH g oo oo oo OO OO oo OO OO oo oo oo oo oo oo oo cq.H «00 o tM 00 ■«*l> r^ t* coo t-r-1 OOS oot osq (MOS OOS cocq 00 OlTji CO 00 tJItH i>o 00 (A ooo QO 22 22 22 ss =?'= rH rH 00 00 ot- C-OO 00 CO rHO rH r^ R9 OO OO OO OO OO 00 00 00 00 00 00 00 00 00 00 r-!r-I (Ccn i!0(M o 0(M 00 en C4 00 •7)^ ai<£> OSrH OOS 00 T* •rH 00 O'* o»o OrH OrH (M'* 29 OO OO ° 00 c-t- 00 '0 rHr-I IMO os^ COrH §s A OS 00 CO 00 OS t-co 00 QDO lOt- (Mcq • O'**" (M • rHrH eqo 00 tHrH rHCq 00 (MiH rH - rHrH • coo (M(M 00 OO ° OO ° 00 00 00 00 00 00 0-* COI> (MO CON tt: xo r-o 000 oso 000 coo 01 C^ coirt tHCO COtH (MtJ* Oi-l iH rHrH coo 000 coo COO r-l(M 04 (» OO 00 iHi-l (MO rHO OOS rHCO 00 OO OO OO OO 00 00 00 00 00 00 00 rHrH tnoa C40 «« <* (MO (MO iHrH (3> OI> rHCO CO(M OS CO COrH OrH 00 H-^ iOtr- OO o rHO 0-* t-(M (Moo 00 00 eo cq • t-o 00 03 C3 OO !NIN o OO (MrH coo rHO t^r^ ^(M • 00 00 OO OO OO o OO OO 00 00 00 00 00 ■<*o 00 00 T-irA iAtI< iHOi OJOO ■WtH i-ICO OOi Oi ooo 00 oco 'fUT^ rH t- 00 ■^Ttl TjffS OO t-QD tXXt ^0 oso ooo rHrH (MO 00^ 00 coco eqco OO OO 00 (M OrH t^S cq(M 00 00 00 OO OO OO OO OO OO 00 00 00 00 00 coco 00 00 00 rHrH 01i-4 t-00 «^ OO OO r-(00 T»IC^ rHO t-oo OTil rHCO 000 Ooo 00 ^-lO coo OO 01 CO 000 00 00 (MOO rHO 00 MCO CM CO OO OrH OrH 00 NrH r-i 00 ss OO OO OO OO OO 00 00 00 00 00 (M(M So 00 00 1-irA i>.o OiO t^N OO ooo ::!«* tno Oi» OrH 0(M 00 00 oq(M 00 t-OS (MOi rHrH ON ooo (MOD iHO 00 r-00 ot- coco 000 OCJ cqcq OO tH>-H OO o-* coo 00 00 00 COrH rHO 90 OO OO OO OO OO 00 00 °' 00 00 00 rHiH 00 00 00 rHrH CO 00 t-in cqt^ a>'# ooo coo tHO COtH 00 OS 00 000 (Mt- oso 00 is^ t-CD fHOI Oi-* (M03 ocq 00 rH OS T-i rH ■*o tH ^^ t-co as fr '^SE OOS 00 -*CO -*C0 00 CO 00 o»oo r-QO ■*{pl oco O'* ■<*00 00 ^ coo 00 rHO o-* 00 t- r>co 00 eoc1-< (Ml> ■*« 00 -* OO ooo r-o OI> 0(M OCO OOS 00 fHH OO OrH (Mi-i f-llM t-t- 00 00 t-{M COrH 00 OO OO OO OO OO OO 00 00 ° 00 00 00 00 00 00 00 rHrH ' iHO COi-H OO 00 o> 00-* co ■rH^OO 00 o^ OO I-lTjt t-o (MO t-TtI rH coo rHO go T-i COCO rHO OS(M t-o OOrtl So ooo OOO) 00-* Or-) t-o ooa OS rHO rH 00 00 eoiM oJ^ OO Oi-i (M(M rH(M coo rH rHO 00 00 rHCq COrH cqcq OO OO OO OO OO OO 00 00 00 ° 00 00 ° 00 rHO 00 00 rHrH ' iHrH t>- CXM OCO rtiO CO "cq~ (Mt* OS 00 (M oqo (M(M 000 00 So Or-C 00 171 t-o t>o OJO coco 00 ot- t>(M rHO 00 CQIM OtH Oi-l OrH iHi-t tHCO s rH 00 00 cocq ■<*rH MrH 22 OO OO OO oo OO OO 00 00 00 00 00 ° rHO 00 00 rHrH — ob» C30 00 00 i-Hl> iNt> OO (^^Til t-o 00 rH OSrH rH t-rH ta-rp 04r-i ■00 IMrH ooo cseo coo OnH r-IO 01 r- 00 ts ^^ THrH ow 00 CO 00 oan Or-I rHr-( T-ir-l iH(M T*lO 00 00 oco OrH rHrH 22 - 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Oti-i -^o MI> i>o ■^ (MCO 00 Oco oco rH 00 taj-* oco 00 ^vb D-OS OtH OO 00 i> 00 (M ■rf"* 00 00 (MrH 00 OOS 00 OO (TQi— 1 OO rHiM ■>*■* cq 00 00 ■^1-t r^ 00 OO op OO OO OO OO 00 00 00 r^ 00 00 ° (MO 00 00 rHrH l>^ lOffl r-o i-(0 OrH too cot- C-CX) or- 00 0(M (MO Oco "*0 00 OT03 OOi eoD- o-* rHr-l oco 00 iH w oco ■* IM(M (MrH OEM ■d<^ 0)00 §s OO OO COrH OO (30 r-l(M coo rH 00 00 00 00 coTh COrH 00 OO OO OO OO OO 00 00 00 00 00 ° (MO 00 00 rHrH II OrH coo • OS CO CO rH COtJI (M 000 iM (MO 00 rH, o-* r-1rH rHCq 00 COI> r-lrH tH rH(M rH lO ■*rH' rHO tH^ 00 OO ^co OO 2 22 S3 CO 22 2° rHrH 00 OO OO OO OO OO OO 00 00 00 00 00 00 (MO 00 00 r-IrH "is" ■■ »ft^ i-tOi (MO Oi-i 1-1 C-- iHI> 000 CO (MCO CO "* T*(r- 000 cqeo (Xlt- ^00 00 rt*-^ ^iH i>o OS iHCS r-00 t> 00 (MrH ass (MO ■*-* ^(M CO 00 eOTji (NfiO OO (MCO TH&q 22 00 rHO 00 00 rH OO OO OO OO OO OO 00 00 ° 00 00 00 00 rHrH 00 rHrH*. 00 00 i>o oc- 00 250 000 00 t-00 00 00 00 Sot ^I> rHCq oso (MOt coco 00 rH rHCO 00 OO OO iHr-l 1=12 22 (M-* 1~i1-1 00 OO OO OO OO OO o 00 00 00 00 00 00 r-1r^ 00 r-JrH irteo AGO ss coc^ ^Oq OSt- rHCO 122 t>oi 0(M (M(M 00 ss OO ooo 0-* -*o 00 OS OS t-n 00 OO OO OO liJS? =!2 29 ^^ rHrH 00 OO r-(0 OO OO OO 00 00 00 00 00 00 rHrH 00 So c-loq r-40 t>l> o COrH lie COrH (MCO (MCO oso 00 000 00 ooo ^o ooo ■iH 1H1H 030 OrH 522 S!S9 CO 00 CiCO 00 00 •dio OO • o y-t^ 2*^ 22 00 00 rHO rHO 1-^1-A 00 OO ^^ OO •OO o 00 00 00 00 00 00 rH?H 00 rHrH S8 gg 1— lO COt- OO oaeo CO 00 eo« Ost- COO oco rHOS OS(M 00 OO tH-* t-00 rH(M (Mcq ■*rH coo ot- 00 OO oq^ OO ■=2 22 00 22 22 rHO 00 03 CO 00 So MW OO OO OO 00 00 00 00 00 00 00 00 rHrH rHiH iHOa gg OO (Mcq o» coeo Ei 00 cocq t-o 00 (M t-o t-co 00 si cq^ or- rHO 00 00 os» ^t- fr- 1> OO tHO 2 22 t-o -*■«* rHrH 1-ty-i 00 OO (MOq OO OO OO 00 00 00 00 00 coco 00 00 00 rHrH Sn Sp^ SfH 3n !^P^ ^[^ ;^fh Sph fH Sn N fe Sfe ^^ s^ Sh SfH ^fR :^^ ^1^ Sn s if 1 1 h ll i H 1 1 II 3 .23 II li i ll f li a . ! 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(30 oq C-rH COO in 00 rH>A o t^ia OSC4 oo oo ■^ffi ifliM rHOT i>m »I> rHM rHrH WrH T-fy-i (MM OOO rH s rH "" m(S rHrH m{D Mao ^eo OM MCM (M-* MO Cii-i MI> mm iH rH r-m rH rH rn" ri'*""' (M°0 rH S9S eoco OS^ SR3 oo lO^Ji 1>00 312 COCO insD ■^O to OS tM rHrH rH M rH_rH r-T oTco" rH^m ori> lOOT COtH t>M rHOO <£lr^ 00 rH r-^i^i M (MCO (30 M ■*OS MCO oo oo OrH OOT I>00 rHt* 00 r- CO 00 00 50 rH mos rHrH MOT rH (3rH rn" M rH co''m'' 9« i>o (MOO rHQO (MM MrH i>i5q MiO I>0 ift iftM (M ■^ i>© i>in fHO oo OO r-lOT OCO i>o> OrH 0(M iflin C-00 i>m rH(M T-i OT lO m-* m''m I>M COCO r-t rHO oo OO OS-* COiA mo oom mm MM (M(M r^r^ 00 (M SS OS (M "* 1-IrH 5S3 MOT oo oci oo LA 00 I-HCO NCO rH moo i-HrH CD rHrH S OS (M 00 rH rH coco* -hTm" oqin i>o ■ OO rHM ME- MfM Mm OTOS OS iAtH t- OS 00 00 (M(M TiirH oo OO CS-* COOS MtJI ■*rH OT-* iH OS 00 I> rH COrH mtM (M(M o^eo COtM" oo {M(M OO) tH M-* M«5 Tl-M OOS oo OO ost- oo OSI>- mM (M^ rHrH (Oi> -*-* ^S3 (MO im''(m" Cqo U3CO o I>1> o o r^rH QOM coos oo OO I>iO oo lOrH OOS OO (MrH iH OS OS OS oo oo t*rH oo C»0 mo4 oo t*rH OSOi com OO OrH 00^ iHO ■^o ■^Ci ■^o sOO ■<4<(M o 0(M o o 0£- eota COiO oo oo ^I> oo rHD- TtirH OO iHlO fi rHrH tHiH »Ortl LAM rHrH OOO rHrH rHQO COTjt OOO i>'^ om iftiO ■*M 00 ■*M o o tHOO MM OCO oo oo ■«ilO oo MIA 00 OS OO >O00 -*'* 01 rH 00 »« MM !MrH MM SS fes^ £S OOO 1-t Sh ^p^ a^ g^ :aii. Sn Sfe !^;^ pR Sn ^ PR ;^ph Sf^ Sn Sn !^l^ gh gfH Sh a^ Sfq o 1 i: i ■11 ^1 ^1 s i! Si ii 1 1 "o-a Is i' 5 a Ij 1 M p •3 • ■ la ■'3 s 1 ■^ •a ta <§ : !i ^1 X gg rH g lit ill -t- s OS rH u M o fi s -«! M » o < <^ PM *^ n *-* '-' oS > ^ c -flPH o«ooi*i©i-*t»o ing for migration (on which g SSSg SgSRSSffKSgg the results are based). co««o9ot o3oo«cop3«oomcoimoo Total Deaths (Males) on which gSISS S8§SSiS§S3S?S8 results are based ....<» o>. ", ". ^ « » t- lO S ^1 S o § § S en Aqe at Death. Aqe at Death. TOTAL 1 2 3 1 2 3 4 5 6 7 8 9 10 11 week weeks weeks mth. mths. mths. mths, 1 mths. mths. mths. mths. mths. mths. mths. No. Females— Cause. Under and and and Under and and and and ' and and and and and and and Under I under under under 1 under under under under under under under under under under under 1 week. 2 3 1 Mth. 2 3 4 5 6 7 8 9 10 11 12 Year. weeks weeks mth. mths. mths. mths. mths, , mths. mths. mths. mths. mths. mths. mths. 8 Whooping Cough 1 2 20 24 132 398 310 202 162 85 108 122 152 122 106 106 2,008 28 Pulmonary Tuberculosis . . 1 2 8 3 3 8 14 8 14 8 11 8 85 29 Acute Miliary 3 8 3 8 22 30 Tubercular Meningitis 14 22 22 27 19 22 25 33 42 39 264 31 Abdominal Tuberculosis "1 "3 5 16 14 25 16 19 3 17 8 22 3 151 34 Tuberculosis of Other Organs 1 3 3 6 35 Disseminated - Tuberculosis 5 I 3 3 11 37 Syphilis '22 12 '20 'u i76 135 65 107 57 1 41 58 28 17 11 25 17 739 61 Meningitis 40 27 14 7 222 87 102 128 140 140 135 169 150 172 139 137 1,715 71 Convulsions 286 116 43 29 1,151 146 117 87 82 93 94 75 69 103 70 67 2,157 1,756 2,357 1,657 18,456 89 Acute Bronchitis 15 40 57 35 379 357 250 158 132 80 72 72 69 81 50 47 91 Broncho-Pneiunonia 14 28 33 32 283 372 277 191 153 154 163 122 164 194 153 132 92 Pneumonia 25 22 26 17 228 173 106 137 118 146 149 136 103 133 136 95 104 Diarrhoea and Enteritis . . 37 96 106 95 835 1,432 1,876 2,144 2,197 1,937 1,666 1,544 1,328 1,368 1,159 992 109 Hernia, Intestinal Obstruction 18 13 9 99 32 11 27 57 77 77 53 44 25 19 11 530 150 Malformations 597 122 62 40 2,003 203 128 104 99 92 80 47 36 61 56 59 2,970 151 Congenital Debility, Icterus and Sclerema 5,579 809 518 289 17,375 1,418 878 710 419 352 265 243 155 142 128 114 22,226 152 Other Diseases peculiar to early infancy 1,329 147 64 24 3,750 73 38 i 3,863 79 153 Lack of Care 30 74 5 Other Causes 343 152 130 58 1,670 649 343 355 293 223 254 252 283 306 328 306 5,262 8,337 1,586 1,095 675 28,391 5,483 1 4,536 4,394 1 3,967 j 3,485 3,173 2,902 2,620 1 2,773 1 2,436 2,141 Population of fematesat the be- ^SS^'S'SSSKSS""^ _ 9 ginning of each period allow- » S § S 3 S g 3 S S S fe S S S 2 ing for migration (on which § S S e: £ g g" S 2 S" -f '-' o- °>" oo 00" the results are based). SSoSSS SggSSgSSgg^g TotaIDeaths(Females)onwhich gSgS SSS'-'-°<=>'aaoiaao'a results are based .. .. § S § K o » i 3 S S S 1 1 g g S *"* eq rH r-t iH ri r-l iH s MORTALITY. 423 {(016- 152 Wosf— t— I Wosl— J— Hz:35i Irf— 150. % 3 61 9 lit . ^j 004- t T -71 00 1 00 1 1 8S 61 ■^ = 3 6 9 li ;00^'4 -t 92 — L r -jY 0004- o' f ^ / "v. , . ■^ 1 a? N ^•^ V - — m i 3-6- 9 izmths. .QO''^'*' 1 1 nn\ ■iv"! L s^ -*; =». ■0004 Fig. 104. The 13 figures ruled into rectangles are death-rates for the first 12 months of life, the rates being shewn by the figures on one of the horizontal lines. The 38 figures ruled into smaller squares shew the death-rates for all ages of the diseases indicated by the numbers. For the index to the above curves see next page. 424 APPENDIX A. Index to Coives in Figure 104. Death-bates foe all Ages. +1. Typhoid Fever 6. Measles. 7. Whooping Cough. 8. Diphtheria and Croup. 9. Influenza. 12. Other Epidemic Diseases. 13. Tuberculosis of the I.ungs. 14. Tuberculous Meningitis. 15. Other forms of Tuberculosis. 16. Cancer and other Malignant Tumours. 16a. Other General Diseases. 17. Simple Meningitis. 18. Cerebral Haemorrhage and Softening. 18o. Other Diseases of the Nervous System. 19. Organic Diseases of the Heart. 190. the the Other Diseases of Circulatory System. 20. Acute Bronchitis. 21. Chronic Bronchitis. 22. Pneumonia. 23. Other Diseases of Kespiiatory System. 24. Diseases of the Stomach. 25. Diarrhoea and Enteritis (all ages). 26. Appendicitis and TypUitis. 27. Hernia, Intestinal Obstruc- tion. 28. Cirrhosis of Liver. 28a. Other Diseases of the Digestive System. 29. Acute Nephritis and Brigfat's Disease. 30. Non-cancerous Tumours of Female Genital Organs. 30o. Other Diseases of the Genito-urinary System. 31. Puerperal Septicsemia. 32. Other Accidents of Preg- nancy and Labour. 32ff. Diseases of the Skin and Cellular Tissue. 326. Diseases of the Organs of Locomotion. 33. Congenital Debility and Malformations. 34. SenUe Debility. 35. Violent Death (Suicide ex- cepted). 36. Suicide. 38. Unknown or Hl-de fined Diseases. fli Broncho-Pneumonia. 150. Malformations. 92. Pneumonia. 151. Congenital Debility. 104 Diarrhoea and Enteritis. 152. Other Diseases peculiar to 109. Hernia and Intestinal Ob- Early Infancy. struction. A. Other Causes. Death-bates fob First Yeae op Life. •8. Whooping Cough. 37. Syphilis. 61. Meningitis. 71. Convulsions. 89. Acute Bronchitis. * These numbers, on Fig. 104, are identical with those of the "Detailed Nomenclatures of Diseases" of the International Ojmmission, Session July 1909, at Paris. t These numbers, on Fig. 104, are identical with those in Table CXLIX. to CLI., and where not marked "a" are those of the "Abridged Nomenclature" of diseases of 1909, where " a " or " b" added it denotes that the balance for the class in question is included. The form of the mortality curves during the first year are given on the upper part of Fig. 104 ; see the Index thereto. 32 . Annual fluctuation of death-rates. — The frequency of death from particular causes, and therefore generally, is afiEected by the season of the year, and though in the aggregate of deaths from all causes the seasonal effect is somewhat masked, it is not whoUy obhterated. To ascertain rigorously the character of the annual periodicity, either generally or from a particular " cause," of death it is necessary to obtain the rates for smaU units of time, say equalised months ; thus the rates 8i, 82, ... . 812 must be obtained : these are sensibly iadependent of the fluctuations in the deaths and population during the month. Inasmuch, however, as deaths occur very rapidly in the first few days of hfe, any periodicity in birth-rate involves the death-rate ; that is to say, the constitution of the population is not quite homogeneous, and a correction is — ^theoretic- ally — ^necessary. The correction, however, is so small that it may be neglected. These last observations apply, mutatis mutandis, also to deaths from certain particular causes. The annual fluctuations of birth- rate, and the mode of solving have been indicated at length in Part XI., §§ 14-19, pp. 166-174. General factors for reducing the values given for calendar months to the values for equahsed months must be so apphed as to have regard to the average values at the beginning and end of the months. Table CLIII. depends upon a total of 252,443 deaths of males ^, and 185,367 deaths of females occurring in an aggregate population of > For example there were 3529 deaths from typhoid in the 9 years, of which 473 occurred in the month of January. These, when corrected, for the growth of population during the year, and altered so as to give the result for the exact twelfth of the mean length of the year, gave the basis for the calculation of the results in the table. MORTALITY. 425 over 21,000,000 males and nearly 20,000,000 females. The numbers given in the table correspond to a population of 10 millions in each case. In Table CLIV. the proportions of deaths occurring in months of equal length, when the population is constantly the same, are given. Algebraically if b and e be the equalising corrections at the beginning and end of the month to D, the number of deaths, and P be the sum of the populations of the corresponding month for the whole period under review, the results in Tables CLIII. and CLIV. are respectively : — (650). .8 = {D+b+e)/P ■ (651) p = 128 / US. TABLE CLIII. — Shewing Average Number of Deaths due to Various Causes, per 10,000,000 Males, and per 10,000,000 Females respectively of all Ages during each Equalised Month of the Year. Based upon 9 Years' Experience (1907-1915) in Australia. Cause op death. Sex Jan. Feb. Mar. April. May. June. July. Aug. Sept. Oct. Nov. Dec. Year. Typhoid Fever M 221 231 247 223 167 108 63 54 60 51 92 164 1,671 F 12S 152 156 118 97 80 41 26 24 25 50 88 986 Whooping Cough M 77 64 43 45 61 63 66 79 85 74 77 77 811 I'' IOC 83 71 63 64 63 92 94 102 102 94 89 1,017 Diphtheria and Croup M 81 76 99 123 149 153 142 109 102 85 83 89 1,291 i' 95 88 86 147 151 165 145 128 116 90 107 83 1,401 Influenza M 39 27 29 38 54 69 99 163 188 140 93 49 988 H' 48 27 28 36 45 56 94 160 191 146 97 59 987 Tuberculosis M 613 581 585 58C 646 641 713 701 692 669 640 590 7,651 If 527 484 489 514 511 512 533 590 558 585 496 516 6,315 Cancer M 637 659 604 638 603 594 571 595 619 643 622 658 7,443 i!' 613 628 613 615 623 593 578 587 588 610 615 605 7,268 Diabetes M 58 56 62 65 66 86 78 85 73 73 65 68 835 It 74 74 68 80 82 94 102 98 109 99 90 93 1,063 Organic Diseases of the . . M 855 784 802 832 903 995 1,052 1,070 994 925 884 794 10,890 Heart F 725 613 650 667 697 834 950 891 780 744 629 670 8,850 Diseases ol the Eespiratory M 757 646 743 844 1,000 1,250 1,500 1,594 1,519 1,197 1,042 830 12,922 System \f 519 472 471 581 723 895 1,083 1,217 1,088 895 726 609 9,279 Diarrhbea and Enteritis . . M 1,021 941 866 764 503 265 203 166 185 338 782 1,069 7,103 V 894 820 787 678 457 264 164 127 137 309 644 895 6,176 Infancy M 663 697 695 703 686 719 734 656 683 614 666 680 8,196 F 58C 543 571 608 562 579 616 561 504 528 521 532 6,705 Old Age M 692 664 629 671 754 857 905 873 836 733 722 732 9,068 M 629 567 548 566 631 .697 748 726 688 628 581 570 7,579 Total all Causes 10,406 9,681 9,469 9,633 9,604 9,881 10,411 10,309 10,215 9,570 9,984 10,146 109309 i' 8,152 7,667 7,391 7,724 7,702 7,897 8,279 8,411 7,895 7,697 7,802 7,967 94,584 TABLE CLIV. — Shewing for each Equalised Month the Average Relative Frequency of Death due to Various Causes, the Population being Constant throughout the Year. Based upon 9 Years' Experience (1907-1915). AustraUa. Cause of De.ath, Sex Jan. Feb. Mar. AprU. May. June. July. Aug. Sept. Oct. Nov. Dec. Year. Typhoid Fever M 1.589 1.657 1.773 1.599 1.199 .779 .449 .391 .357 .367 .663 1.179 12.000 H' 1.567 1.844 1.905 1.436 1.178 .979 .494 .320 .288 .304 .612 1.071 12.000 Whooping Cough . . M 1.139 .942 .630 .664 .910 .930 .975 1.176 1.258 1.095 1.138 1.144 12.000 1'' 1,185 .978 .835 .742 .761 .744 1.082 1.107 1.205 1.206 1.105 1.052 12.000 Diphtheria and Croup M .752 .706 .921 1.141 1.381 1.426 1.316 1.015 .948 .790 .776 .829 12.000 F .81C .752 .739 1.258 1.295 1.414 1.241 1.098 .994 .774 .914 .711 12.000 Influenza M .468 .332 .349 .463 .657 .840 1.199 1.982 2.281 1.707 1.134 .593 12.000 it .58C .332 .345 .434 .550 .676 1.140 1.942 2 325 1.779 1.182 .714 12.000 Tuberculosis M .961 .912 .917 .910 1.012 1.006 1.118 1.100 1.086 1.049 1.003 .926 12.000 F 1.001 .919 .929 .976 .971 .973 1.013 1.121 1.061 1.112 .943 .981 12.000 Cancer M 1.027 1.062 .975 1.029 .972 .958 .921 .959 .998 1.036 1.164 1.061 12.000 F 1.011 1.037 1.012 1.015 1.029 .979 .954 .969 .971 1.007 1.016 .998 12.000 Diabetes M .839 .811 .888 .927 .952 1.235 1123 1.224 ' 1.047 1.041 .938 .975 12.000 F .834 .837 .767 .901 .929 1.056 1.148 1.102 1.232 1.124 1.012 1.052 12.000 Organic Diseases of the M .942 .864 .884 .917 .995 1.096 1.159 1.179 1.095 1.O20 .974 .875 12.000 Heart F .983 .831 .882 .904 .945 1.130 1.288 1.208 1.058 1.008 .853 .909 12.000 Diseases of the Eespiratory M .703 .599 .690 .784 .929 1.161 1.392 1.480 1.411 1.112 .968 .771 12.000 System F .671 .611 .610 .752 .935 1.157 1.400 1.573 1.407 1.157 .939 .788 12.000 Diarrhoea and Enteritis M 1.725 1.590 1.463 1.290 .850 .448 .344 .281 .312 .570 1.321 1.805 12.000 F 1.737 1.594 1.530 1.317 .889 .512 .319 .246 .266 .600 1.252 1.739 12.000 Infancy M .971 1.020 1.017 1.030 1.004 1,053 1.075 .961 .999 .899 .975 .996 12.000 F 1.038 .971 1.002 1.089 1.005 1.037 1.102 1.003 .902 .945 .933 .951 12.000 Old Age M .916 .878 .833 .888 .997 1.134 1.197 1.155 1.106 .970 .955 .969 12.000 F M .996 .897 .868 .896 .999 1.103 1.184 1.150 1.089 .995 .920 .903 12.000 Total all Causes . . 1.047 .974 .952 .969 .966 .994 1.047 1.037 1.027 .963 1.004 1.020 12.000 F 1.034 .973 .938 .980 .977 1.002 1.050 1.067 1.002 .977 .990 1.011 12.000 426 APPENDIX A. Fig. 105. The distances from the centres of the circles shew the average ratios of the death- rate per month to the average rate for the entire year, the ratios for males being denoted by firm lines, and those for females by dotted lines, the succession of months being clockwise. In the case of absence of fluctuation the sector-boundaries would all be on the circle marked " 1," e.g., " Cancer." In the case of " Influenza" it will be seen that the September rate is more than double the average for the year. 33. Studies of particular causes of death : voluntary death. — ^Although the study of particular causes of death might appear not to belong to the general theory of population, it is really an essential. For example, if diseases, the incidence of which is characteristic of earlier life, be com- batted, the consequence will be an increase in deaths from those which MORTALITY. 427 characterise later years {e.g., tuberculosis and cancer). Again statistics of voluntary death or suicide, are of special importance, inasmuch as they disclose the regularity of human conduct even in matters which might be thought to be peculiarly under individual control, and be imagined to lie outside regular law. But suicide follows well-defined laws, and even as regards the mode of death the regularity is remarkable, as the following table shews : — TABLE CLV.— Mode Of Voluntary Death. Australia 1907-1 5. J Number of Suicides. * o EH Range. Mode of Death. 1907. 1908. 1909. 1910. 1911. 1912. 1913. 1914. 1915. 1 • 1 r Poison Asphyxia Hanging and Stiangulation ■ Drowning Firearms Cutting Instruments Precipitation from Height Crushing Other 57 2 71 37 129 61 6 3 19 88 1 68 31 146 54 4 6 15 70 2 67 24 138 74 7 5 11 79 72 42 134 79 3 8 15 93 2 69 43 133 65 2 6 33 128 4 79 34 168 76 8 17 127 2 79 25 163 88 6 10 16 121 2 72 30 201 76 4 2 26 105 84 38 196 89 4 8 13 868 15 661 304 1,408 662 36 56 165 .2079 .0036 .1583 .0728 .3373 .1586 .0086 .0134 .0395 57 67 24 129 54 2 11 96.4 1.7 73.4 33.8 156.4 73.6 4.0 6.2 18.3 128 4 84 38 201 89 7 10 33 92.5 2.0 75.5 31.0 165.0 71.6 3.5 6.0 22.0 Total, Males . . 385 413 398 432 446 514 516 534 537 4,175 1.0000 385 464 537 461 ^Poison Asphyxia Hanging and Strangulation Drowning Firearms Cutting Instruments Precipitation from Height Crushing Other 32 12 19 3 5 1 2 2 35 15 14 7 6 2 2 3 54 9 19 6 5 1 3 34 10 19 6 13 2 52 1 10 13 9 9 2 2 70 12 11 10 8 1 6 76 1 22 14 9 4 2 1 2 61 15 17 4 3 4 2 3 64 1 18 21 5 6 3 2 2 478 3 123 147 59 59 14 11 25 .5201 .0033 .1338 .1600 .0642 .0642 .0152 .0120 .0272 32 9 11 3 3 2 53.1 0.3 13.7 16.3 6.6 6.6 1.6 1.2 2.8 76 1 22 21 10 13 4 2 6 54.0 0.5 15.5 16.0 6.5 8.0 2.0 2.0 4.0 Total, Females 76 84 97 84 98 118 131 109 122 919 1.0000 76 102 131 103 Ratio of Females to Males Ratio of Males to Females .197 5.07 .203 4.92 .244 4.10 .194 5.16 .220 4.56 .230 4.36 .254 3.94 .204 4.90 .227 4.40 .220 4.54 .194 3.94 .219 4.60 .254 5.16 2.37 4.55 * It is worthy of note that the mean of the highest and lowest number of suicides in any year is sensibly equal to the arithmetic mean. The male population increased about 18.40 per cent, on the period covered, and the female 21.82 per cent. The ratio of the total females of age 16 and above, to the total males of 16 and above, was about 1.10904, and of 21 and above was 1.12391. This would indicate a frequency of 4.097, or 4.042 to 1 for male, as com- pared with female suicides. But this relative frequency is very variable. On the whole it is rapidly increasing. The ratios of the death-rates of males and females according to age are as follow, viz. : — Age. Ratio of Death Bates Smoothed Ratio 10-14 1.7 .74 15-19 0.97 1.37 20-24 2.15 25-29 3.01 2.62 30-34 3.25 3.25 35-39 4.32 3.88 40-44 5.09 4.51 I 45-49 50-54 6.17 3.93 5.13 5.76 55-59 5.14 6.39 60-64 7.86 7.30 65-69 10.67 8.80 70-74 9.07 11.50 75-79 14.27 15.60 80-84 28.79 28.80 85-89 These results shew that the ratio of the rate of suicide by men to that of suicide by women increases about 0.125 per annum till about age 60, when it becomes more rapid. The general result is, that this rate p can be expressed between the ages 10 and 57.5 as : — (652). .p = 0.1256 {X — 6.63) 428 APPENDIX A. after which the points he upon the curve indicated by the numbers 6.39, 7.30, etc., in the preceding result as smoothed. The annual fluctuation of suicide is fairly well-defined . By correcting the results so as to make them represent what would have been furnished by records of equal months, and a constant population ^ (as at the middle of the period), the following values are obtained, viz. :— TABLE CLVI. — Number of Suicides per diem in a Population of 1,000,000 Persons. Australia, 1900 to 1915. Period. Jan. Feb. Mar. April. May. June. July. Aug. Sept. Oct. Nov. Dec. 1900-1909' 1907-1915 .359 .376 .371 .356 .335 .326 .336 .335 .310 .306 .284 .262 .301 .346 .326 .295 .307 .351 .353 .358 .323 .356 .345 .381 Mean . . .367 .364 .330 .335 .308 .273 .324 .310 .329 .3'56 .340 .363 These results are given by 0.3291 + 0.0354 sin (x + 72° 4') — 0.0117 sin 2 (x + 73°.22') I sin 3 (x + 12° 49')— 0.0142 sin Hx + 40° 520—0.0131 sin 5 (x+fy.W) + 0.0104 sin urn. Boy. Soo. N.S.W., xlv., p. 99. + 0.0031 am a va: -f- la as ;— u.ux*a sin 6x : Journ. Boy. Soo. N.S.W., xlv., p. 99. The final mean results probably do not define the curve representing an indefinitely large number of cases. The results given are based upon only about 10,000 oases', and at least 10 times this number would be necessary to get satisfactory results. The distribution is more likely to be of the form. * (653). .y= A+ B sin x-\- G cos x= A+b am {x+P)+c COS {x+ y) (654) . . .4=(2'i» y)/n ; B=b cos ^— c sin y ; 0=b sin ^+c cos y. '■ The population records give for the population at the middle of each month the following results, 00 omitted : — Jan. Feb. Mar. April. May. June. July. Aug. Sept. Oct. Nov. Dec. 209,686 210,012 210,338 210,662 210,983 211,305 211,657 212,039 212,421 212,834 213,278 213,723 Females — 194,153 194,513 194,873 195,054 195,055 195,056 195,442 196,211 196,981 197,766 198,567 199,369 ' See " Studies in Statistical Representation" (Statistical Applications of the Fourier series), by G. H. Knibbs, Journ. Roy. Soc, N.S.W., xlv., pp. 76-110, IQll- in particular pp. 97-110. XVI.— raGRATION. 1. Migration. — The effect of immigration, and indeed of migration generally, is to modify the age, sex, and race constitution of a community, and these facts are well illustrated in the statistics of any new country (e.g., the Commonwealth of AustraUa). Concentrations of population due to seasonable or similar influences, or from other causes, may also become a factor of importance from particular points of view. Por ex- ample, statistics of morbidity or of mortahly, the object of which is to differentiate between urban and country hygienic conditions, may be materially affected even by temporary concentrations of populations in cities ; for example, by the fact that serious impairments of health may lead to transfer to the cities for special treatment, with a consequent increase of the mortality and morbidity rates ; and so on. Certain obvious economic consequences may, too, arise from such concentrations. For these reasons statistics for particular purposes are often Hmited as regards precision. In countries where the migration of adults is a striking characteristic, the constitution of the population according to age ceases to be normal ; but the aggregates obtained by inclusion of the group of countries between which the migration takes place, tend to restore the normality. In AustraJia financial arrangements between the component States have, among other things, led to records being kept (a) of oversea migration, (6) of interstate migration by sea, and to a partial record (c) of overland migration. All of these shew fluctuations of annual period. Records of overland migration by road are not kept, but such migration is assumed to be in balance, that is to say, the immigration and emigration are supposed to be equal. It will be seen later that over- land immigration by rail virtually balances the overland emigration. 2. Proportion born in a country.- — The correlation of birth-place and age in any population is of sociologic importance. ^ In the following results, from the 1911 Austrahan Census, the " unspecified" cases (as to whether the birth-place was Australia or outside of Austraha) have, for each age-group, been distributed in the proportion of the numbers given as born in and out of Australia, respectively. The results are as shewn in Table CLVII. hereunder and in Fig. 106. These disclose the fact that the initial preponderance of persons born in Australia diminishes very rapidly with age ; this of course being due to the fact that the commence- ment of colonisation was at a point of time nearly identical with the birth of the present oldest inhabitants. 1 An analysis of the Australian population will be found in the Census Report, Vol. I., pp. 120-125. 430 APPENDIX A. TABLE CLVn. — Shewing according to Age and Sez the Proportion of Persons Living in hut not Born in Australia.^ Proportion not Born in Proportion not Proportion not Proportion not Born in Age Age Bom in Age Born in Age last Australia. last Australia. last Australia. last Australia. Birth- Birth- Birth- Birth- day. day. 1 day. day. Males. Females Males. ^Females Males. Females Males. Females .0036 .0036 15-19 .0403 .0243 50-54 .4536 .3244 85-89 .9792 .9756 ■ 1 .0106 1 .0103 20-24 .0699 .0513 55-59 .5875 .4915 90-94 .9781 .9886 "2 .0160 .0166 25-29 .1866 .1100 60-64 .7014 .6485 95-99 .9569 .9449 3 .0202 1 .0193 30-34 .2290 .1531 65-69 .7572 .7181 00 and .9143 .8966 i .0215 .0207 35-39 .2538 .1806 70-74 .8880 .8653 over 5-9 .0249 1 .0242 40-44 .3007 .2083 75-79 .8952 .9318 10-14 .0239 .0232 45-49 .3834 •2673 80-84 .9731 .9637 The results in the table are graphed in Fig. 106, the Curves M, M' and F, F' denoting respectively the results for males and females. The Proportions bom in Australia. l-(ll j 1 1 ) 1 [ II 1 ) Mil ij^^fij 1° z Vi * : V a J - 2 „ t- 3 H V* ^1 .3"' _i:?_ : t- '2 - (x) = SiA'e'^), or = 2'(4a;»»e-" in other words, it may be regarded as the sum of a series of curves of one or both of the types shewn, see formulae (23) to (39a), pp. 22 to 24, and formulse (.147) to (156), pp. 52 to 55. Like nearly all statistical curves it will probably not conform exactly to any simple expression. The variation with time will ordinarily be considerable in new countries. The characteristics of the annual fluctuations are not quite identical for the sexes : hence each of the components (T^ and Tf ) may be analysed separately, or the total ( T^, + T/ ) may be analysed, and the fluctuation of the sex-ratio, determined for individual months, may be analysed. 5. Migration-ratios for Australia. — ^The migration-ratios for Aus- traha, determined as indicated by formulae (655) to (659), are as follow : TABLE CLVin. — Shewing the Migration-ratios for Australia and the Sex-ratios of the Migration for Oversea and Interstate Sea Migration and for Migration by Railway. Oversea Migration, 1909-1913. Interstate Sea Migkaiion, 1909-1913. Interstate Migration by Railway, 1914-1916. To (I) or from (B) Males. Fe- males. Per- sons. Eatio Males to Total. Males. Fe- males. Per- sons. Ratio Males to Total. Males. Fe- males. Per- sons. Ratio Males to Total. N.S.W. I B Vic. I -B Qld. I B S. Aus. I B W. Aus. I B Tas. I B .05237 .03654 .02195 .01376 .02284 .00928 .02199 .01130 .05561 .02308 .02129 .00910 .02644 .01751 .01336 .00788 .01472 .00267 .01048 .00372 .03502 .00963 .01104 .00566 .04003 .02748 .01763 .01080 .01912 .00626 .01632 •00757 .04676 .01730 .01630 .00742 .68549 .69666 .61875 .63298 .64953 .80570 .68336 .75729 .67788 .76057 .66986 .62842 .04557 .04394 .07226 .07313 .05516 .05028 .05873 .05502 .07460 .08202 .22592 .24873 .03251 .03095 .04807 .04766 .03251 .03137 .03534 .03463 .07288 .07593 .16258 .18082 .03935 .03775 .06009 .06032 .04482 .04165 .04720 .04497 .07386 .07940 .19505 .21564 .60666 .60976 .59749 .60242 .66902 .65623 .63097 .62041 .57566 .58876 .59379 .59163 .18966 .19104 .19580 .19071 .16804 .16970 .22646 .23406 .09635 .09447 .07766 .08047 .11287 .11476 .10576 .10170 .14426 .14406 .13582 .13474 .14238 .14413 .16490 .16655 .67506 .68094 .70974 .69681 .63129 .62984 .67288 .68857 The table shews that as regards oversea migration, immigration is preponderant : in interstate sea migration it is also generally preponder- ant, the exceptions being— Victoria, " males" and " persons" ; Western Austraha, " males," " females" and " persons." Interstate migration by railway shews an approximate equaUty between immigration and emigration, the balance on either side being variable. That these results have very accordant values from year to year will appear from the following table ; — ■ 434 APPENDIX A. TABLE CLIX.— Interstate Imm^ation by Sea, 1909-1913. 1 Migration-ratios. PjitioofMale Migrants to Total Migr'nts s Mlgration-iatios. 1 Mlgration-iatlos. 1^1 Year ^^^'■' mills} Z^: Males. Fe- males. Per- sons. Males. Fe- males. Per- sons. 1909 1910 1911 1912 1913 • CO .0401 .0408 .0487 .0488 .0454 .0278 .0294 .0347 .0359 .0320 .0343 .0354 .0420 .0427 .0390 .6134 .6026 6066 .6008 .6110 1 > .0651 .0670 .0747 .0763 .0714 .0407 .0442 .0504 .0526 .0491 .0528 .0556 .0625 .0644 .0603 .6099 .5994 .5967 .5915 .5931 ■6 a? .0575 .0521 .0559 .0522 .0557 .0323 .0323 .0323 .0311 .0320 .0460 .0430 .0452 .0425 .0448 .6801 .6571 .6714 .6641 .6717 1909 1910 1911 1912 1913 1 .0536 .0555 .0605 .0610 .0578 .0315 .0349 .0371 .0362 .0337 .0427 .0453 .0490 .0488 .0458 .6356 .6209 .6278 .6352 .6351 1 .0693 .0790 .0783 .0709 .0698 .0691 .0757 .0753 .0707 .0664 .0692 .0776 .0770 .0708 .0683 ■5705 .5808 .5811 .5694 .5756 1 .1886 .2017 .2210 .2465 .2464 .1331 .1396 .1606 .1792 .1829 .1614 .1713 .1862 .2139 .2158 .5961 .6007 .5886 .5935 .5915 Excluding Federal Territory. TABLE CLX. — Shewing ior the Years 1909 and 1913*, the Ratio of Male Migration to the Total Migrationt, and the Proportion of Males, Females and Persons, under 12 Tears of Age, to the Total Number of Emigrants. Australian Interstate Migration by Sea.t states from N.S. Wales. Victoria. Queensland. S. Australia. W. Australia. Tasmania. Masc.M..066 Masc. M. .061 Masc. M. .086 Masc. M. .161 Masc. M. .096 » .059 „ .063 „ .070 .126 J, .087 To .607 F. .095 .665 F. .112 .591 F. .109 .573 #. .212 .518 F. .079 N.S. Wales. .600 „ .085 .667 „ .118 .599 „ .103 P. .095 .598 ^ .163 .499 ., .098 P. .078 P. .078 .188 P. .087 „ .069 „ .082 „ .083 '• .141 Masc.M..061 Masc. M. .084 Masc. M. .065 Masc.M .182 Masc. M. .061 J, .063 ,. .085 „ .063 .155 „ .059 To .612 F. .086 .649 F. .119 .647 F. a22 .539 #. .196 .618 F. .088 Victoria. .578 ., .082 .600 „ .122 .677 „ .099 if. .085 .550 , .182 .600 ,, .096 P. .071 ^. .096 P. .189 P. .071 „ .071 „ .100 „ .075 » .167 Masc.M..059 Masc. M. .085 Masc. M. .033 Masc. M. .106 „ .063 „• .081 „ .109 .270 To .687 F. .126 .649 F. .136 .831 If. .160 .610 f. .067 Nil. Queensland. .678 „ .116 P. .080 .642 „ .130 !^. .103 .567 „ .155 ¥. .054 .525 ^, .263 .001 „ .080 „ .098 „ .129 » .267 Masc.M..095 Masc. M. .063 Masc. M. .018 Masc. M. .112 Masc. M. .024 „ .078 „ .052 ., .156 ,106 ,, .000 To .601 F. .054 .659 F. .113 .829 F. .089 .629 #. .189 .971 F. .053 S. Australia. .588 „ .128 .674 „ .109 P. .080 .427 „ .070 P. .030 .635 ,, 161 .348 ,, .000 P. .111 i. .140 P. .029 „ .098 „ .070 „ .107 .. .126 „ .000 Masc.M..162 Masc. M. .178 Masc. M. .058 Masc. M. .120 Masc. M. .097 „ .143 „ .148 .543 F. .184 „ .103 .340 F. .121 „ .112 „ .417 To .556 F. .193 .607 iF. .191 .633 F. .056 W. Australia. .566 „ .174 „ .170 !&. .181 .439 „ .181 if. .100 .620 „ .158 .343 ., .087 ¥. .176 P. .148 P. .082 „ 156 „ .157 „ .091 „ .130 „ .200 Masc.M..080 Masc. M. .054 Masc. M. .100 „ .081 „ .053 „ .111 To .483 F. .072 .620 F .082 .409 F. .IOC Nil. NU. Tasmania. .494 „ .087 .612 „ .096 P. .065 „ .154 P. .076 P. .IOC „ .084 „ 069 „ .136 • The upper figures are for the year 1909, the lower for the year 1913. t The masculinity of the migration in the table is the ratio of males to persons. J Based upon the departures front and arrivals in the States indicated, MIGRATION. 435 6. Periodic fluctuations in migration. — Periodic fluctuations of migration are exhibited alike by oversea migration, by interstate migra- tion by sea, and by migration overland. The following tables give the variations for the first and second for Australia. Table CXLI. shews also the monthly variations of the sex-ratio (or masculinity) of the migration. To express these results by Fourier series, see Part III., § 5, pp. 38-40, and also Part XI., § 16, pp. 169-171. TABLE CLXI.^Shewing Oversea Migration into and from Australia during the period 1909-1913, and its Fluctuations for " Persons" during the Year. (For equalised months and a constant population). I or B I £ I B Totals tor 1909 -13. Jan. Feb. Mar. AprU. May. June. July. Ajg. Sept. Oct. Not. state. Persons. Males. Females Dec. N.S.W. 337,997 232,056 .6856 .6967 231,634 161,666 106,303 70,390 8.^9 1.002 .697 .692 1.110 1.101 .665 .670 1.180 1.096 1.227 .679 .680 .989 1.101 .708 .712 .992 .884 .708 .707 .792 m .848 .723 .721 .876 .806 .689 .735 .920 .761 m .694 .709 .985 .896 .679 .716 1.091 .976 .659 m .712 1.070 1.316 1.082 Masc. .698 .642 m .689 .705 Vict. I B I B 116,603 71,425 .6187 .6330 72,148 45,211 44,455 26,214 .883 1.189 .632 .663 1.040 1.288 .612 .620 .990 1.529 .914 1.386 .637 .601 1.105 .974 .652 .631 .856 .801 .659 .652 .756 .766 .645 .658 .832 .627 .636 .689 .994 .602 m .628 .664 1.024 .720 .613 .658 1234 .877 .594 .649 1.372 1.241 Masc. .593 .580 m .565 m .626 Qld. I B I B 58,507 19,161 .6495 .8057 38,002 15,438 20,505 3,723 .850 1.125 .650 .890 .717 m 1.244 .670 .854 1.114 1.434 .743 1.243 .677 .702 m .963 1.022 .672 .748 1.165 .801 .688 .777 .855 .697 m .637 .775 .989 .709 .628 .812 1.172 .777 .711 1.264 .895 .626 .848 .938 1.062 .670 .863 1.230 .991 Masc. .595 .840 .592 .841 .865 S. Aust. I E I E 33,496 15,529 .6834 .7573 22,890 11,760 10,606 3,769 .902 .939 .733 .833 1.004 1.305 .713 .769 1.037 1.699 .918 1.480 .721 .691 1.038 1.084 .654 .761 .854 .808 .652 .811 .840 .719 .698 .791 .731 m .723 .762 ,826 .816 .684 m .747 .789 1.050 .726 .708 .802 1.796 .880 .568 m .744 1.014 .953 Masc. .714 .680 m .659 .737 W. Anst. I B I E 7 B I E 67,168 24,846 .6779 .7606 45,532 18,897 21,636 5,949 1.389 .932 .715 .793 1.169 1.058 :703 .750 .702 1.436 .847 1.298 .724 .709 1.059 1.198 .682 .718 .860 .895 .662 .758 1.095 .796 .654 .772 .594 m .797 .708 .801 .842 .641 m .612 ..806 .811 .679 .653 .788 1.556 1.076 .897 .610 m .781 1,873 Masc. .746 .683 .707 .838 Tas. 15,633 7,121 .6698 .6284 10,472 4,475 5,161 2,646 .841 1.518 .622 m .685 1.105 1.582 .641 .580 1.279 1.169 1.427 .705 .583 1.029 .755 .704 .612 1.036 .639 .696 .652 1.002 .499 m .689 .628 .840 .561 .671 .637 1.040 .590 .675 .671 .838 .952 .814 .722 .821 m' .625 .624 .607 1.005 1.732 1.120 Masc. .686 .553 m .648 .690 The quantity underlined is the greatest, and that marked m the least during the year. The two upper figures in each section are the relative average magnitudes of the migration for the month, the monthly average for the year being unity. The two lower figures are the migration-ratios tor the correspondlng'months, viz. the ratio o£ the migrants to the population of the State, 436 APPENDIX A. In Table CLXII. hereunder the fluctuations of interstate migration by sea are shewn, and the " migration-ratios" are also shewn. TABLE CLXn. — Shewing the Fluctuations for " Persons" in the Interstate Migration by Sea in Australia for the Period 1909-1913. (ForequaUsed months and aconstant population and the migration ratios xl.OOO.OOO.) State. FLUCinATION RATIO (TOIAI = 12.000) AND MlGRATIOK-KATIOS FOB PERSONS. Mi- grants. Jau. Feb. Mar. April. May. June. Jjly. Aue. Sept. Oct. Nov. Dec. To— ' Victoria Q'land S. Aust W. Aust. Tas. N. Terr. 137,916 16,344 109,542 12,982 20,788 2,464 16,218 1,922 33,617 3,972 825 98 1.645 1.254 1,707 1.011 1,094 1.194 245 1.263 200 1.868 615 .60 5 1.045 1,428 1.059 1,146 1.485 1.157 1,575 1.225 1,325 1.325 272 1.366 .891 1,214 1.318 .663 903 1.256 1,359 .776 159 .809 129 .437 j» 145 .108 9 .595 m 810 .952 1,030 .622 128 .683 109 .460 152 .102 8 .641 873 .883 955 .592 m 122 .636 m 101 .448 148 .87 7 .797 1.085 .782 846 .744 153 .763 122 .661 219 .69 m 5 1.164 1,586 .706 m 764 .860 176 .858 137 .730 242 .93 8 .965 1,314 .711 769 .879 180 .871 139 .827 274 .78 6 1.283 i 00 2,105 1.195 1,293 1.243 255 1.068 170 1.991 1,748 .902 CO n 1,425 1.096 225 1.333 212 .498 165 .128 10 976 1.184 305 1.270 202 1.200 397 .97 8 243 1.160 ^ 218 .907 300 .217 18 185 1.983 659 .83 7 656 .93 8 Total .. 318,806 37,781 1.425 1.228 3,866 1.106 3,482 1.178 3,708 1.032 3,251 .859 2,704 .710 2,237 .701 m 2,207 .772 2,429 .925 2,911 .852 2,682 1.212 4,489 3,816 ■■N.S.W. 145,326 21,973 1.354 1.274 2,333 1.343 2,458 1.297 2,376 .958 1,755 .788 1,352 .629 1,151 .610 m 1,117 .743 1,361 .709 1,298 1.088 1,992 1.267 2,479 8,302 t^ Q'land. 25,828 3,905 .935 304 .909 296 .987 321 1.187 387 1.467 477 1.515 1.272 414 1.031 336 .760 247 .606 m 197 .662 212 .679 493 221 to S. Aust. 28.006 4,235 1.266 1.231 435 1.233 435 1.212 428 1.083 382 .787 278 .701 247 .631 m 323 .779 275 .775 274 1.060 371 1.262 II ft,- 447 442 W. Aus. 46,031 6,960 1.082 627 1.339 777 1.438 1.307 758 1.220 708 .926 537 .716 415 .729 422 .710 412 .703 m 407 .864 501 .966 f 835 560 1 Tas. 163,688 1.614 1.263 1.041 1.009 .683 .694 .663 .602 .748 .786 1.112 1.905 ^ 23,220 3,126 2,425 2,013 1,952 1,322 1,149 1,264 1,164 1,447 1,520 2,151 3,686 £ N. Terr. 166 .92 .10 m 1 1.46 .39 2.46 1.16 .77 .92 1.38 .31 1.60 .64 - 24 2 3 1 5 2 1 2 3 .6 3 1 Total 398,915 1.390 1.246 1.207 1.174 .925 .758 .695 .649 .745 .736 1.041 1.434 60,317 6,986 6,266 6,066 5,901 4,649 3,810 3,493 3,264 3,745 3,697 5,230 7,211 MIGRATION. 437 TABLE CLXII.^Shewing the Fluctuations for "Persons" in the Interstate Migration by Sea in Australia for the period 1909-13 — continued. FLtrCTUATION RATIO (TOTAT. =■ 12.000) AND MIOBATION-RATIOS FOK PEMOlfS. State. Mi- grants. Jan. Feb. Mar. April. May. June. July. Aug. Sept. Oct. Nov. Dec. 'N.9.W. 106,280 1.140 .957 .976 .890 .811 .781 .788 .905 1.036 1.057 1.037 1.622 U3 03 34,731 3,300 2,769 2,825 2,578 2,347 m 2,261 2,282 2,619 2,998 3,058 3,001 4,693 i" Vict. 19,664 1.088 .914 .818 .959 .721 .773 .785 1.039 1.131 1.205 1.071 1.496 6,426 582 490 438 514 m 386 414 420 556 606 645 573 801 II 9. Aust. 593 1.23 .69 .93 .92 1.78 .63 1.11 .97 .69 1.03 1.06 .97 ft. 194 20 10 20 14 29 10 17 36 11 16 17 16 g W. Aust. 325 1.11 1.22 .85 2.14 1.85 .85 .52 .59 .92 .89 .29 .77 1 116 10 11 8 19 16 8 5 5 8 8 3 7 J Tas. 62 0.0 4.8 4.7 2.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 20 8 8 4 N. Terr. 629 .79 .93 .95 1.47 .89 1.09 .77 1.16 1.11 1.04 1.18 .67 L 173 11 13 14 21 13 16 11 17 16 15 16 9 Total 127,463 1.130 .952 .953 .908 .804 , .780 .788 .926 1.049 1.078 1.040 1.692 41,651 3,924 3,302 3,307 3,151 2,791 m 2,708 2,736 3,213 3,640 3,743 3,610 5,526 N.S.W. 22,116 10,774 1.285 1,153 1.422 1,277 1.494 1.231 1,105 .956 858 .694 623 .591 531 .560 m 503 .772 693 .870 781 .866 768 1.270 00 1,341 740 i Vict. 25,828 12,583 1.222 1,281 1.445 1.188 1,245 1.069 1,121 .770 807 .793 831 .673 706 .626 m 657 .749 786 1.105 1,160 1.027 1,077 1.333 II ft, 1,515 1,397 Q'land. 1,019 .77 1.21 .84 1.07 1.17 1.25 .98 .84 1.18 1.03 .66 m 27 1.00 i 496 32 50 35 44 48 52 40 35 49 43 42 1 W, Aust. 43,341 1.128 1.275 1.317 1.185 1.049 .869 .800 .783 m 1,377 .786 .911 .896 1.001 Total 21,115 1,984 2,243 2,318 2,086 1,845 1,529 1,408 1,383 1,603 1,577 1,760 a & 92.303 44,909 1.188 7,941 1.357 1.318 7,501 1.163 6,681 .950 5,568 .810 4,617 .717 4,331 .686 m 4,455 .777 5,614 .956 6,862 .920 6,954 1.158 8,348 10,528 'N.S.W. 20,370 1.292 1.350 1.232 .967 .884 .804 .685 .579 .830 .930 1.062 1.385 CO 14,180 1,527 1,595 1,456 1,142 1,045 949 809 684 981 1,100 1,255 1,637 to CO Vict. 45,690 1.283 1.310 1.135 .978 .828 .609 .519 .608 .796 1.077 1.093 1.763 iH 31,806 3,400 3,476 3,007 2,592 2,195 1,615 1,375 1,612 2,110 2,852 2,898 4,674 ft, Q'land. 531 1.08 .79 1.22 .93 1.45 1.22 .68 .86 .77 1.02 .95 104 .X 370 33 24 38 28 45 38 21 26 24 31 29 32 4 S. Aust. 47,205 32,860 1.088 2,980 1.182 3,236 1.096 3,000 1.050 2,877 .833 2,281 729 m 1,996 .775 2,122 .776 2,125 .913 2,499 1.032 2,827 1.012 2,772 1.514 4,145 &=■ N. Terr. 266 0.0 1.17 0.0 .267 .13 1.22 27 .45 0.0 3.39 .00 2.71 1 185 18 41 2 26 4 7 52 42 u ^ Total 114,062 79,401 1.200 7,941 1.262 8,348 1.134 7,501 1.010 6,681 .841 5,568 .698 4,617 .655 m 4,331 .673 4,455 .848 5,614 1.037 6,862 1.061 6,954 1.691 10,528 438 APPENDIX A. TABLE CLXn. — Shewing the Fluctuations for "Persons" in the Interstate Migration by Sea in Australia for the Period 1909-lZ— continued. Fluotuation Eatio (Total = 12.000) ahd Mioeation-eatios foe Peesons. State. Mi- grants. Jan. Feb. Mar. April. May. June. July. Aug. Sept. Oct. Nov. Dec. 00 l-H N.S.W. 37,786 3,939 1.566 1.474 4,837 1.482 4,865 1.442 4,735 1018 3,343 1.044 3,427 .826 2,708 .553 1,815 .735 2,411 .520 m 1,709 .613 2,013 .728 to 5,140 2,388 II Vict. 168,563 17,572 1.606 23,512 1.619 1.432 20,970 1.223 17,911 .821 12,014 .686 10,053 .693 10,149 .680 m 9,952 .709 10,389 .876 12,831 784 11,485 .871 fX, 23,700 12,756 rt S. Alls. 339 .57 .67 4.01 3.33 .81 .78 0.0 .07 .28 .71 .60 .17 s a 35 17 20 118 98 24 23 2 8 21 18 5 i W. Aust. 1S8 0.0 .46 3.49 6.68 1.06 .31 •■ 1 1. 16 6 48 92 15 4 Total 206,846 21,564 1.695 1.690 28,563 1.447 26,001 1.271 22,839 .867 15,396 .752 13,506 .716 12,857 .656 m 11,768 .718 12,809 .810 14,560 .762 13,^516 .843 28,669 15,150 N.S.W. 412 .87 .64 .61 .68 .75 1.11 .93 .41 1.72 1.81 1.49 1.08 M CO o 39,800 2,900 2,100 2,000 1,900 2,500 3,700 3,100 1,300 5,700 6,000 4,900 3,600 II Vict. 210 .67 1.66 1.37 .51 .40 .40 .67 .91 .80 1.49 1.26 2.06 B, 20,300 9,700 2,800 2,300 8,700 6,800 6,800 9,700 1,500 1,300 2,500 2,100 3,500 ^' Q'land. 387 .75 .63 .44 m 1,300 .84 1.02 .93 1.12 1.39 .66 1.77 1.24 1.33 Tl 37,400 2,300 1,600 2,600 3,200 2,900 3,600 4,400 2,000 5,500 3,900 4,200 ^ W. Aust 161 0.0 1.57 0.0 1.49 1.49 .75 1.12 .30 .37 2.01 .62 2.38 s 15,600 2,000 1,900 1,900 9,700 1,400 3,900 4,800 2,600 6,800 3,100 S Total 1,170 .66 .91 .61 .78 .88 .87 .96 .81 1.02 1.76 1.23 1.62 113,200 6,200 8,600 5,700 7,400 8,300 8,200 9,000 7,600 9,600 1,600 1,200 1,400 The upper figures are the relative average magnitudes of the migration for the month, the monthly average for the year being unity. Those underlined are the maximum-values and those marked " m" the minimum values during the year. The small figures are the number of migrants (" persons") per 1,000,000 population in the State from which the migration takes place. That Land Migration also shews marked periodicity is evident from Table CLXIII. It is worthy of notice that the total immigration for a year is sensibly equal to the total emigration for the same period though the want of balance for individual months may be considerable. TABLE CLXIII. — Shewing the Periodic Fluctuation of Overland Migration (by Railway) for equalised months and a Constant Population. Australia, 1914-1916. ("Persons.") Month. AEKIVAIS or iMMIQEATIOIf. DBPAETUEES OE EMir.EATION. N.S.W. Vic. Q'land. S. Aust. N.S.W. Vic. Q'land. S. Aust. January February March AprU May June July August September . . October November . . December 1.1517 .d824 1.1091 1.2085 .9584 .8034m .8389 .8734 -.9394 .8700 .9091 1.3549M 1.0855 .9812 .9427 1.1153 .9353 .8360 .8269 m .8985 1.0494 1.1441 .9884 1.1967 M 1.5297M 1.0387 1.2059 1.3487 1.0189 .7474 - .7472 .8454 .7266 .7610 .7117 m 1.3188 1.1928 .9737 .9944 1.1649 .9761 .8007 m .8528 .9445 .9602 .9047 .9350 1.3002 M 1.2452 1.0044 1.0339 1.1811 .94.18 .7888 m .7999 .8938 .9432 1.0000 .8809 1.2790 M 1.0973 .9437 1.0848 1,1745M .9928 .8533 m .8921 .9179 1.0177 .9115 .9970 1.1174 1.2142 .9738 1.1270 1.2167 .8907 .7072 .6995 m .8231 .8349 .8703 .8219 1.8207 M 1.2352 1.0580 1.0250 1.2827M 1.0278 .8643 .8904 .8488 m .8893 .8970 .8725 1.1091 Mean No. for equalised mnth. Aggr. Popn. . . 67,102 48,188 24,278 18,063 67,007 47,804 24,516 18,244 M denotes the maximum and m minimum value. M [ORATION. 439 7. Migration and Age.— If the ages of migrants of each sex are re- corded at the moment of entry into or exit from any community, it is possible to know continuously the constitution of the population accord- ing to sex and age, once a population Census has been taken. Results forwarded to the compiling authority only at long intervals require cor- rections, of the type referred to in Part XI., §§ 7-9, pp. 152-160. The deduction of ages is best effectuated by referring all the results to the one point of time, say the end of the calendar year. 8. Defects in migration records and the closure of results.— Not- withstanding that elaborate care was taken as regards the record of emigration, it has been found in Australia that errors occur therein of considerable magnitude. From the 1901 Census and the intercensal records up to the Census of 1911, it appeared that, if the discrepancy were attributed wholly to this source of error, it would amount, in the case of males, to 0.1459 of the whole recorded male migrants outward (de- partures) and in the case of females to 0.0995 of the whole recorded female migrants outward. A still more extraordinary result was that apparently the island-continent of Australia was rapidly losing females.^ Suppose that a statistical element Eq is accurately ascertained at anypoint of time {e.g., as at a Census) and after n years is again accurately ascertained and found to ba En ; and further that the intervening changes are e^, 62 , ... en. Then : — (661).. En=S!„+k (61+62 H-. . ..+en); or k={E„~E„)/{ei+ez+ . .+e„) The quantity k may be called the coefficient of proportional linear adjust- ment, and El, E^, etc., may be found by the successive additions, viz., of kei, ke^, etc., instead of the unadjusted change. We may, however, correct the results as indicated in (662), that is : — (662) . . En= ^o+ei+e2+ • • +e«+€ =Eo+{ei+K)+{es+K)+ . . +(e„-f/c). in which last expression K = e/n, the total defect of closure, e, being divided equpUy among the changes. Thus in this case Ei, E^, etc., may be found by successive additions, viz., of bi+k, e^+K, etc. This may be called simple linear adjustment. The question as to whether one or the other or either is legitimate, must always be decided by the nature of the case, and obviously no general rule can apply. 1 Upon a change being made on the system as between State and State, such that the aggregate of the State -increments of population gave the increment of population to the Commonwealth, this peculiarity vanished. XVn.— raSCELLANEOUS. 1. General.' — It is proposed in this part to refer to a number of miscellaneous matters, which have not been included in previous parts, and which either do not fall under any particular heading, or have been omitted from ear her consideration. 2. Subdivision of population and other groups.^ — The values of group-subdivisions, which are obtained by dividing groups bj'^ the middle ordinate, are given earUer, see Part VI., § 4, pp. 80-81. These formulae are not always applicable. Two questions often arise, viz. (i.) the value of the subdivisions or (ii.) of their ratios to each other. Considering firsfc the subdivision of a group g into two parts, let it be supposed that the function, representing a series of groups, viz., g^i, g_i, g,gi. . . .g^, is a-\-bx-\-cx^-\- etc., then we shall have^ :— (663) g^i =ig - gig [61(sri-g_i)- 44:(g^-g-2)+ 19(g3-?-3)-3M9'4-?-4)+etc.] gr_j denoting the portion of the group g on the negative side of the middle ordinate of that group. This formula is in general suitable about maxima and .minima values, but may, of course, be inappropriate ii g^. — g./^ increase more rapidly than the coefficients diminish. It may often be employed, however, when pairs of terms in the square brackets are sharply convergent. Another process of arriving at values for the subdivision of groups into halves is the following :■ — Let the values of the successive groups be C, B, A,M, A',B' and C", and M, the group to be divided. Then the portion next to .4 is ^ :— (664)..Af'=JM-2^[201(^'-.4)-44(£'-£)+5(C"-C)— ..] which in many cases gives substantially the same result as (663), though it is not an identical formula, and apparently might be regarded as not in agreement therewith. 1 This is deduced by finding, in terms of the groups themselves, the values of the constants a, 6, etc., of the curve : and then integrating between the limits which give the first half of the group to be subdivided. 2 This is easily derived from the usual formula for interpolation into the middle, viz., F(ii = 'F -\- \ a' — \b„ -\- ,-§g d„ — xi^jj/o + etc., by regarding the aggregates G,0-\-B, G-\-B -^ A,0+B-\- A-'rM, etc., as successive totals represented by ordinates represented by a -f px -{- yx^ -\- etc. ; finding the value to the middle ordinate of group M and subtracting C-\- B-\- A. MISCELLANEOUS. 441 In the case of groups rapidly increasing or rapidly dimiaishing in amount — as for example the numbers dying at the beginning of life in 0-1, 1 to 2, etc., days, weeks, months or years, the following method of subdivision may be followed : — Let it be required to divide each of a series of larger groups A, B, C, etc., for equal limits of a variable into s smaller groups, viz., aj, «£ > • • • ■ as ; 6i, . . . 6s ; Cj, . . . Cg ; etc., and suppose that £=m^ ; C=m'B =min' A ; etc. Then if m'=m, etc., it is 6bvious that the successive values of the smaller groups will be : — (665) . . (oi + noi + n^Oi +...) + (6i + w6i + w^&i + • • •) + etc. = in which n is the sth root of m and m'. The brackets shew the groups, the sum of which give the original values A, B, G, etc. Since from each of any three adjoining groups an equal quantity Q may be cut off or added, so that the altered values A', B', C will be A', m^ A', ml A', we can constitute the group-divisions by adding a common value Q/s to each of a series of quantities of the type of (665) above, n^ in this case being the sth root of Too . Hence we have : — (666) a {I + w + w2 + ... n»-i) = A' = A — Q ; from which, since n is known, a can consequently be determined, and the series o, na, n ^a, etc., to which, if a comrnon quantity q= Q/s is added we obtain ax, «£ . ©tc. Thus :■ — (667) ai = A'/{l + n + n^ + ... + n'-^) + q = a + q ; 02= na -\- q; a^ = n^a + q ; etc. In applying this method practically, any group may be subdivided by treating it as B, and dividing it according to the indications of the groups on either side A and G. 3. The measure of precision in statistical results.— Statistical results, expressed without regard to their possible or probable error, often suggest the attainment of a precision far beyond that which the data can furnish. For example, if the ratio of the survivors after one year be given (as in life-tables) to 5 decimals, the results imply for Australian data an average precision of og-e for the first year of 1.1 hour, or at its terminal of 0.4 hour. For other countries it will be much the same. Again, in the case of the instantaneous rate of mortality at the end of the first year, the expression to 5 places of decimals implies a precision, in the time or epoch to which it may be deemed to apply, of 8 days. In both cases the apparent precision is illusory, 1 forasmuch as the recording of the facts and their actual 1 See Census Report, Vol. III., p. 1215, and also p. 1212. 442 APPENDIX A. variableness does not conform to this order of precision. For example, births and deaths are not recorded as regards age to 0.4 hour per annum even on the average : nor can the point of time to which they may ap- propriately be referred be deemed to be ascertained to 8 days or its equivalent in a decade. Actuarial tables are often carried to 7 places of decimals. A unit in the last place is (on the average) for ages 1 to 2 about equivalent to an age-difference of 2 miuutes, and, owing to the diminution of death-rate with the lapse of time, also to about the same as to the poiat of time to which the result is presumed to apply. Let u and y denote respectively fuifctions of time (t) and of age (x), then if : — (668) Au = Idt; Ay = JAx; or I = du/dt ; J = dy/dx in the Umit, / and J are the ratios of relative importance — as compared with the units of u and y — of precision in the units respectively of t and x. These ratios serve as guides in fixing the relative accuracy required in the data giving the two co-ordinates. If iu graphing results, the units on the axis of abscissae are, respectively, I and J times the units on the axis of ordiaates, then the curve wiU make an angle of 45° with either axis, and this, in so far as it is practicable to foUow it, is the best scale-relation between ordinate and abscissa for any graph intended to be used for analysis. The life-tables published in connection with the Australian Census of 3rd April, 1911, foUow the usual tradition as regards the number of figures to which the results are expressed. It is not, however, implied that the precision indicated is realised, they merely are followed for the sake of oonsistencj^ in the results. By suitable combinations of arithmetical and graphical methods results can be obtained to any required degree of practical precision. ^ 4. Indirect relations. — ^It is often necessary to establish statistical relations which reaUy depend upon some intermediary statistical relation. For example, the average num.ber of children bom to an individual, or " average issue" may be related to age of " mothers," and such a relation would, of course, be a direct one. For certain purposes, however, (e.g., social insurance) the average issue may be required as related to the age of fathers. The later relation, though physically indirect, is a regular and important one. Nevertheless, it is one which may be deduced by means of certain data from direct relations ; at the same time it is not prefer- able to obtain it in this way. The relations according to " wives" and " husbands " are both given immediately by the Census, and the relation so given is, in general, to be preferred to the deduced relation : see Fig. 107. 1 If the value of / or J is not between the Umits J to 4, the natural scale for both co-ordinates is not ordinarily satisfactory in graphing a function ; however the mode of variation of the greater co-ordinates will assist in the determination of a truly smoothed curve. MISCELLANEOUS. 443 Fertility Curves. A B =s.=rs-iu^ = s 40Averag6s SO 60 C F Fig. 107. Curve A shews the ratio, according to age, of first bkths to all births. Curve B shews the probability, according to age, of a nuptial birth ; see also p. 242 and p. 243. Curve Ca shews the probability, according to age, of an ex-nuptial birth on the assumption (1.) that they are attributable wholly to the never-married. Curve Cb shews the probability, according to age, of an ex-nuptial birth on the assumption (ii.) that they are attributable equally to the never-married, widowed, and divorced. Our-e D shews the average issue, according to age, of wives at the Census of 1911. Curve E shews the average issue, according to age, a&related to husbands at the Census of 1911. Curve ITB ' shews the average interval, according to age, between marriage and first-births. Curve F6 shews the average interval, according to age.between marriage and first-births, occurring within 1 year of marriage. 5. Limits of uncertainty. — The limits of an uncertainty in any deduced quantity may be due to possible errors in the numbers upon which it is founded, or upon an uncertainty as to the particular quantity which should be employed. The first cause of uncertainty is sufficiently illustrated by the ratio of, say, first births to all births : for prediction purposes the smoothed numbers in Table CLXIV. are really more probable than the crude numbers : see Fig. 107. The second cause of uncertainty is illustrated in the following example : — If the " never married," the " widowed" and the " divorced" are regarded as a homogeneous class, the probability of a case of ex-nuptial maternity during one year is found by dividing the number of births in one year by the sum of the average numbers in the three 444 APPENDIX A. classes. If, however, they are not homogeneous as regards this proba- biUty, a more accurate result might be obtained by dividing by the never married. The general probabiUty must lie between the two results : see the curves marked Ca and Cb on Pig 107, and the results in columns marked I. and II. respectively in Table CLXIV. It may be noted that the characteristics of a variation may be wholly changed by restriction within limits. This is seen by taking the interval according to age between marriage and a first birth, when the consideration is restricted to the lapse of 12 months, or is indefinite : see the curves FG and PE' respectively. TABLf CLXIV —Shewing Rates of First to All Births, and Probabilities of Ex-nuptial Maternity . AustraUa. 1907-14. ProbahUity Ex- Probability Bx- Ratio oJ First nuptial Maternity Ratio of First nuptial Maternity to all Births. hased on to all Births. based on Age. assumption Age. assumption Crude. Smooth- I. n. Crude. Smooth- I. II. ed. ed. 12 1.0000 .000015 .000015 13 1.0000 .9970 .000062 .000062 34 .0994 .1040 .01325 .0118 14 1.0000 .9930 .00037 .00037 35 .0923 .0920 .0130 .0115 15 .9404 .9715 .0016 .0015 36 .0817 .0825 .0127 .0110 16 .9407 .9430 .0042 .0042 37 .0703 .0730 .0122 .0103 17 .9130 .9035 .0085 .0085 38 .0640 .0640 .0110 .0094 18 .8602 .8450 .0131 .0131 39 .0583 .0560 .01045 .0084 19 .7627 .7627 .0157 .0162 40 .0524 .0485 .0095 .0072 20 .6594 .66j4 .0172 .0173 41 .0437 .0425 .0076 .0058 21 .6912 .5912 .0180 .0179 42 .0381 .0370 .0059 .0044 22 .5285 .5170 .01835 .0181 43 .0352 .0338 .0043 .0030 23 .4534 .4485 .0181 .0179 44 .0351 .0310 .0030 .0018 24 .3360 .3960 .0176 .0174 45 .0349 .0285 .0020 .00123 25 .3482 .3482 .0169 .0161 46 .0244 .0295 .0012 .00080 26 .3098 .3080 .0160 .0154 47 .0360 .0338 .00085 .00040 27 .2722 .2710 .0154 .0147 48 .0255 .0428 .00055 .00022 28 .2352 .2370 .0149 .0141 49 .0769 .0555 .00036 .00013 29 .1)87 .2065 .0146 .0135 60 .1333 .0790 .00022 ,00009 30 .1800 .1795 .0143 .0130 51 .0909 .1230 .00012 .00006 31 .1555 .1560 .0140 .0126 52 .1429 .2500 .00004 .00004 32 .1324 .1355 .0137 .0128 53 .00002 .00002 33 .1160 .1182 .0135 .0120 54 .00001 34 .0994 .1040 .01325 .0118 55 I. denotes the ratio of birtlis to the never -married ; 11., the ratio of births to the aggregate of the never-married, widowed and divorced. 6. |The theory of " happenings" or " occurrence frequencies." — ^Ih order to establish a rational theory of, and to completely interpret, the frequency curves met with in the various elements of the statistics of population, a theory of the frequency of occurrences of various kinds is a first requisite, and the type-forms of distribution estabhshed by Prof. K. Pearson and his co-workers are a contribution thereto, based upon the application of the theory of probabihty, plus certain empirical assumptions by means of which assymetrical forms of various kinds are deduced. Recently a foundation has been laid of a perfectly general theory of the frequency of occurrences, by Prof. Sir Ronald Ross. This latter seems to have had its birth in an attempt made in 1866 by Dr. Parr to develop a definite theory of an epidemic (cattle plague) i. In 1873-5 Dr. G. H. ' Dr. William Farr, "On the Cattle Plague," Journ. Soc. Sci., 20th Mar., 1866. MISCELLANEOUS. 445 Evans endeavoured to extend Fair's theory to other epidemics, i The subject was again reopened by Dr. J. Brownlee^ in a series of very significant contributions, and later, by Ross. Quite recently the last- named has put forward a definite theory, the fundamental elements of which are outlined in this section.^ Although the main object was initially the determination of a basis for a theory of epidemics, the results are entirely general, and may be called the theory of " occurrences" or " happenings." The, differential equation of independent occurrences, reduced to its simplest expression, may be deduced as follows : — Suppose a population P to consist of two parts, viz., A a part which is unaffected, and Q a part which* is affected*, by any " happening, "° so that P= A-\-Q. Suppose also that some portion, viz., hdt, of the unaffected part becomes affected in the time dt, and also that a portion rdt of the affected part reverts to the unaffected part in the same element of time, so that the element of increase of the affected part is (h — r) dt ; and finally let bdt, mdt, idt and edt denote in the unaffected part, the rates of birth, death (or mortaUty), immigration and emigration respectively ; and Bdt, Mdt, Idt and Edt denote the similar rates in the affected part. Obviously therefore :• — (669). .dP= A (b—m+i—e)dt + Q{B—M+ I~E)dt = ( Av+ Q V)dt ; {Wl{i)..dA=A{b-m-\-i-e-h)dt-^Q(B-\-r)dt={A[v-h)+Q{B+r)]dt; (611).. dQ = Ahdt+ Q(—M+ I~E—r)dt = \ Ah+Q{ V—B-r)}dt; ^Dr. G. H. Evans, " Some arithmetical considerations on the progress of epidemics," Trans. Epidemiol. Soc. London, Vol. 3, Pt. III., p. 551, 1873-5. 2 Dr. J. Brownlee (i.) Theory of an Epidemic," Proc. Roy. Soc. Edin., Vol. 26, Pt. IV., p. 484, 1906; (ii.) "Certain considerations on the causation and course of epidemics," Proc. Roy. Soc. Med., Lond., June 1909 ; (iii.) " The mathematical theory of random migration and epidemic distribution," Proc. Roy. Soc, Edin., Vol. 31, Pt. II., p. 261, 1910 ; (iv.) " Periodicity in infectious disease," Proc. Roy. Phil. Soc, Glasgow, 1914; (v.) " Investigations into the theory of infectious diseases, etc.. Public Health, Lond., Vol. 28, No. 6, 1915 ; (vi.) " On the curve of the epidemic," Brit. Med. Journ., May 8, 1915. ' Lieut. -Col. Prof. Sir Ronald Ross, (i.) " The logical basis of the sanitary policy of mosquito reduction." Cong. Arts and Sci., St. Louis, U.S.A., Vol. 6, p. 89, 1904, and Brit. Med. Jouin., May 13, 1905 ; (ii.) The prevention of malaria in Mauritius," Waterlow and Sons, Lond., 1908, p. 29-40 ; (iii.) The prevention of malaria, J. Murray, Lond., 1910 ; 2nd Edit., 1911 ; Addendum on" the theory of happenings," 1911 ; (iv.) Some quantitative studies in epidemiology. Nature, Lond., Oct. 5, 1911 ; (v.) " Some a priori pathometric equations," Brit. Med. Journ., Mar. 27, 1915 ; (vi.) " An application of the theory of probabilities to the study of apriori pathome- try" ; Proc Roy. Soc, Lond., Vol. 92, ser. A„ July 14, 1915, pp. 204-230. See also H. Warte, " Mosquitoes and Malaria," Biometrika, Lond., Oct. 1910, Vol. 7, No. 4, p. 421. * The affection may be of any nature, such as a disease, etc., and the supposition is quite general. "■ The " happening" is the becoming affected, and is equally general with th? preceding supposition. 446 APPENDIX A. and writing v and F for the algebraic sum of the quantities in the brackets in (669), the final forms of the preceding equations are given as is necessary of course, dP=d A-\-dQ. It may be noted that only a A and d Q contain terms representing the happening (h) and reverting elements (r), and that QBdt appears in (670) but not in (671), because, in general at least, the progeny of the affected part are not affected at the instant of birth. Although the variation elements b, m, i, e and B, M, I, E will, if long periods are considered, generally be functions of time, they may be re- garded as constant when short periods only are under review. Consequently for elementary cases mean values may be taken without sensible error, ^ similarly in regard to the reverting elemeni.^ The most important element is the happening element, h, which it is to be clearly understood ordinarily falls on both groups ( A and Q) alike. Should, however, it fall upon individuals already affected, it merely reaffects them and does not cause them to pass from one group to the other. Hence, though the total number of " happenings" is P.hdt= {A-\-Q) hdt, the number Qhdt are already affected and must not be taken account of. The actual number of new cases Gdt, say, is thus only Ahdt. Thus :— (672) Odt/Phdt =^ ; or G = hA = h{P ~ Q) " Happenings" may be divided into two classes, viz. : — (a) those in which the frequency of the happening is independent of — ^and (b) those on which is dependent upon — ^the number of individuals already affected.^ In independent happenings h and G are constants, in dependent happeninge they are functions of Q. * If , as is often the case, the " happenings" have no effect on the birth, death and migration rates, then we may have b= B, m=M, i= I, e= E, and consequently v= V, which may also occur fortuitously though the several terms differ. In general 6 is less than B in marriages, m than M in accidents, while in certain alarming epidemics (e.g., cholera, plague, malaria) i is greater than 7, and e less than E, in which case v is greater than V. In fatal accidents M= 1, and B, I and E are all 0, which value may also be assigned when considering happening among the same individuals. If a surrounding population be not affected 1=0 ; if affected indivi- duals cannot move S=0. " In the case of " independent happenings" — see later, rdt denotes merely the proportion of affected individuals who may become reaffected in the time dt. {e.g., by divorce in marriage). In " dependent happenings" it implies loss of capacity for affecting others (e.g., in infectious disease it implies both immunity and loss of infectivity). In some diseases r may be zero (e.g., leprosy and organic diseases, fatal accidents) ; it may be of small value (e.g., many zymotic diseases) ; it may be of high value (e.g., snake-bite, heat-stroke, etc.), and it may be imity (e.g., alight accidents). » To the former belong cases which are attributable to what may be called external causes (e.g., accidents, non-infectious diseases, etc.) ; to the latter belong all cases attributable to propagation from individual to individual (e.g., infectious diseases, etc.). MISCELLANEOUS. • 447 In independent happenings, therefore, the happening falls upon the same proportion Qidt) of the population in every element of time. Put x= Q/P and P— Q for A, then equations (669) and (671) give :— (673) dP/dt = vP — {v — V) xP (674) d {xP)/dt = hP {1 — x) + (V — B- r) xP and by difEerentiation : — (675) d (xP)/dt = xdP/dt + Pdx/dt. From these three last equations, we have after dividing by P, and eliminating d {xP)/dt and dP/dt : — (676) dx/dt = h -{h + v — V +B + r)x+{v — V)x'^ which gives one form of integral if v— V=0, and a different one if v and V are unequal. When the sum of the variation elements of the affected group is constant the case may be called the equivariant case, the total population is unaltered. 1 Putting :-^ (677) K = h+ B+ r; L = h/K ; y=L—x; hence (678) dx/dt= — dy/dt = K {L—x)=Ky ; dy/y= — Kdt ; which gives on integrating :■ — (679) log y=~Kt+C,OTy = y„ e-^\ yo being the value of y at the beginning of the " happening." Con- sequently, since y„ = L — Xq : — (680) X = L — {L — Xo)e-^t viz., the proportion of the total population affected at the time t, the pro- portion being Xq when ( = 0. * When V is not equal to V, we have the general case of independent happenin^gs which involves the integration of (676). This may be written in the form : — (681). . . .dx/dt = K (L—x) L'—x) = K (a—^—x) (a+^—x) 1 An example would be the occurrenee of slight accidents in which case r=l, .or the attainment of a oertaia standard of wealth tending to diminish simultaneously the birth, death, and migration rates of the affected by an equal decrement. If the progeny of the affected are also aSected B should be omitted from (670), and in- serted in (671), and will disappear in. (674) and (676). ' Obviously in (673) if v — F=0, a differential equation of the sajne form as (678) is obtained, hence P= P|,e"', formula (2), p. IQ herein. 448 APPENDIX A in which a={h+B+r+ K)/2Ka,nd j8 = ^(a^ — h/K), the roots L= a — j8 and L'= a+jS, being always real and positive when v > V. This gives : — (682).... x=L-(L'^L) (I^z,)/l(L'-x„) e^^P'~{L-x„)\ which simphfies sUghtly if x^=Q. The relative number of the affected depends upon whether K, that is whether v— V is positive or negative, the former being usually the case in injurious happenings and the latter the case in beneficial ones. This gives :■ — (683). .P=P„e'". Le-^i'^/(L'—L) ; or Poe"". ~Le-^^'''/(L'—L) the former expression being appropriate when K is positive, the latter when it is negative. Among dependent happenings the case of proportional happenings is important as a first approximation to the study of the infection of a com- munity. In this instance A is a function of Q and consequently of t. If each affected individual affects c others in a unit of time the total happenings in the time dt will be cQdt. The number of new cases per element of time may be taken as probably : — (684) Gdt/cQdt = A/P ; or G = cQ {I — x) ; h = ex. This gives : — (685).. dx/dt =Za;(i;-a;), in which K=o—v+ V; L=l-{B+r)/K, from which may be obtained : — (686) x=L/{ 1+ {L/xo- 1) e-K^"f. This gives regular bell-shaped curves : x^ and Qg can never be zero. Sufficient has been indicated to shew the value and reach of Prof. Ross's analysis of the question, and to render evident the fact that it is the foundation of a rational theory of "occurrences" of any kind, which can be numerically defined, in a population. 7. Actual statistical curves do not coincide with elementary type loims. — The importance of a rational theory of " happenings" does not consist in the fact that the curves deduced from elementary suppositions, meticulously correspond to actual statistical frequencies, but in the fact that deduced types give the general configuration. Since in actual cases what may be called the frequency of initiation is variable, the deduced forms of frequency at any given moment are only partially applicable to actual cases. Moreover any assymetrical and polymorphic curve, and indeed even any regular curve, can be built up in an infinite number of ways. The dissection of a, curve into additive components is therefore, MISCELLANEOUS. 449 in general, purely empirical. Although this is so, when extra-mathe- matical reasons exist for the acceptance of an hypothesis of constituent elements, whose origins, and general characters, are known, it may be possible to effect an analysis into components which yields a real and not merely a formal interpretation.^ In general, type-curves, the interpretation of which is impossible and is ignored outside certain selected points (e.g., the points where they meet the axis of abscissae) are logically unsatisfactory. The function of a " theory of happenings" and of the " theory of probability," is therefore one of guidance ia interpretation, and of deciding as to the applioabihty or otherwise of particular types of mathe- matical expression for the representation of the change of frequency with change of the variable. Mere arithmetical tests of the " goodness of fit" of particular mathematical expressions are significant or otherwise according as they conform to what is known a priori, or is deducible from a priori considerations, and these must certainly be taken in conjunction with the observations over the whole range of experience. * 8. International norm-giaphs and type-curves. — The function strved by the creation of norms has been indicated in Part VIII., § 6, p. 102. When norms for every important population-character have been computed, it is desirabk that they should be graphed and used internationaUy. This could be done by printing squared graphed paper, with the norm shewn thereon, say in pale colour (or by a very fine line). The graphing of the same character on such paper for any particular population, would then immediately disclose the nature of its deviation from the normal. In this way the population phenomena could be graphically studied in their comparative relationships. An extension of the system would be for each country to shew by pale tint not only the international norm, but also its own norm for (say) the previous decade. Type-curves for international use would also greatly assist in the work of a better technical reduction of statistical results. The forms desirable or necessary would doubtless be more readily recognised when the international norms had been obtained. ^ For example in the harmonio analysis of tides, the forma and periods of the components are determined by celestial positions (i.e., of the sun, moon, " anti-sun," " anti-moon," etc.), and the elements to be ascertained are merely the epoch of each component and its amplitude. * For example, to systematically vary the representation of facts in order to agree with some adopted mathematical expression to which it is thought they oitght to conform, is only to delude oneself. The character of terminal conditions is often known a priori, and the mathematical expression representing the facts should not be merely one in substantial arithmetical agreement with the frequency, but one which expresses as accurately as may be the law of its change. Similarly, the adoption of an expression which disturbs the observed critical values of the frequency, vitiates tjie results, 450 APPENDIX A. 9. Derivative elements from population-theory — It is beyond the purpose of this monograph to discuss the various derivative branches of the theory of population ; such, for example, as the estimation from probate-records of the aggregate of private wealth ; of the economic value of an average man or woman ; of the economic value of different classes of persons ; the cost of; and economic value of, education, etc. The present increasing length of life tends to give a higher average economic value other things being equal — to an individual : the average wealth possessed per individual is probably also increasing. Although all that relates to population may, in a comprehensive view, be regarded as belonging to its theory, it is quite appropriate that purely economic questions should be separated out. Therefore, while results obtained by means of the development of the population-theory are essential and are of the first order of importance, in any attempt, for example, to reach decisions as to the economic aspects of population, the questions that arise are so extensive that they must be treated independently. Nevertheless, the value of a suitably developed theory of population is not seen until it is viewed in the light of all its applica- tions among which the economic is but one. Similar observations apply to the anthropometric elements of the population. These are probably correlated with elements treated here- inbefore ; nevertheless, it is preferable to deal with them independently. 10. Tables for facilitating statistical computations Mathematical tables of various kinds have been prepared for faciUtating statistical computations, among which may be specially mentioned "Tables for Statisticians and Biometricians," by Prof. Karl Pearson, F.R.S., etc. In this monograph the following tables are solely for facUitating the computation or illustrating the mode of deducing quantities which enable required quantities to be found by inspection : — Tables I. VI. XVII. & XVIII. XXXVI. & XXX\^I. XL. LXV. LXVI Pages 20 77 123 159 163 217 219.220 Tables LXVII. LXXV. LXXXI. CXLIV. Pages 221-222 247 266 398 11. Statistical integrations and general formulae.— Reference has already been made in Part VI., §§ 6-8, pp. 82-84, to statistical integra- tions, and references were given to various tables, see p. 82. The integ- ration of functions of a single variable is the subject of one of the Cambridge Tracts in Mathematics and Mathematical Physics, No. 2. This and the works previously mentioned will enable most integrations occurring in practical cases to be effected. For convenience the following are given ;— MISCELLANEOUS. 451 Table of Integrals and Limits. /^ .log^. lx>9(i + f )•.y^^=ilog(a.±6):/.^|^ = ^:.^-~?logC«^S): 2a. "'^ 20.^^9 9 2axt6v3 2a 'i'^ cu^Fp ^Tp at^o^+o; «-^.jj^^/o, >/ — oicTS 2a. a.' ^ J? ''=''■ ' "^ " c^ /x'dc.^'.-^{^'.az^f)(^'-a^^f), etc. . f^^a^^m^) = /^,imt/3.6^-4«4 /^.iloa.^'-A /:^,s«eabove! /^ =i_ A^]oq#V %2«?/"-# , see above : fiisn{ajc*h)da- — i-]ogcos(aa;+8J :ycot(aj5+S) . Tu^'l oq j: ) i nx. n'sc' i . n.'j'QoqaO 'iJ. Tut n-'j:' K. ^"'■'"•cte-yV'^il*'urloga;ti,(7vx3oga:)V^,(.7ijr:log«)'+- } d» : •av5iA.= e(cos?*l^isia«) : ^i-logx cte- ^(Jogx-i^g) . true aUo if^O. I 1 f dx- ■ Joa a 1 i 1 . 1 1 i. ■'l7H.-l)e/x(at»x;'^' (m-i)B0"'-' Xm-i)a4l(pi-Z)/>"'"' (m-3)ajzi"'-" ^ (m.-V^a'^^'^-*' tar- '■fi'^ '5^>-(^9-)"-^ = ^{Oo9xr-^,(togxr.|^aio3x)-- . -^^^^^ ) : ^3^4^,1^ = -. i'„^^-^'^'^"'''''.I'J^-l = J)enofc]ogCl.j:c)Tori,g'x; 1 (l+a)» = e : I tlva:)- =1:1 (l-^f- I Cl**y)' - e*j 1 «Ji -Jcqa. .-. X £^=i I(log*Ca>fl) -log*a:)= : I (H-|+|*- ti -logx) - y- 0-5772156G490--- -EuIex's canst. 1. K l*i+A--4Vlo9 =c} = 1:1 cC/x. -oo : I a7x-"'= oo if»77T. lie + :I e^. S ; LxaoQJ:-0: I. Jogx/x»-=€x. : X«»loqj:--oo : I ^. : I "t(TO-i^..-l7n--7t+l) Iix*'=l,.-. 0°'l (nDtiroanafi^r) r l^'^'^l :i a:"""^ 0: I x.^i.e^-^ = e -. 114"= e^ "'""^ «-+" ""^o '-+0 *-*' J£-flie espisaeteani I(l+3o9»)==-^ = e: I Ci'"<-2'V-'».')/'a.'"-^: L {(a-i)'+(a:+2)%- +{a.-v770'}/77i.'*H.i 3IaillBCrum.~^Qu£S. T(x=X-4€16521-j»0'8556O32 ■• : a:* fonar- 0-3678794- =0G922OO7. 1 (l+^i.)''=I [U-+^x)'tI ;if j^