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Readers are asked to report all cases of books marked or muti- lated. Cornell University Library HG8783 .H49 Mortality laws and statistics ........ 3 1924 032 540 928 olin 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/cu31924032540928 MATHEMATICAL MONOGRAPHS EDITED BY Mansfield Merriman and Robert S. Woodward. Octavo, Cloth. No l. History of Modern Mathematics. By David Eugene Smith. Ji.oo net. No. 2. Synthetic Projective Geometry. By George Bruce Halsted. #i.oo net. No. 3. Determinants. By Laenas Gifford Weld. $i.oo net. No. 4. Hyperbolic Functions. By James Mc- Mahon. $i.oo net. No. 5. Harmonic Functions. By William E. Byerly. $l.oo net. No. 6. Qrassmann's Space Analysis. By Edward W. Hyde. Ji.oo net. No. 7. Probability and Theory of Errors. By Robert S. Woodward. #i.oo net. No. 8. Vector Analysis and Quaternions. By Alexander Macfarlane. $i.oo net. No. 9. Differential Equations. By William Woolsey Johnson. $i.oo net. No. 10. The Solution of Equations. By Mansfield Merriman. $i.oo net.' No. 11. Functions of a Complex Variable. By Thomas S. Fiske. Si.oo net. No. 12. The Theory of Relativity. By Robert D. Carmichael. $1.00 net. No. 13. The Theory of Numbers. By Robert D. Carmichael. $i.oo net. No. 14. Algebraic Invariants. By Leonard E. Dickson. $1.25 net. No. IS. Mortality Laws and Statistics. By Robert Henderson. $1. 25 net. No. 16. Diophanttne Analysis. By Robert D. Carmichael. §1.25 net. PUBLISHED BY JOHN WILEY &SONS, Inc., NEW YORK. CHAPMAN & HALL, Limited, LONDON. MATHEMATICAL MONOGRAPHS EDITED BY MANSFIELD MERRIMAN and ROBERT S. WOODWARD No. 15 MORTALITY LAWS AND STATISTICS BY ROBERT HENDERSON, Actuary of the Equitable Life Assurance Society of the United States FIRST EDITION FIRST THOUSAND NEW YORK JOHN WILEY & SONS, Inc. London: CHAPMAN & HALL, Limited 1915 Copyright, 1915, BY ROBERT HENDERSON THE SCIENTIFIC PRESS ROBERT DRUMMOND AND COMPANY BROOKLYN, N. Y. PREFACE The present work is designed to set forth in concise form the essential facts and theoretical relations with reference to the duration of human life. A description is first given of those mortality tables which have had the greatest influence on the development of the science of life contingencies or on its appli- cation in this country. A few chapters are then devoted to the mathematical relations between the various functions connected with human mortality, to the analysis of probabilities of death or survival, so as to lead to their simplest form of expression in terms of the mortality table, and to the general mathematical laws which have been proposed to express the facts of human mortality. The connection is then established between the mortality table and mortality statistics and some investigation made of the corrections which must be allowed for in interpreting such statistics. The methods of constructing mortality tables from census and death returns and from insurance experience are then taken up. The methods adopted for the purpose of adjusting the rough data derived from experience are next described and their theoretical basis investigated. Some of these methods of construction and graduation are then illustrated by a new mortality table now first published. In the Appendix ten useful tables are given. The scope of the treatise is confined to fife contingencies excluding all monetary applications, so that the combination of the theory of compound interest with that of fife contin- gencies is not touched upon. A warning may not, however, be amiss that the present value of a sum of money payable IV PREFACE at death cannot properly be calculated by assuming it to be payable at the end of a definite period equal to the expectation of life, nor can the present value of a life annuity be calculated by assuming it to be certainly payable for that period. R. Henderson. New York, March i, 1915 CONTENTS PAGE Chapter I. Mortality Tables i II. The Mortality Table and Probabilities Involving One Life jy III. Formulas eor the Law oe Mortality 26 IV. Probabilities Involving More than One Life 34 V. Statistical Applications 45 VI. Construction of Mortality Tables 51 VII, Graduation of Mortality Tables 68 VIII. Northeastern States Mortality Table 95 Appendix. Data from Various Mortality Tables 100 MORTALITY TABLES Dr. Halley's Breslau Table 3 Deaths and Population, Northeastern States, 1908-1912 96 The Northampton Table 100 The Carlisle Table 101 Actuaries', or Combined Experience, Table 102 American Experience Table 103 Institute of Actuaries' Healthy Male (H m ) Table 104 British Offices O m[si Table 105 National Fraternal Congress Table 106 Northeastern States Mortality Table, 1908-1912 107 Rates of Mortality per Thousand According to Twelve Tables. . . . 109 Death Rates per Thousand According to Various Tables no DIAGRAMS 1. Comparison of Aggregate and Analyzed Rates of Mortality .... 67 2. Comparison of Graduated and Ungraduated Rates of Mortality. 93 3. Rates of Mortality by Various American Tables 99 v MORTALITY LAWS AND STATISTICS CHAPTER I MORTALITY TABLES i. The subject of human mortality is one which, from its nature, is of widespread interest to mankind. It has always been recognized that it is impossible to predict the duration of any individual life and that the only thing that could be taken as certain on the subject was that death would come sometime to each one. In other words it has been recognized that the date of death of any individual is subject to chance. The scientific study of the subject of chance is, however, a comparatively modern development of mathematics and con- sequently the science of life contingencies is also comparatively modern. 2. Like all other events, whether considered as chance events or as certainties, .the death of any individual or his survival to any specified date is the necessary result of those forces which have been operating upon him. The cause of our ig- norance regarding the result in the case of an individual is the limitation of our knowledge regarding the forces operating and their effects. We do know, however, that among the important ones affecting the result are climate, sanitary conditions, medical attendance and habits of life. These vary in their tendency and effectiveness as we pass from one locality to another or at different times in the same locality and may even differ in a recognizable way as between different individuals in the same locality and at the same time. The results of the observations made with respect to human mor- 2 MORTALITY LAWS AND STATISTICS tality under any given set of circumstances are frequently set forth concisely in the form of a Mortality Table showing the number surviving to each age out of a given number living at some selected initial age. A brief description is here given of some of the Mortality Tables which have had a relatively im- portant part in the history of the science of life contingencies. The Bkeslau Table 3. This table is of importance because it represents the first attempt to construct a mortality table from which to deduce the probabilities of survival and the values of life annuities. It was formed by the celebrated astronomer, Dr. E. Halley, from returns of the deaths in the City of Breslau, Silesia, during the five years 1687 to 1691 inclusive. Owing to the fact that the births during the five years only slightly exceeded the deaths (the numbers being 6193 and 5869 respectively) he assumed that the population might be considered a sta- tionary one. He therefore appears to have graduated by inspection the average number dying per annum at the various ages and assumed that this gave the decrements of the table. Dr. Halley published his table in two columns, the first headed " Age Current," and the second " Persons," and it appears from the explanation given to have been in modern notation equivalent to a table of values of L x ^ 1 , the population at age x next birthday. 4. On the basis of this table Dr. Halley solved various prob- lems regarding survival and calculated annuity values, thus laying the foundations of the science of life contingencies and preparing the way for the transaction, on a scientific basis, of the important business of life insurance, although it was not until nearly seventy years had elapsed after the publi- cation in 1693 of this table that the first company to operate on a scientific basis was established. Mr. E. J. Farren, writing in 1850, said regarding this table: " With respect to its form, as has already been stated, no improvement has as yet been adopted, beyond inserting MORTALITY TABLES 6 the column of differences or deaths, and choosing higher num- bers for exemplification. Of its two principles of construction, viz., as to the number of living being deducible from the num- ber of deaths, by aid of the assumption of a stationary pop- ulation; and as to the number of deaths at contiguous ages after childhood being allied in number; the former principle was generally prevalent in the construction of such tables, until the appearance of Mr. Milne's Carlisle Table in 1815, but is now as generally abandoned; the latter characteristic is still operative and considered as valid in all the best tables." The mortality indicated by this table was considerably higher than that shown by more modern tables. 5. The table as given by Dr. Halley is as follows: Age Cur- rent. Per sons. Age Cur- rent. Per- sons. Age Cur- rent. Per- sons. Ag. Cur- rent. Per- sons. Age Cur- rent. Per- sons. Age Cur- rent. Per- sons. I 2 3 4 S 6 7 1000 855 798 760 732 710 69? 8 9 10 11 12 13 14 680 670 661 653 646 640 634 15 16 17 18 19 20 21 628 622 616 610 604 598 592 22 23 24 25 26 27 28 586 579 573 567 560 553 546 29 30 31 32 33 34 35 539 531 523 515 5°7 499 49° 36 37 38 39 40 4i 42 481 472 463 454 445 436 427 Age Cur- rent. Per- sons. Age Cur- rent. Per- sons. Age Cur- rent. Per- sons. Age Cur- rent. Per- sons. Age Cur- rent. Per- sons. Age Cur- rent. Per- sons. 43 44 45 46 47 48 49 417 407 307 387 377 367 357 5° 51 52 53 54 55 56 346 335 324 313 302 292 282 57 58 59 60 61 62 63 272 262 252 242 232 222 212 64 65 66 67 68 69 70 202 192 182 172 162 152 142 71 72 73 74 75 76 77 131 120 109 98 88 78 68 78 79 80 81 82 83 84 58 49 41 34 28 23 20 Age. Persons 7 5 547 14 4584 21 4 270 28 3 964 35 3 6 °4 42 3 178 49 2 709 56 2 194 63 1 694 70 1 204 77 692 84 253 100 107 Sum Total 34000 The column on the right hand is evidently a summary of the preceding figures in groups of seven years; with an additional item giving 107 as the population at ages 85 to 100 inclusive. MORTALITY LAWS AND STATISTICS The Northampton Table 6. This table was first published in 1783 in the fourth edition of Dr. Price's work on Reversionary Payments. It was constructed by Dr. Price on the basis of a return of the deaths in the parish of All Saints, Northampton, England, during the forty-six years from 1735 to 1780 inclusive. These deaths by ages were as follows: Ages. Number of Deaths. 0-2 1529 2-5 362 5-10 201 10-20 189 20-30 373 30-40 329 40-50 365 50-60 384 60-70 378 70-80 358 80-90 199 90-100 22 Total 4689 7. Owing to the fact that the total number of baptisms during the same period was only 4220, or almost exactly ten per cent less than the number of deaths, Dr. Price, apparently believing the baptisms to correctly represent the births, as- sumed a stationary population supported by births and immi- gration at age 20. He supposed that the immigration was sufficient to supply thirteen per cent of the total deaths. The actual process followed appears to have been to transfer a sufficient number of deaths from age group 20 to 30 to age group 30 to 40 to equalize the numbers in the two groups. The number of deaths in the various groups was then pro- portionately increased so as to make the total number 10 000. Thirteen per cent of this number or 1300 in all were then de- ducted pro rata from the groups above age 20. All the groups both below and above age 20 were then increased pro rata so as to bring the deaths above age 20 up to the same figure MORTALITY TABLES 5 as before. This increased the total deaths to n 650, which was taken as the value of Iq. A subtraction of the deaths in the successive groups gave the values of h, h, ho, ho, etc., the intermediate values being afterwards inserted. 8. From the above explanation it will be seen that in this case, as in the case of the Breslau table, only the death returns were available, without any enumeration of the pop- ulation, and the resulting table indicated rates of mortality, especially at the younger ages, which, in the light of subsequent experience, appear unduly high. In fact, instead of being stationary, the population and the births had been regularly increasing and the population and deaths at the young ages were consequently proportionately higher than would have been the case in a stationary population. Thus when it was assumed that the number attaining any given age was equal to the number dying above that age the result was an under- statement of the former number by the amount of the total increase during that period in the population above that age, subject to adjustment for immigration or emigration. This understatement of the denominator of the fraction determining the rate of mortality, of course, overstated that rate. 9. The Northampton Table was adopted by the Equitable Society as a basis for its calculations immediately after its construction. This fact, combined with the success of that Society, caused its adoption for many purposes for which it was not suitable. An outstanding illustration of this is the fact that the British Government based their rates for the sale of annuities upon it and consequently sustained a serious loss because the longevity of the annuitants proved much greater than was indicated by the table. Until within a few years the Northampton Table with five per cent interest was the basis prescribed by the court rules in New York State for the valuation of life interests and dower rights, but the Carlisle Table has now been adopted instead. MORTALITY LAWS AND STATISTICS The Carlisle Table io. This table was constructed in 1815 by Dr. Milne and was the first table to take both the deaths and the corre- sponding population into account. It was based on two censuses of the population of the parishes of St. Mary and St. Cuthbert, Carlisle, taken January 1, 1780, and December 31, 1787, or an interval of eight years, and on the deaths for the nine years 1779 to 1787. The following schedule shows the data: Age Last Birthday. Popu ation. Deaths 1779 to 1787. Jan. 1780. Dec. 1787. 1 ' 39° 1 173 2 1029 1 164 • 128 3 70 A' 5i 5" 9 908 1026 89 10- 14 715 808 34 IS" IQ 675 763 44 20- 29 1328 1 501 96 3°- 39 877 991 89 4°~ 49 858 970 11S 5^ 59 588 66 S 103 60- 69 438 494 173 70- 79 191 216 152 80- 89 58 66 98 90- 99 IO 11 28 100-104 2 2 4 Total 7677 8677 1840 11. The figures for the deaths are derived from a record kept by Dr. J. Heysham, and the population as of January, 1780, is also derived from an enumeration by him, taking account of the ages. The population as of December, 1787, appears to have been merely enumerated in gross and then distributed by ages in the same proportions as had been found in 1780. It will be seen on examination that the proportions on the two dates are the same, which is a condition scarcely likely to be realized in two actual enumerations. It was assumed MORTALITY TABLES 7 that the average population during the period covered by the observations could be represented by the mean of the two censuses. The method followed in deducing the rates of mortality will be described in Chapter VI. This table presented a more accurate statement of the probabilities of death at various ages than any preceding table and was widely used for insurance calculations. It has now been largely superseded for this purpose by tables more recently constructed from the experience of insured lives. It is, however, still in use for special purposes. Owing to the small extent of the data on which it was based and the graphic method adopted in redistributing the pop- ulation and deaths into individual ages, the rates of mortality were somewhat irregular, particularly at the older ages, and various regraduations of the table have been made with the idea of removing the irregularities. The English Life Tables 12. At various times tables of mortality have been con- structed on the basis of the census returns and registration of deaths in England. On account of the fact that an exten- sive series of monetary tables was based on it the most widely known of these tables is the English Life Table No. 3, which was constructed by Dr. Farr on the basis of the censuses of 1841 and 1851 and the deaths of the seventeen years 1838-54. Separate mortality tables were constructed for male and female lives starting with radices of 511 745 and 488 255 respectively or for the combined table, 1 000 000 at age zero. The method followed in constructing these tables will be described in Chapter VI. 13. At about the same time as these tables were constructed other tables, known as the Healthy Districts Life Tables, were also constructed from the census returns for 1851 of the sixty- four English districts having at that time an average death rate below 17 per thousand, and the deaths in the same dis- tricts during the five years 1849-53. These tables were pre- 8 MORTALITY LAWS AND STATISTICS sented in threefold form, the radix at age zero for the male table being 51 125 and for the female 48875, the two added together constituting a mixed or combined table with a radix 100 000. The Healthy Districts Male Table with certain modifications was used by the Committee of the Actuarial Society of America in charge of the Specialized Mortality Investigation as a basis for the comparison of the mortality in the different classes. 14. The more recent of the series of English Life Tables are designated as Nos. 6, 7, and 8. The English Life Tables No. 6 were based on the census returns of 1891 and 1901 and the deaths of the ten years 1891 to 1900 inclusive and, while not originally prepared by Mr. Geo. King's method described in Chapter VII, have been readjusted -by that method. The English Life Tables Nos. 7 and 8 have just been published and were prepared by Mr. Geo. King by his method. The former set are based on the census returns of 1901 and 191 1 and the deaths of the ten years 1901 to 19 10 inclusive, while the latter are based on the census of 191 1 adjusted for increase to the middle of that year and on the death returns of the three years 1910 to 1912 inclusive. The special feature of these tables is that not only do they indicate an improvement in mortality as compared with the earlier tables of the series, but, when compared with one another they indicate that the improvement was still progressing. The No. 7 Tables show a lower mor- tality throughout than the No. 6 and the No. 8 Tables a lower mortality at practically all ages than the No. 7. The Actuaries', or Combined Experience, Table 15. This table, also known as the Seventeen Offices' Ex- perience Table, was prepared in 1841 by combining the experi- ence, by lives, of the Equitable and Amicable Societies with the experience, by policies, of fifteen other companies as con- tributed in 1838 to a committee of actuaries. It was thus the first example of a mortality table formed by combining the experiences of different insurance companies into one MORTALITY TABLES 9 general average. It appears to have covered in all 83 905 policies or lives of which 13 781 were terminated by death, 25 247 were terminated otherwise, and 44 877 were in existence and under observation when the observations closed. The total of the numbers exposed to risk, for one year at each age, was 712 163 indicating an average duration of 8.5 years. 16. Probably owing to the mixed nature of the data, which as above stated, was partly by lives and partly by policies, and to the fact that the average duration of the experience contributed by the companies other than the Equitable and the Amicable was only 5.5 years, this table was never widely used in Great Britain for insurance purposes. It was, however, prescribed by the State of Massachusetts as the basis for the valuation of the reserve liabilities of life insurance companies. The example of Massachusetts was later followed by New York and other states with the result that for many years the Actuaries' Table with four per cent interest was the accepted valuation standard in the United States, although the pre- miums actually charged by the companies were as a rule based on a different table. The Healthy Male (H m ) Table 17. This is the most important of the group of tables published in 1869 and representing the results of the Insti- tute of Actuaries' Mortality Experience, 1863. They were based upon data contributed by twenty British life insurance companies regarding their experience up to 1863 on insured lives. The H M table was based on the experience of male lives insured at regular premium rates, and duplicate policies on the same life, whether in the same or in different companies, were carefully eliminated. This table represented a much broader experience than that upon which the Actuaries' Table had been based, confirmed in a general way the results of that experience and obtained immediate acceptance as a fair representation of the average mortality of insured fives. The official H M Table was graduated by Woolhouse's formula 10 MORTALITY LAWS AND STATISTICS but it was subsequently regraduated by King and Hardy according to Makeham's formula, with a modification at the younger ages, and extended down to age zero by means of rates of mortality taken from the Healthy Districts Male Table. This graduation of the table is published in Part II of the Text Book of the Institute of Actuaries. 18. In the construction of this table all lives of the same attained age were included together without regard to the period elapsed since medical examination. But an analysis of the experience indicated that the rate of mortality among lives recently insured was much less than among lives of the same attained age who had been insured for a longer period. Accordingly a second table, known as the H M (5) Table, was formed by omitting the experience during the calendar year of issue and the next four calendar years. This table was taken as representing the ultimate rate of mortality after the effects of selection had worn off. The rates of mortality at the young ages are considerably higher in the H M (5) table than in the H M , but the two rates gradually approach one another and coincide at the extreme old ages where there are no recently selected lives. 19. These two tables used together were adopted by many British companies for the valuation of their liabilities, and the H M Table was for many years prescribed for that purpose by the laws of Canada. It will be noticed that the rates of mortality according to the H M Table are lower than those for the same ages in the Actuaries' Table except for ages 46 to 50 and ages 73 to 85 inclusive and ages 95 and over. The difference is not, however, important except at the young ages, where it is considerable. The British Offices' Life Tables, 1893 20. These tables represent the experience on insured lives of sixty British life insurance companies during the thirty years from the policy anniversaries in 1863 to those in 1893. The data were compiled under the joint supervision of the MORTALITY TABLES 11 Institute of Actuaries and the Faculty of Actuaries in Scot- land and was classified into male and female lives and accord- ing to the plan of insurance issued. The M Table represents the experience of male lives insured on the Ordinary Life plan with participation in profits. The total number of lives under observation was 551 838, of whom 149 566 were insured prior to 1863. Of these 140 889 died, 148 392 withdrew and 262 557 remained insured in 1893, the total number of years of risk being 7056863. The M (5) Table represents the same ex- perience, omitting the first five policy years and covers 5 324 862 years of risk and 129 001 deaths. These tables were grad- uated by Mr. G. F. Hardy. The M (5> table was first grad- uated by the application of Makeham's formula, the dif- ferences in the values of log p x by the two tables being then graduated by the use of a double-frequency curve. A select or analyzed table was also prepared from the same data and is known as the [MI Table. In this table separate rates of mortality are indicated for each age at entry for the first ten policy years, merging into an ultimate table at the end of that time. This select table was also graduated by Makeham's formula, different constants being used for the different policy years. The rates of mortality by the M table are lower throughout than those in the H M table as graduated by Make- ham's formula and also, with unimportant exceptions, than those in the official H M table. The M (6) Table is the basis at present prescribed for the valuation of policies in Canada. The American Experience Table 2r. This table was constructed by Mr. Sheppard Homans and was first published in its present form in 1868. No com- plete record has ever been made public of the method adopted in its construction, but it has always been understood that the mortality experience of the Mutual Life Insurance Company of New York was used as a basis. As that experience covered only a few years and therefore did not include any exposures or deaths at extreme old ages it must have been supplemented 12 MORTALITY LAWS AND STATISTICS from other sources. The table is a very smoothly graduated one and evidence has been discovered which seems to indi- cate that the author first constructed a table of values of the reciprocal of the rate of mortality showing the number of lives out of which one death would be expected at each age. From these values the usual columns of the mortality table were then formed. 22. The first publication of the table was in the schedule of an act prescribing it as a basis of valuation in the State of New York and although it was temporarily abandoned in that state for the sake of uniformity it is now the legal standard in practically every state of the Union. The table as originally published was found to conform very nearly to Makeham's law, and was subsequently regraduated in accordance with that law for use in connection with joint life calculations. The American Experience Table has been widely used in America as a basis for insurance premiums even when another table was prescribed as a legal basis of valuation, as it presented a conservative view of the mortality after the effect of selec- tion had worn off. The rates of mortality shown were higher than those in the Actuaries' table for ages 30 and under and for ages 78 and over, but lower between 30 and 78. Com- pared with the H M Table, which was published about the same time, it gave higher rates of mortality for ages under 36 and over 80 and slightly lower for the intervening years. Compared with the M(6> it shows higher rates of mortality for ages 40 and under and for ages over 70 and lower values for the intermediate ages. It will thus be seen that in general the American Experience Table seems to give relatively low rates of mortality for the central ages and high rates for the young and old ages. The National Fraternal Congress Table 23. This table was constructed by the Committee on Rates of the National Fraternal Congress, an association of Fraternal Societies m the United States of America, and was presented MORTALITY TABLES 13 in its original form at the annual meeting of that association in 1898. It was based on the experience up to that time of the societies connected with the Congress. It was subse- quently regraduated by Mr. Abb Landis and reported in its amended form the next year. Compared with the American Experience Table the rates of mortality are lower throughout, although the difference is proportionately smaller at the older ages than at the younger. Compared with the M Table the rates of mortality are higher at ages 20 to 27 inclusive and for ages 81 and over and lower at the intervening ages. 24. This table of mortality with interest at four per cent is prescribed as a basis for minimum rates of contribution in fraternal orders by the laws of several States. It is worthy of note that a subsequent investigation was made of the experi- ence during the year 1904 of 43 societies. This experience covered 2 880 166.5 years of exposure and 19 414 deaths, and the rates of mortality in the resulting table were lower than those in the National Fraternal Congress Table for ages up to 52 inclusive and for ages 79 and over, but higher for the inter- vening ages. The M. A. Table of the Medico-Actuarial Mortality Investigation 25. This table was constructed in 191 2 by the joint committee of the Medical Directors' Association and the Actuarial Society of America in charge of the Medico-Actuarial Mortality Investigation into the relative mortality of special classes of risks. It was intended for use as a standard with which to compare the mortality of the special classes. It was there- fore based on the experience of the same companies as con- tributed to the special class experience on policies issued during the same period and observed up to the same date. The data used were based on the experience of the companies on policies issued during the month of January in odd years and July in even years from 1885 to 1908 inclusive, observed to the anni- versaries in 1909. The total number of policies was 500 375, 14 MORTALITY LAWS AND STATISTICS the total years of exposure 2 814 276 and the number of policies terminated by death 20 222. The table is shown in the form of analyzed rates of mortality for the first four policy years with an ultimate table for the fifth and subsequent years. A special feature of this table is that the difference between the rates of mortality in the early policy years and those shown for the same attained ages in the ultimate table is relatively small. This has been explained on the theory that an improve- ment in general mortality conditions was going on during the time of the observations and that, owing to the fact that the observations in the early policy years were on an average made at an earlier date than those for the longer durations, this partly concealed the true effect of selection. This theory was confirmed by investigating separately the experience on policies issued in the years 1885 to 1892 inclusive, those issued in 1893 to 1900 inclusive, and those issued in 1901 to 1908 inclusive. A progressive improvement was shown in passing from one group to the next. In the ultimate part of the M. A. Table the rates of mortality are throughout lower than those for the same age in the American Experience Table, but prac- tically equal at age 69. Compared with the National Fraternal Congress Table, they are lower at ages under 55 and over 80 but higher between those ages. For ages under 70 the rates of mortality are lower than in the ultimate part of the British Offices' [M1 Table, but after that age they agree exactly with that table. 26. This table was constructed in a special way for the special purpose above indicated and is not recommended by its authors for any other purpose. The experiences, however, of some individual companies which have since been investi- gated appear to confirm substantially the ultimate part of the table as a fair representation of ultimate mortality of in- sured lives in American and Canadian Companies transacting a normal business. mortality tables 15 McClintock's Annuitants' Mortality Tables 27. These tables were constructed in 1899 by Dr. McClin- tock on the basis of experience of fifteen American companies, collected and analyzed by Mr. Weeks. The data comprised the entire experience of the companies on annuities up to the anniversaries of the contracts in 1892. Separate tables were constructed for male and female lives, the number of lives taken into consideration being 4365 males and 4821 females. Although this was an experience of American companies only about one-fourth of the number of annuitants were actually American lives, the remaining three-fourths representing an- nuities granted abroad by the companies. The experience was taken out strictly by lives, all duplicates being carefully eliminated, and in the case of deferred annuities only the ex- perience after the annuity became payable was considered, owing to the uncertainty with regard to the date of death during the deferred period. 28. Each table was graduated by Makeham's formula, (Art. 50), the same value of c being used for the two tables and in consequence of this fact the principle of uniform seniority may be used, although in a modified form, even where the lives are not all of the same sex. The formula adopted was colog ^z = log b+c x log h, where log £ = .04 and for the male table log b = .003 2 and log log ^ = 5.55; for the female table log b = .001 s and log log h = 5.43. The rate of mortality is higher throughout the male table than for the same age in the female table, the difference being proportionately greatest at the young ages. The rate of mortality in the male is higher than in the American Experience Table up to age 62 and lower above that age. In connection with these tables it should be remembered that at the young ages they are purely theoretical, there being only two actual deaths at ages under 40 in the male experience and three in the female. These tables are now prescribed by the law of New York State as the basis for the valuation of annuity contracts issued by life insurance companies. 16 mortality laws and statistics The British Offices' Life Annuity Tables, 1893 29. These tables are derived from the experience of British Offices in respect of life annuitants, male and female, during the period 1863 to 1893, including the British Annuity experi- ence of three American companies. Both select and aggregate unadjusted tables were constructed, duplicates being separately eliminated for each. After the final elimination of duplicates for the aggregate tables the total number of male lives involved was 6728, the number of years of risk 53 599 and the number of deaths 3503. For the female table the number of lives was 18 951, the number of years of risk 173 519 and the number of deaths 9107. 30. The graduated tables constructed from these data were shown in the select or analyzed form with separate rates of mortality for each of the first five contract years, merging into an ultimate table at the end of the fifth year. The male table was graduated by Makeham's formula (Art. 50), mod- ified for duration, the value of logioc being .038. The female table could not be graduated as a single series by that law. A second series was therefore introduced and it was assumed that l lx]+t = l ix]+ t+l lx]+l , where l [x]+l and lf x \ +t each con- formed to Makeham's formula modified for duration. The rates of mortality in the ultimate part of the male table are lower than in McClintock's table for ages under 50 and over 82 and higher for the intervening ages. In the ultimate female table the rates of mortality are higher throughout than in McClintock's table. The value at 3! per cent interest of an annuity at date of issue is somewhat higher by the British Offices' Male Table throughout than by McClintock's Table. By the female table the value at date of issue is lower than by McClintock's table for ages under 62 and from a— / 1 1 g4+j 1 1 " 4+j 1 1 dlx+t 1 1 g 4+t 1 , 2 « 4 r 12 d^ 48 rfr "4+(+i _d>l x +t \ _d i x +i , 1 tf 4+t I ifl 4+t I , ^ J /^+ 2 ^ I + 1 1 I+{ * W1 ' ^" 12\dX dx 72o\ ax 6 ax 01 / etc. ...... Whence, summing and remembering that at the upper limit l x+t and all its differential coefficients may be assumed to vanish, we have I l x + tdt = 2'z~T"2i l x + t~\ j T^"r e tC, J Q ' 12 dx j2oax 6 = ?it2i l x +t i x n x -— r+etc. 12 j 20 ax 6 1 dH Hence, if we neglect the term — f and all higher differ- j2odx 3 ential coefficients, we have "x^x — 2 "x "T -"1 "x+l '\~ i i}xP-Xi — ~%v%~T*xfix lawj or e x = h + e x -^x (19) This shows that the correction to the first approximation is approximately —-fan s . As the value of this correction is very small except at extreme old age it is usually neglected and the first approximation used for e x . MORTALITY TABLE AND PROBABILITIES INVOLVING ONE LIFE 25 46. From a table of the expectation of life it is possible to derive directly the corresponding rates of mortality, for we have L x c x = /z+i~Mx+2~r4:+3 _ re'tc. ) '1+1(1 -\-e x +i) = /:t+l~r'z+2~Mz+3~r e tC., so that we have l x e x =l x+ i(i-\-e x+ i), or e x = ~ :} (i+e x+1 )=p x (i+e x+1 ), . . . . (20) l x whence, and P*=—, . i+e I+ i gx ^- {e r ex+i) c«) i+e x+ i Eq. (21) gives the rate of mortality in terms of expectations and Eq. (20) gives a rule for determining e x from p x and e x+x . By the successive application of this formula, beginning at the oldest age, the expectation of life for all ages may be computed from the rates of mortality without constructing the l x column. 47. The value of \x x may also be expressed in terms of l x dx, from Eq. (18). Differentiating, then, with respect to x we have, after changing sign, de 7 ° 7 "-frj 7 l^ X l' X € x i x - i x , ax or 7 O nJe* = i +-f = 1 -i(e x -i-e x +i), approximately whence i—-|(gs-i — g»+i) CHAPTER III FORMULAS FOR THE LAW OF MORTALITY 48. Before the various labor-saving devices now in use in connection with the calculation of monetary values from the mortality table had been invented, the desirability of reducing, if possible, the mortality table to a mathematical law in order to facilitate such calculations was especially evident. The first attempt of this kind was made by DeMoivre in his " Treatise of Annuities on Lives," for the purpose of passing from the expectation of life to the value of a life annuity. His assumption was the very simple one that the value of d x was the same for all ages, or in other words, that l z decreased uniformly up to the limiting age. The equa- tion for l x in terms of x can therefore be written l x = a(w — x), (1) where x is less than w. Whence dl x dx and therefore, also ix=V-x-- a(w — x) w — x' (2) lx°Cx= J l x dx= I a(w — x)dx= — [(w — x) 2 ] =-(w — x) 2 Jx Jx 2*2 whence °e x = h{w-x) (3) This equation may also be stated in the form w = x-\-2°e x , which shows that, if the formula applied, the function x + 2e x would be a constant. The calculation of this function for 26 FORMULAS FOR THE LAW OF MORTALITY 27 two or three ages at intervals, or the examination of the d x columns of any mortality table based on actual experience will show that DeMoivre's hypothesis is only a rough ap- proximation to the truth. While it accomplished its purpose of enabling approximate life annuity values to be calculated from the expectation it cannot be accepted as a statement of the true law of mortality. 49. In the next attempt the problem was approached directly by an investigation of the causes of death. It was made by Benjamin Gompertz, who assumed that the force of mortality increases in geometrical progression with the age. This may be written as follows : v-x = Bc x , (4) where B and c are constants for a given mortality table, but may have different values in different tables. From this equation we have d log e l x _ B(f _ dx Whence, integrating with respect to x, we have B . l0g e k = l0g e k- loge C or k = ktf, ( S ) 7? where log e g=- 1 , or B = - log e g ■ log e c. We may thus log e c express p. x in terms of the constants of the mortality table by substituting this value for B, so that we have fl x =- (loge g T0ge C)C X (6) 50. Gompertz's formula constituted a genuine approxima- tion to the law of mortality, but it was found that it did not apply to the period of childhood, and that even at adult ages it would not cover the complete range without a change of constant at an age in the neighborhood of 50 or 60. To remedy this Mr. Makeham proposed to modify Gompertz's formula 28 , MORTALITY LAWS AND STATISTICS in a way actually suggested by the reasoning of Gompertz himself, who had stated that "It is possible that death may be the consequence of two generally coexisting causes; the one chance, without previous disposition to death or deteri- oration; the other a deterioration, or increased inability to withstand destruction." The modification consisted in adding a constant to the expression for the force of mortality, which became !x x =A+Bg* ( 7 ) = -loge-S-Oogeg- log, C>f (8) by putting A = — log c 5 Substituting then ge - for n x and integrating, we get log,, l x = log e k+x log, s+c x log e g, where log e k is the constant of integration, or l x = ksY x (9) This formula has been applied to various mortality tables with considerable success, reproducing them very closely from about age 20 to the end, but not covering the period of infancy. Certain other tables, however, cannot be reproduced by this formula. 51. The simplest way of determining the constants in Makeham's formula is from four equidistant values of log l x . We have. log 4 = log k +x log s -\-c x log g, log l x+t =log k + ix+t) log s+c x+t log g, log 4+21 = log k + {x + 2t) log S+C x+2t log g, log Z I+3! = log k + (x+$t) log s+c x+m log g. Taking now the differences, we have log l x+t -log l x =t log s+c x (c'-i) log g, log 4+21 -log 4+< =tlogs+c x+t (c'-i) logg, log 4+31 -log 4+21 = t log s + c x + 2t (c' - 1) log g. FORMULAS FOR THE LAW OF MORTALITY 29 Taking differences again, we have logl x+2t -2 log 4+, +log4 = c :c (c ! -i) 2 logg ] log k+3*-2 log k+2*+l0g l x + t =C X + '(c'-l) 2 log g. Dividing the second by the first, we have j fog h+3t-2 log 4+2i+lQg lx+l ■ ■"*-■■ log 4+21-2 log / x + , + k>g l x From this equation c is determined and then in succession log g, log s, and log k. 52. For example, according to the Makehamized American Experience Table, we have, log & + 20 logs+c 20 log g = log ho= 4.96668 log &+40 log s+c 40 log g = log ho= 4.89286 log k+bo log 5+c 60 log g = log ho= 4.76202 log &+80 log 5+c 80 logg = log /go= 4.16122 20 log s+c 20 (c 20 — 1) log g = — .07382 20 log 5 + C 40 (c 20 — i) log g= —.13084 20 log 5 + C 60 (c 20 — l) k>gg= —.60080 c 20 (c' 20 — i) 2 log g = — .05702 C 40 ^ 20 -!) 2 log g = -.46996 Taking logarithms 20 log c + 2 log (c 20 — i)+log log g = 2. 7560372 40 log c + 2 log (c 20 — i)+log log g — i.6']2o6n 20 log c = . 91603 log £ = .0458015 . . (a) Hence c 20 = 8.242 log (c 20 -i) = . 85986 20 log C + 2 log (c 20 -l) = 2.63575 log log # = 2.7560372- 2.63575 =4-I2028w log £=-.00013191 . (b) 30 MORTALITY LAWS AND STATISTICS 20 log c+log (c 20 -i)+loglogg = 2.756o3«-.85986 = 3.896i7» c 20 (c 20 -i) log g = -.00787 but 20 log s+c 20 (c 20 - 1) log g = - .07382 .-. 20 log 5 = - .06595 log 5= -.OO32975 . . (c) 20 log c+log log g = 3~.0363iw c 20 log g = — .00109 20 log S + C 20 log g = — .06704 but log & + 20 log S + C 20 log g= 4.96668 .-. log&= 5.03372 . . (d) 53. It is interesting to compare the values thus calculated with those from which the table was constructed. The dif- ferences arise from the fact that in the mortality table the value of l x is expressed to the nearest unit and that we have used five-figure logarithms throughout. The comparison is as follows : Exact Value. Calculated Value. log c .04579609 .045802 logs - log g log& 5 OO3296862 —.003298 OOOI3205 —.OOO13191 033701 16 5 .03372 The value of log c in the Makehamized American Experi- ence Table is one of the highest which it has been found nec- essary to use. The range of values lies between .036 and .046 with a general average close to .04. 54. Unfortunately for the widest usefulness of Makeham's formula it is not possible to evaluate the integral J ks x g cX dx otherwise than approximately, so that it does not serve the original purpose for which a mathematical law was sought. Any mathematical law, however, gives a very smooth series FORMULAS FOE THE LAW OF MORTALITY 31 which enables formulas of approximate summation or inte- gration to be used which greatly reduce the labor of cal- culating the values of complicated benefits, and Makeham's law in particular offers great advantage in the calculation of probabilities of survival involving more than one life, on account of the form of the expression for log „p x . This point will be discussed further in connection with those probabilities. The advantage thus secured is so great, however, that it is considered that a mortality table which is to be used for monetary calculations should be adjusted so as to conform to Makeham's law if it can be accomplished without departing too seriously from the facts upon which it is based. The general subject of adjustment or graduation will be taken up in a later chapter. Modifications have been proposed to Makeham's formula for the purpose of making it fit certain tables more closely. These modifications consist in adding terms to the expression for p x . One modification assumes t i x =A J rHx+Bc x , adding the term Hx to Makeham's expres- sion, and another takes the form jx x = ma x -\-nb x . Both of these modifications sacrifice a considerable por- tion of the advantage which can be secured from the use of Makeham's formula in its original form. 55. Another formula has been proposed by Wittstein, which is intended to cover the entire range of fife from infancy to extreme old age. He assumes that the values of q x may be expressed in terms of x as follows : fc = a-w-«"+i-fl-<«*>*. m From the form of this expression it is evident that the first term becomes equal to unity when x = M and that where a is greater than unity and n is positive the value of this term increases regularly up to that value. It appears therefore that M+i must equal the limiting age in the table. Also for n I infant mortality we have q = a~ M -\ — , showing that the m 32 MORTALITY LAWS AND STATISTICS probability of death during the first year after birth is slightly greater than — . Also m dq. MM-x) n - 1 a- m - x)n -n(mx) a - 1 a- imx)n) i lo §' «• dx Now it is evident that this vanishes when M . mx = (M — x) or %-- m- and it will be found that for all cases arising in practice this represents a minimum value of q x . In applying this formula to mortality tables it is found that for normal mortality in temperate climates a is approximately 1.42, n is approximately .63, M is between 95 and 100 and m is not less than 6. With these values it is evident that the second part of the expression for q x decreases rapidly as x increases and becomes negligible at about age 25, so that the first term may be taken to rep- resent adult mortality and the second term to represent the additional mortality of infancy. This formula does not possess the practical advantages of Makeham's and consequently has not been much used in practice. 56. Still another method was adopted by Prof. Karl Pearson, who took the numbers dying at the various ages and analyzed the series into the sum of five frequency curves typical respect- ively of old age, middle life, youth, childhood, and infancy. The table selected was that known as the English Life Table No. 4 (males) and the expression which he deduced for l x fi x was as follows: lxu.x = 15-2(1- ' 5 ) c J»«(«-7Xjn ) + c4e"'- 05S!4(I "" u - t)|! J_ 2 _gg-[-09092fct-22.5)F +8.5(x-2); 3271 e-- 3271fa - 3 > +4i5.6(x + .75)-- 5 e-- 75 < I +-7s> FORMULAS FOR THE LAW OF MORTALITY 33 In the first four curves the maximum values are at ages 71.5, 41.5, 22.5, and 3 respectively, while the fifth theoretically extends below age zero, the ordinate becoming infinite at age — .75. The method has not, however, been applied to other tables and it is difficult to lay a firm foundation for it, because no analysis of the deaths into natural divisions by causes or otherwise has yet been made, such that the totals in the various groups would conform to these frequency curves. CHAPTER IV PROBABILITIES INVOLVING MORE THAN ONE LIFE 57. In calculating probabilities involving more than one life it is usual to assume that the probabilities of survival or death of the various lives involved are independent of one another, so that the probability of a compound event is found by simply multiplying together the elementary probabilities of which it is composed. For example, the probability that two lives now aged x and y respectively will both be alive at the end of n years is found by multiplying the probability that the life aged x will be alive, n p x , by the probability that the life aged y will be alive, „p v . For the sake of brevity it is usual to write (x) for a life aged x. If then the probability that both (x) and (y) will survive n years be denoted by n p xv , we have nyxy nY% nj^y , , \IJ lx 'ly Similarly, where more than two lives are involved, we have nPxyz ' • • npz'npy'nPz • .... y2J 58. These probabilities of joint survival are the elementary forms to which other probabilities are usually reduced. It is interesting to investigate the form which they take when Makeham's law applies. We have log n p x = log l x+n -\og l x , = {logk + (x+n) log s+c c+a log g} — {log k+xlog s+ =(a"-i) la a log r + (6"-i) ft" log s = l log n p w . From this it follows that for the m lives (x), (y), (z), etc., we may substitute I lives of equal ages w. The difficulty is that I is not usually integral and it would, in practice, be found necessary to determine any required value by a double inter- polation because w also is usually not integral. 63. We have seen that for a single life (x) we have the relation e x = ~S n p x . Similarly out of a large number N of groups of m fives aged respectively x, y, z, etc., we find N\p xv2 . . . (m) complete the first year, N-2p X yz ...&») complete the second, and so on, so that if we denote by e xyz . . . (m) the average number of years completed during the joint continuance of the m lives, we have ^xyz . . . (m) = ^npxyz . . . (m) , 1.4/ Consequently where any expression occurs involving a sum- mation with respect to n of the probabilities of joint survival, we may substitute a joint expectation. 64. Heretofore we have dealt with the probability that all of the fives involved shall survive. Similar reasoning will, however, show that the probability that every one of the m fives will be dead at the end of n years is obtained by multiplying together individual probabilities of death. This probability is expressed by Or I npxyz . . . (m) • 38 MORTALITY LAWS AND STATISTICS We have therefore, \n1xyz . . . (m) = \riQx ' \nQy ' |ra?z . . . = I 1 — nPx) \J npy)\I n Pz) • \S) In this symbol the bar over the letters denoting the ages of the lives involved signifies that the last survivor of the fives is in question, the probability designated by \ n q xyz . . . (m) being that the last survivor of the m lives shall have died before the end of the nth year. 65. The complementary probability is nPxyz ...(m)~I V 1 nPx)\^ npy) • • • , and is evidently the probability that at least one of the lives will survive n years. By expanding the product and reducing the equation may be written as follows: npxyz . . . (m) ^npx ^nPxy\ ^nPxyz etC. . . yO) In this equation the summation extends over all probabilities similar to the one under the 2, that is, involving the same number of lives. 66. Let us now investigate the probability that exactly r out of the m lives will be alive at the end of n years. This probability is designated by „p m The probability xyz . . . (m) that r particular lives, (x), (y), etc., are alive and the remain- ing (m — r) lives (2), (w), etc., all dead is evidently nPxy . . . (r) -\n(}zm . . . (m-r) Or n p xy . _ . M ( I — B p 2 ) ( I — n p a ) . . . and the total probability sought is the sum of these probabilities for all the combinations rata time of the m fives, or nP W = ^nPxy . .. m(l-npz)(l-nPw) ■ ■ ■ (7) xyz . . . (m) From the form of the expression it is evident that it may be expanded in a series of probabilities of joint survival involving from r up to m fives; also that each probability involving more than r lives will appear more than once in the expression because it will appear once for each combination r at a time of the lives involved in it; also that the sign of any probability in- PROBABILITIES INVOLVING MORE THAN ONE LIFE 39 volving r+t lives is positive or negative according as t is even or odd. Thus we have . = ^nPxy . . . (r) — T+lCr^nPxu ■ ■ ■ fr + D "T r+2Cr^nPxy . . . (r + 2) — etc. = 'Snpxy . . . (r) — (/ + l) %nPxy . . . (r+1) ' 1-2 S B /> W . . . (r+ 2) — etc. . (8) 67. This may be verified by supposing all the ages x, y, z, etc., to be equal, in which case 2 n p xvs . . . (r+ o becomes equal to m c T+tn px T+t , because m c r+t is the number of terms included in the summation and each term becomes equal to n p/ +t . The whole expression therefore reduces to ■rr&TnPx T+lCr'mCr+1' npx + r + 2 c t ' mfir + 2 npx " CtC, ~ rrfir' nVx m^r' m — r^l' nPx ~\~m^T'm — r^2'nPx etc. — mCr'nPx(.I nPx) This is evidently the proper expression for the probability in question, because the probability that any particular r of the m lives aged x are all alive and the remaining (m — r) all dead is n px T (i—npx) m ~ r , and there are m c T different groups of r lives included among the m. 68. A very convenient symbolic notation is sometimes used to condense the form of Eq. (8) by substituting Z' for "Znpxyz . . . (o , when the equation takes the form «P m : xyz . . . (m) ^Z r -(r+i)Z r+1 + ^ +l) ^ +2) Z r+2 -(r+i)(r + 3 )(f+3)^ + , +etc 1-2-3 =Z'(i+Z)-< r+1) (9) In this connection it is to be remembered that the expres- sion is purely symbolic and that no operations can be performed upon it which in any way disturb the meaning of Z l . 40 MORTALITY LAWS AND STATISTICS 69. Let us now investigate the probability that at least r out of m lives will survive » years. This is denoted by „p r and it is evident that we have xyz . . . (m) nP I =np W + n p lr+11 +. . . „Pxyz...(m)- (io) zyz . . . (m) 2:1/3 . . . im) xyz . . . (m) From this we see that the expression may be written in the form „ p r = ?, n p xm . . . m +ai~S n p xvz , . . u+v+azZnpzyz . . . ( , +2) +etc. xyz . . . (m) where at, 0,2, etc., remain to be determined. But from Eqs. (8) and (10) we have 0-t = I — r+«Cl+ r+ (C2— r+( C3 + . . ■+(-l)r+A, = r — (l +r+J- lCl) + (r+t- lCl +r+(- 1C2) — ( 7 + t -iC2 +t+i- 1C3) + • ■ ■ + (-iy(r+t-lCt-l+r+t-lCt) Therefore, we have rfV-l - 1) nP = ^npxyz . . . (r) ~t ' ^npxyz . . . (r + 1) ~\ ^npxyz . . . (r+2) xyz ... (7?i) 2 — etc., = Z'(i+Z)"', („) where the same meaning is assigned to Z T as before. 70. It is to be noted that although the relation is purely symbolic and the function of Z has no meaning except as expanded in ascending powers and then interpreted, we have the following relation : »P 1 =Z T (i+Z)- r =Z'(i+Z)-«+»\i-Z(i+Z)- 1 }-i xyz . . . (m) =Z'(i+z)-< r + 1 Mi+z(i+z)- 1 +z 2 (i+z)- 2 +. . .! =Z r (i+Z)-< r+I >+Z r+1 (i+Z)- (r + 2 >+Z r + 2 (i+Z)-< r + 3 > + . . . etc., =»P !rl +np "•+» + n p ir+2] +etc xyz . . . (m) xyz . . . (m) . xyz . , . ( m ) as in Eq. (10). PROBABILITIES INVOLVING MORE THAN ONE LIFE . 41 71. Also if we have m lives aged respectively x, y, z, etc., the expected number of survivors at the end of n years is ~2r„p m where the summation extends over, all values xyz . . . (m) of r from unity to m. Expressed symbolically this becomes, from Eq. (9), Z(i+Z)- 2 + 2Z 2 (i+z)- 3 + 3 Z 3 (i+Z)- 4 +etc., =Z(i+Z)- 2 {i + 2 Z(i+Z)- 1 + 3 Z 2 (i+Z)- 2 +etc.}, =Z(i+Z)- 2 {i-Z(i+Z)- 1 }- 2 , =Z = S„^ (12) This may be also verified by reasoning similar to that by which Eq. (11) was deduced. We thus see that the expected number of survivors out of any group of lives is found by adding together the individual probabilities of survival. 72. Another class of probabilities involving more than one life relates to the order in which the deaths occur. The probability that (x) will die in the nth year from the present time is &x+n-l "z + n-l "x+n , , nPx~\ , 7 ~" 7 —n-lPx nfx 7 npxt '■x l-x px-1 and the probabihty that (y) will be ahve at the end of the nth year is n p y . The probability therefore that ix) will die in the nth year and (y) will be ahve at the end of that year is »^-l _ J, \ f, = nPx£l 1 y_ . nfx \nfy nifxy Px-1 I Px-1 Summing this function for all values of n from unity up, we get the total probabihty that (y) will be alive at the end of the year in which the death of (x) occurs This sum is I v ft= m _y" =0 ° k — * _ •%,_! nfx-l : v i . = 1 »?"«~7 e x-l:v e xv . Px-1 . ' Px-i Similarly, the probability that (x) will be ahve at the end of the year in which the death of (y) occurs is e x ■ ^zi — e xy , Py-l 42 MORTALITY LAWS AND STATISTICS and the probability that both deaths will occur in the same year is the complement of the sum of these probabilities and is there- fore, , _ i i I \ 26 X y C x ~ i : y @x : y — 1- Px-1 Pv-1 It may be assumed that where both deaths occur in the same year the chances are even that the death (x) will occur before that of (y) . The total chance therefore that (x) will die before (y) denoted by Q\ y is \Jxy = \ ~ @x-l : y &xy \ \ 1 I [2^ ; ?x~l : y " "x : y-\ ( (Px-1 J 2[ Px-1 P V -1 J '-{i-" ' -e7 ex:y-^l\ (13) , . ^x-1 : y 2 I Px-\ Py- 73. The same probability may be otherwise expressed in terms of the infinitesimal calculus by indefinitely reducing the intervals considered. The probability that the death of (x) will occur in the interval of time between t and t+dt will be Y^dt/l x , and the probability that (y) will be alive (Mr at that time is —■. The total probability therefore that Ly (y) will be alive when the death of (x) occurs is Qlv= ~Tjyj 4+( ir^' — tt I %+« , ai, l x'yJo ax = ~TT"d~r I b+th+tdt, i_ d_ (1 , o s U v 'dx Kxv6xv) ' = --—(le ) ■l x dx Kxxvh __l/o dl x yidezA l x \ %v dx x dx /' PROBABILITIES INVOLVING MORE THAN ONE LIFE 43 _I/o J _,< _ o de xll ~ Mxv dx ' dx /' = vJzy+^(ex-i:v-e x +i:v) approximately. (14) 74. Similarly, where m lives, (x), (y), (z), etc., are involved, the probability that (x) will die first is Qxyz . . , (ml — —J-, I , h + fH+t . . . dt, vxh . . . Jo dt Hx^xyz . . . Cm) T 2 \^x — \:yz... (m) &x+ 1 : yz . . . (m) / • • V ■*- 5 / 75. Also from the fact that the probability that (x) will die rth in order out of the group of m lives may also be stated as the probability that, when {£) dies, there will be exactly m — r survivors of the m — i lives other than (x), we may express this probability by the use of Eq. (9) in terms of probabilities of dying first. In fact, if we denote by Y l the sum of the values of Q l xvz ... for (x) along with all groups t at a time of the (ra — 1) lives (y), (z), etc., we have o [i x+l tPxZ^ii+zy >d*. where the summation included in Z covers only the to — 1 lives (y), (z), etc. p x+t ,p x Z'dt = Y 1 for all values of I, Therefore expanding, integrating, and condensing, we have &.... w = P-'(i + F) -<•»-'+». . . . (16) For m = 3 we have \Jxyz = Px xyz\~2\Vz-l:yz &x+l : yz) , • ° ° V 1 ?/ $„ z = F(i + F)- 2 = F- 2 F 2 , = Q l xy + Qlz-2Q 1 xyz, ...... (l8) & = (i + F)" 1 = i-F + F 2 , = i-Qxy-Qlz+Ql„z (19) 44 MORTALITY LAWS AND STATISTICS 76. Where Makeham's law is assumed to hold, the prob- ability Ql v takes a special form. We have generally Qxy—— 7T I 'c+ ! j/ TT I h+Jx+tHx+tdt, t'xi'vja m ''xhjo = 4^ t p xt 4t+B'x+i = i' X td x , we have for the population at age x last birthday denoted by L x the following L x =£(l x -td x )dt = l x -\d x = \(l x +l x+1 ). . . (1) Also the deaths occurring per annum between the ages x and x-\-dx will be l x \x x dx = — — - dx. Therefore, the total deaths dx per annum at age x last birthday or between ages x and x + i will be I — T 1 dx=l x — l x+ i=d x . Summing this for all ages J x dx x and over we see that the total deaths for those ages is l x , so that the aggregate number of deaths per annum at all ages 45 46 MORTALITY LAWS AND STATISTICS will be Iq, which is also the number of births. The population is therefore constant in total number and also in age com- position. 79. The total population at age x and over will be Z"Z I} which is usually denoted by T x , so that we have The total population at all ages will be T = loe 0) (3) The general average death rate is obtained by dividing the total number of deaths per annum by the total population, its value is therefore equal to h/To — h/heo = i/et or the re- ciprocal of the complete expectation of life at birth. Similarly, the average death rate at ages x and over will be 'x/ J- x = V»A = 1/ e x- The total number of deaths per annum between ages x and x+n will be l x — l x+n , and the total population at the same ages will be T x — T x+n , therefore, the average death rate between those ages will be — * — * + " . When n is equal to 1, this be- *- x J- x-tn comes the central death rate for age x last birthday and is denoted by m x , so that we have "x vx+1 &x U X (L x / \ mx ~ T —T T - A/7 4-/ \~l IT' ' ' W - 1 x 1 x+1 J->x 2\ t 'Xlh;+lJ h 2 a x If we divide both numerator and denominator by l x , we have m x in terms of q x , as follows: Qx 1 1 1 r \ »« = ^T- or — = § (S) i—%q x m x q x 80. Assuming uniform distribution of deaths within each year of age, the sum of the ages at death of all those dying in a year at ages x and over will be 2z (X-Tzjdx = Z, x (X-\—% )\"x ''x+l), = (x+^)l x + ^ +i l x =xl x + 2™L x = xl x + T x , = \X~\-e x )l'X' STATISTICAL APPLICATIONS 47 Since the total number of deaths at those ages is l x it follows that the average age is x+e x . Putting x equal to zero, we see that the general average age at death is e - The aggregate of the ages at death of those dying between ages x and x-\-n is evidently (xl x + T x )-{(x+n)l x+n + T x+n }=(T x -T x+n )+{xl x -(x+n)l x+n \ and the number of deaths is l x — l x+n , so that the average age is \-L x J- x + n) \Xl x \X--\-Vl)l x + n ■ i x ■* x + n ^x+n T~-i / -/ ' "x "x+n "x l, x+n 81. We have seen that the population at age x last birthday is L x , and the total population is T , so that the proportion of the total population at age x last birthday is L x /T and the proportion between ages x and x+n is (T x — T x+n )/T . Sup- pose, for example, that all young men are required to serve in the army from age 18 to 21, then, assuming a stationary pop- ulation, the proportion of the total male population so serving will be (Tig — 7V)/r . 82. We have hitherto assumed that we are dealing with a stationary population. A consideration, however, of the ques- tion leads to the conclusion that such a condition never exists, but that, owing to various disturbing factors, the percentages of the total population at the various ages will not be exactly the same as in the assumed stationary population derived from the mortality table representing the actual death rates experienced. It is evident that, if for any reason we have in one community more than the normal percentage of the pop- ulation at those ages where the death rate is low and in another community less than the normal percentage at those ages, then, even though the death rate at every individual age might be the same in the two communities, the general average death rate in the first will be less than in the second. It cannot be assumed, therefore, that a higher average death rate nec- essarily means a more unfavorable mortality experience. A correction must first be made for the difference in age dis- tribution. 48 MORTALITY LAWS AND STATISTICS 83. One method of making this correction is to construct the mortality tables representing the observed death rates^ analyzed by ages, in the two communities and calculate from such tables the complete expectation of life at birth. From the fact that in a stationary population the general average death rate is the reciprocal of this expectation it is readily seen that this amounts, in effect, to substituting for each actual population the stationary population corresponding to its actual mortality. It is readily seen that this method may be applied to the death rates for ages above any assigned age, or within given limits, by constructing the corresponding por- tion of the mortality table and calculating the average death rate in the stationary population. For example, it might be desired to compare the mortality in two communities for ages 15 and over or for ages 15 to 64 last birthday inclusive. The corrected death rate for the former would be 1/I15, and for the latter — - — |r- = „H f ■ The labor of constructing a J- 15 — i- 65 2 15 L;z mortality table is, however, considerable and other methods of correction are usually followed. 84. Although the stationary population is largely of theo- retical interest the notation derived from it is useful with cer- tain modifications in connection with actual population statistics. For this purpose 9 X represents the deaths between age x and age x + i, and \ x = 6 x + 6 x+1 +etc, is the total number of deaths at age x and over, but is not equal to the number attaining age x. For the population between ages x and x + i the symbol L x is retained. The symbol T x is also used to denote the total population at ages x and over, so that we have T x = L x +L x+l +etc, as before. The general average death rate is then X /r , but is not equal to i/! except for a stationary population. Simi- larly for the average death rate at age x and over we have \ X /T X but not i/e x , and for ages between x and x+n we have \c A%+n/ J- x J- x + rf STATISTICAL APPLICATIONS 49 85. One method of correcting the death rates of different communities is to analyze each into certain age groups, usually quinquennial up to age 15, then decennial up to age 85, with a final group for ages 85 or more last birthday, the average death rate for each group being used. These death rates are then applied to a standard proportionate distribution of the population into these age groups. One standard which has been used is the age distribution of the population of England and Wales according to the Census of 1801. The general average death rate for the standard population on the basis of the observed group rates for each community is thus calculated and this is considered as the corrected death rate for the com- munity. In this way all communities entering into the com- parison are placed on the same footing with respect to age distribution. The same method may be extended to cover varying proportions of the two sexes by analyzing the statistics for the different communities and also the standard popu- lation in this way. It may, in fact, be extended to cover any factor, such as occupations, considered as having an impor- tant bearing on the mortality to be expected and for which the necessary data can be obtained. 86. Another method of comparison is to use a standard scale of death rates for the different groups into which the actual populations are analyzed. The actual population in each group is then multiplied by the standard death rate and the expected deaths according to the standard are thus cal- culated. The total of the actual deaths in the community is then expressed as a percentage of the expected and these percentages for the different communities are compared. 87. These two methods have been described as applying to a whole community, but it is evident that they apply also to a part, such as those aged x and over, or those whose ages he between x and x+n, or those who are engaged in a certain occupation. In fact, what may be considered as mortality index numbers for various occupations have been formed from the census and death returns [in England. A standard population is taken, analyzed into the five decennial age groups 50 MORTALITY LAWS AND STATISTICS between 15 and 65, the aggregate population being such that the expected deaths according to the general average death rates for occupied males in the various age groups will total up to 1000. The actual death rates for the various age groups in each occupation are then applied to this standard population and the resulting total of expected deaths gives a number whose ratio to 1000 measures the general mortality of the occupation. This is in effect the standard population method above described with the addition that instead of recording the corrected average death rate we record its ratio to an average death rate based on the same standard population combined with standard group death rates. 88. The standard population method is the one most used for the comparison of general population mortality statistics, while the standard death rate method is most used in con- nection with the mortality of insured lives. In connection with such insurance statistics three modifications are made. The first is that the actual experience is usually analyzed into individual years of age and sometimes also into years elapsed since medical examination. The second is that the rate of mortality or probability of dying within one year is usually used instead of the death rate or average force of mortality, and that along with it the exposed to risk of death, which is dis- cussed under the head of construction of mortality tables, must be used instead of the population. The third is that amounts insured or amounts at risk are frequently taken into account instead of lives, so that we compare actual losses with expected losses rather than actual deaths with expected deaths. 89. In this chapter it has been assumed that the period covered by the statistics is one year. Where a period other than one year is dealt with, we must take the average deaths per annum, and in any event whether for a period of exactly one year or otherwise the average population during the period must be taken. The ratio will, of course, be the same if both of these are multiplied by the period, so that we have on the one hand the total deaths and on the other the aggregate number of years of life during the period. CHAPTER VI CONSTRUCTION OF MORTALITY TABLES 90. In the second chapter it was shown that in any mor- tality table we have the relation l x+1 =l x p x for all values of x and that consequently if we have a complete table of the values of p x we can, by starting at the initial age and working forward progressively, construct a complete mortality table. A little consideration also shows us that there is an insuperable practical difficulty in the way of constructing the l x column of a mortality table by taking a large group of lives of a given age and following them throughout the balance of their fives, observing the number surviving to each age. This difficulty arises not only from the length of time that would necessarily be consumed in waiting for the last one to die, but also from the fact that out of any large number some are certain to pass out of the knowledge of the observers and from the moment that any do so disappear the further observations are nullified by our ignorance of the time of their death. A correction is therefore necessary and this correction can be most con- veniently applied by a method which also obviates the neces- sity of waiting until some particular group of lives selected at a young age have all died. This method is to use the rela- tion already quoted and to determine separately the values of p x for each year of age. By this method the observations do not necessarily extend over a longer period than one year, although a longer period is usually taken in order to eliminate the effect of special conditions. In that event the observations at different times for the same year of age are combined. 91. The observations are not, in fact, made directly on the value of p x , but rather on that of m x determined by the relation m x = 6 x /L x , (1) 51 52 MORTALITY LAWS AND STATISTICS where 6 X represents the deaths observed at age x, last birthday, and L x is the corresponding population. But we have in terms of the mortality table m x d x lx — lx+i _T-—px 2 lx+h + \ h+lx+1 I+Px from which we have and P* = 2—m x 2 +m x g x = i- -P*- 2m x e x 2+m x i-^x\2 ~8x (2) (3) 92. In connection with population statistics it has been usual to calculate m x from the data and then to pass to q x and p x . In connection with observations on insured lives, on the other hand, the practice has been to determine the value of L x +^d x denoted by E x for each age and so to pro- ceed directly to q x by the equation q x = 6 x /E x . The problem therefore reduces to the determination of the values of 6 X for each value of x, and of the corresponding values of L X or E x . The methods followed vary, of course, with the form in which the facts are presented, and the conditions in connection with general population statistics differ so much from those in connection with insured lives that it is well to take up the two cases separately. 93. In the case of general population statistics the in- formation regarding the deaths is usually derived from the registration returns and it is a necessary condition, for their use in the determination of death rates, that the registration should include all the deaths coming within the scope of the investigation. It is evident that, to the extent that the returns are incomplete, the numerator of the fraction determining the death rate is understated and consequently the death rate itself is also understated. For this reason the statistics can be used of only those countries, states or municipalities in which the laws and their enforcement are such as to secure CONSTRUCTION. OF MORTALITY TABLES 53 substantial accuracy in the death returns. In the United States those states and parts of states which, in the opinion of the Federal Census Bureau, comply with this requirement con- stitute the registration district. The area included in this district is extended from time to time as the registration becomes more complete. Particulars of the deaths in the various com- ponent parts of the registration area are published annually by the Census Bureau. 94. For information regarding the population corresponding to the deaths reported we must depend upon the census results. As a census is made only periodically, some means must be devised of passing from these figures at periodical intervals to the average population by age groups during the interval covered by the observed deaths. The census returns and the death returns are also frequently given only for groups of ages, and we have therefore an additional problem to solve, namely, that of passing from age groups to individual years of age. 95. Let us take first the problem of finding the average population in a certain age group during a specified period. For the sake of simplicity we will first suppose that the total population analyzed by age groups is known for the beginning and end of the period and that the whole period may be con- sidered for this purpose as a unit of time. Let the total pop- ulation at the beginning be Pq and at the end P\, also let the population in any particular age group at the beginning be aP and at the end {a+b)P\. Then, evidently, the sum of the values of a for all age groups must be equal to unity and this is also true for the values of (a +b), so that the sum of the values of b must be zero. Also suppose the ratio of increase of the total population during the period is r, so that we have P 1 =rP . Then it is assumed that at any time t during the interval the population in the age group is Poia-j-bt)/, the sum of the values of which for all age groups is evidently P r l . In other words the total population is supposed to vary in geometrical progression, while the percentage of that total in the particular age group is supposed to vary in arithmetical 54 MORTALITY LAWS AND STATISTICS progression. On these assumptions the average population during the period is P (\a+btydt = aP Q £r'dl+bP j'\r'dt, • ~ t , , „ } t r—i = aP - +bP \og e r \log e r (log e r) 2 j' r—i f , , / r i , 1 1 log 5 f[ \r~x lOgef/j 96. It is evident that if the period covered by the obser- vations were the interval between two censuses, the census returns would give directly the values of P , Pi, a and b. But the dates upon which the census is taken do not usually coincide with the limits of the period of observations. Suppose, therefore, that we have the results of two censuses taken at the times h and t 2 counting from the beginning of the period and that the corresponding total populations are P3 and P 4 , also that the populations in the age group are APz and BP 4 . Then, according to the assumptions already made, we have P 3 =r l 'P or log J P 3 = log J P +^ilogr, Pi=r t2 Po or log.P4=logPo + Mogr, (h-h) \ogr = \ogPi-\ogPz, logr = (logP4-logP 3 )/(fe-/i), ■ . . . ( S ) log P = log P 3 - h log r = (t 2 log P 3 -h log Pi)f(k-h). (6) a+bh=A, a + bh=B, b(t 2 ~h) = (B-A), b = (B-A)/(h-ti), (7) a=A~bh = (t 2 A-t 1 B)/(h-h) (8) 97. The expression for the average population in the age group, when we substitute in Eq. (4) these values of a and b, takes the form CONSTRUCTION OF MORTALITY TABLES 55 p r-i \ hA~hB B-A l r i_ °log e r[ h~-h h-h\r-i log e r = Po r—i\faA A A ir 1 fa — h \r — i \og e r{fa — h fa — h\r—i log„r hB B I r fa — h fa~h\r—i log,, r -APj~ X \ h XogtrXfa — h fa-h\r—i loge r j I Bp r-i f i / r i \ h | , -v Since ^4P r (1 and BP r'° are the numbers shown in the two censuses for the age group in question, it follows that we obtain the average population for any age group by multi- plying the numbers shown in the two censuses by log, r [fa-h fa-h\r-i \og e r and -h r ~ x I T I r 1 \ h \og e r\fa-h\r-i log e r/ fa- respectively, and adding together the products. These factors are the same for all age groups and may be calculated once for all. 98. The average deaths per annum may be obtained by dividing the total deaths during the period by the number of years included, or the same object can be accomplished by multiplying up the average population by the number of years to get the aggregate population or years of life corresponding to the total number of deaths. The latter is the course usuaUy followed. 99. Having, then, the total deaths and the corresponding population by groups of ages the remaining problem is to as- certain the death rates for individual ages. An approximation which was formerly used was to divide the total deaths by the total population, and assume that this represented the force of mortality at the middle of the interval, or, in terms of the 56 MORTALITY LAWS AND STATISTICS notation explained in Chapter V, fi x+ " = -~ — J, + " . Where 2 J- x 1 x + n n is odd this gives directly the value of m x+Hn -D, the two functions being approximately equal and each equal to d x+Hn _ 1) /l x+ in . Thus ?z+un-i> is obtained by Eq. (3) of Chapter VI. Where n is even, however, x-\-\n is an integer. The value of q x +m is then determined on the assumption that during the year p. x increases in geometrical progression at the ratio r determined from the values of fi x for the neighboring groups. We have then since Mz+< = d colog e l x+t dt C0l0g e P x+in = I ll x+ln+t dt = IJL x+in J o fdt, r- = Mz+ in ; , (10) log e r From these values of q x the intermediate values are found by a formula of interpolation. 100. It was always recognized that the quinquennial age group from 10 to 15 required special treatment, and it has recently been shown by Mr. Geo. King that the method under- states the death rate at the older ages. This can be seen by taking any mortality table and, assuming a stationary pop- ulation, comparing the values of ~ — 7 j~ with those of m x+2 . In view of this fact some more accurate method is desirable. Greater accuracy has been attained by distributing the total deaths and population of each age group into individual years of age. 101. In the construction of the Carlisle table this distribu- tion was effected by a graphic method. On a base line dis- tances were laid off consecutively representing the number of years included in the successive age groups. On these bases rectangles were constructed whose area represented the total number (of deaths or of population as the case may be) in the age group. The heights therefore represented the average number per year of age in each group. A continuous CONSTRUCTION OF MORTALITY TABLES 57 line with continuous curvature was then drawn through the tops of these rectangles such that the area included between it and the base was the same in each interval as that of the corresponding rectangle. The base was then subdivided to represent individual years and ordinates erected to the curve so drawn. The area between the base and the curve in the interval representing each year of age then represents the number assigned to that year. And the total should agree with the total for the group. 102. Under this method, however, it was found difficult to read off the diagram with sufficient accuracy and an analyt- ical method of redistribution has been devised. If we take the population for the various age groups and sum from the oldest group downwards we obtain a series of numbers rep- resenting the total population older than the respective ages which are the points of division between the groups. In other words the values of T x are given for a series of values of x. If then the values of T x can be interpolated for unit intervals we can calculate the values of L x because we have L x — T x —T x+X . Also, if the deaths are similarly treated, we have a series of values of \ X = 'S6 I , from which, by interpolation and differencing, the successive values of 6 X may be deter- mined. The successive values of m x , q Xl or p x may then be determined from the relations already given. 103. The intervals used in tabulating death returns and population statistics in the publications of the United States Census Bureau are individual years from birth to age 4 last birthday, inclusive, and five-year intervals thereafter. The returns of some countries, however, give only ten-year intervals, beginning with age 15. This grouping is adopted in order to avoid a transfer of lives from one group to another arising from a tendency to state ages at a multiple of ten years. Where the interval is ten years it is readily subdivided into five-year intervals by the finite difference formula : i6f(x)= 9 \f(x-t)+f(x+t)}-\f(x- 3 t)+f(x+ 3 t)}. . (11) This may be easily demonstrated by expanding by Taylor's 58 MORTALITY LAWS AND STATISTICS theorem each of the functions on the right and assuming that fourth and higher differential cofficients vanish. This formula does not apply to the last interval, where we use instead the equation 4\f(x)+f(x-2t)\=6f(x-t)+f(x- 3 t)+f(x+t), . (12) which may be similarly demonstrated. 104. For the sub-division of the five-year intervals a special interpolation formula is used which ensures a continuous series. The first and second differential coefficients are determined at each point of junction by the formulas I2tf(x)=8{f(x + t)-f(x-t)\-\f(x+2t)-f(x-2t)}, . . (13) I 2 Pf"(x) = 1 6 \f(x + 1) +/(x - t) } - \f(x + »t) +/(* ~2t)\- 3 o/(x) . (14) These formulas may be obtained by expansion as above, except that the fourth differential coefficient is not neglected. For each interval a function is then found such that the values of the function itself and of its first and second dif- ferential coefficient at the beginning and end of the interval will be equal to those so determined for those points. As there are six conditions to be satisfied it follows that if a rational algebraic function is to be used it must be of the fifth degree. It may readily be demonstrated that the following function satisfies the conditions for the interval from x to x-\-t: fix) hHt-hy ,^,, ^(10/2-15^+6/^) fix I t f^-^-ih) , /"(*+*) fffr-fl 3 , . j4 2 ^3 " ' - v 5^ This may be seen by differentiating with respect to h and then putting h equal to and t successively. CONSTRUCTION OF MORTALITY TABLES 59 105. This equation takes a simpler form when expressed in terms of central differences as follows: Let 5 denote an operation such that */(*) =/(*+;) -/(H)' .-. 8 2 f(x) =f( x +t)-2f(x)+f(x-t), 8*f(x) =f(x + 2t) - 4 f(x+t)+6f(x) - 4 f( x -t)+J(x-2t), etc. Then, from Eq. (13) we have i2tf'(x) = -f(x + 2t)+8f(x+t)-8f(x-t)+f(x-2t), = «5 4 /(» - 4 5 2 /(x) - 2 8 2 f(x +t) + 1 2/(*+0 - 1 2/(«) , or */(*) = {/(*+*) -1 5 2 /(x+0 } - j/(x) +| 5 2 /(s) - T V5 4 /W }. (16) Similarly, #'(*+*) = {/(*+*) +|, + SS~ V,+£n,- 2g" ^.-K E x +i=E x + \n x+ i-d x -w x+l +i((T x +(x) representing the number per unit of age at exact age x, so that /(x)=J o 4>(x+h)dk, and ■w x+2 =J o (x+h)dh, or Ws_i= I 4>{x+ GRADUATION OF MORTALITY TABLES 75 Expanding (x+h) and integrating, we have J-W 2 24 ^-3= C(x+k)dk = 5 $(x)-^'(x)+^4,"(x)-^ct>"'(x) J-S 2 24 w x+2 = \'(x) +^"{x) +^(x+h)dh = s0(«) +^V(x) +^|V'(x) +237S, //// } 2 6 24 w I _ 3 +w a;+ 2= io(x)-\-^4>"(x), 3 w^-8 +w I+7 = io0(x) H — ^V'(a;). 3 But /(*-*) = I + V(*+a)<*a= *(*)+— «"(*), J-h 24 and 5 2 w I _ 3 +5% a;+2 = (w I _ 8 -2ie) j: _3+w I _ 2 ) + (w I _3-2w x+2 +ze< x+7 ), = (w J _8 + W I+7 )-(w x _3+W :r+ 2)=250^"(x) 12 = fe-3+w»+2)- .i6s(S 2 w I _3 + 5 2 w x+2 ), = (w I - 3 --i655 2 w I _ 3 ) + (w ;r+2 -.i655 2 w a;+2 ). (3) This covers the case where half of the year of age is in each group. 132. Where the force of mortality is required at the age corresponding to the point of division we have 6o(x) = 'j(w x - 3 +w x+ r i )-(w z -g+w x+7 ), or ioct>{x) = {w x - i -\5 2 w x - S ) + (w x+2 -lb 2 w x+2 ), . . (4) which is the formula to be used for the deaths and population. 76 MORTALITY LAWS AND STATISTCIS 133. For the age corresponding to the center of the group we have IVx- i = + 2) 4>(x+h)dh = si>(x) + ~(j>"(x), 2} " 24 h/*_« ) +w*_,+w :s -m j = ( (x+h)dh = i5(x)+£^-"(x), J -71 2 4 whence I2O0(x) = 26w x - h - (w x - Bi +W x + M ), = 24W z - i -d 2 W x _ i , or 54>(x)=w x - i 5 2 w J _ } (5) 24 134. These adjustments enable us to calculate the values of q x , m x , or p. x for values of x separated by quinquennial ages, and the remainder of the problem consists in interpolating intermediate values. For this purpose a formula of osculatory interpolation will be found most satisfactory. Three such formulas have been proposed. The first, which is the basis of the other two, is the one described in Chapter VI in con- nection with the redistribution of population and deaths, the simplest form for practical application being that given in Eq. (20) of that chapter. J .('-*)*'(?'-s*>jy(«+i) (6) 2/\.t 5 The other two formulas are simplifications of this by omitting some of the conditions, the first differential coeffi- cient only being determined for the points of junction. 135. In Karup's form, which is the simpler, this first dif- ferential coefficient is determined on the assumption of a curve of the second degree for f(x) , the value so derived being 2tf'(x)=f(x+t)—f(x — t). A curve of the third degree is then GRADUATION OF MORTALITY TABLES 77 determined so as to have the required values and first dif- ferential coefficients at the points of junction. The equation of this curve is found to be +*/(*+o-*^«y(*+o ( 7 ) 136. Greater accuracy without material increase of work can be obtained by determining the first differential coefficient on the assumption of a fourth difference curve by Eq. (13) of Chapter VI, namely, I2tf(x)=8{f(x+t)-f(x-t)}-\f(x + 2t)-f(x-2t)}. This condition is then found to be satisfied by the equation f(x+h)J^f(x)-^Sj(x)+ t ^(S^f(x)-^Y(x))] +]\f^+t)-^[sJ(x+t)+ h t (8j(x+t)-^^f(x+t))^.(S) 137. When making this interpolation it is usually necessary to assume the age at which q x becomes equal to unity and the age chosen should be consistent with the data at the old ages. When it has been chosen it will be advisable to arrange the division into groups, if possible, in such a way that the ages for which the function is determined will form with the lim- iting age a regular series of differences. For example, if it is assumed that E z -i—E x -e}, the mean value of the square of which is 20 113 p.2= 20/X2. 125 2 ' " iS 2 5 I2 5 It will be noted that for successive summations the effect on the smoothness of the series of errors does not diminish as rapidly as the effect on the absolute values. 142. The summations do not necessarily all extend over the same number of terms nor is the number of terms in each summation necessarily odd because, while an average over an even number of terms of a series does not give a term of the series but a term midway between two of them, a second average over the same or a different even number of terms will bring us back to the original series. An even number of summations over an even number of terms each may there- fore be introduced and the resulting averages will correspond to terms of the original series. For example, suppose, instead of taking the average in fives three times we take the average in fours, fives and sixes. Then the expression for the resulting error will be ■ih;(.Ex-a+3Ex-6+6Ex-i + ioE x -3+i4Ex-2 + i7Ex-i+i&E x + i';E x +i +uEx+2+ioEx+s+6Ex+i+3Ex+s+Ex+fi), the mean value of the square of which is -^— 7*2, or a little 1 20 2 less than for the third average in fives. The expression for the third difference is Tihi(-Ea;+9 — E x+& — E x+ i— E x+ 3-\-E x -\-E x -i-\-E x _ 2 — E x -s, GRADUATION OF MORTALITY TABLES 81 Q the mean value of the square of which is -fi 2 - We thus I20 2 see that by making the periods unequal a slight increase in weight is obtained on the individual term and a great increase in weight in the third difference. In other words, with unequal summations a much smoother series is obtained and it is at the same time a little more accurate. 143. Thus far we have assumed that the series of true values is an arithmetic series, the general term of which may be expressed by a linear function. It is necessary, however, to take into account the second and higher differential coeffi- cients. The summation graduation formulas ordinarily used contain a correction so that a series the general term of which is of the third degree will be reproduced. Where the function is of the third degree, we have, generally, f(x+h) =/(*) +hf{x) +-/"(*) +~f"(x). 2 If, then, we add together n terms of this series, we have f(x)+f(x + i) + . . .f(x+n-i) = nf{x) + »<•»- J f'(x) + ^-i)(^-i) + n»(tt-i)* r(g) _ 2 12 " 24 But nfl xH ) = nf(x) -\ — ^ /' (x) < n2 - z lf( x +!^i) = w(w2 ~ l} /"(x) + n( " 8 ~ l)( "~ l) r / (»). 24 ' \ 2 / 24 48 n— 1 .-. f(x)+f(x + i) + . . . f(x+n~i)=nf(x 2 If, then, we denote 24 \ 2 /( x--Zi) +/ («_«Z3 +. .. fx+ *=l 82 MORTALITY LAWS AND STATISTICS by [»]/(«), we have, generally, 24 or « 24 144. The operation of taking the average of n successive terms has therefore introduced an error f"(x) into the value 24 of f(x). If, then, we repeat the process with the same or a different number of terms, an additional error of the same form is introduced. We also see that, since according to our assump- tions / IT (x) vanishes for all values of % and consequently f"(x) is linear in form, the error introduced by the first average is carried forward unchanged. If, then, three successive averages cover p, q, and r terms, we have, 24 Where the number of terms in each average is the same and each equal to n this may be expressed as follows : mM =f(x)+ ^ nx) _ 145. A correction must accordingly be introduced to com- pensate for this error. If, then, we use y„f(x) as a short expres- sion ioi f(x+n)+f(x — n) we have y n f(x) = 2 f(x)+nT(x), or y n f(x)-2f{x)=nj"(x). we have, therefore, (ayi+by 2 +c 73 )f(x) = 2(a+b+c)f(x) + (a+4b+gc)f"(x), or (i+2a+2b + 2c)f(x)-(ayi+by 2 +cy 3 )f(x) =f(x)-(a+ 4 b+gc)f"(x). GRADUATION OF MORTALITY TABLES 83 If, therefore, we substitute, (i + 2a + 2b + 2c)f(x)-(ay 1 + by 2 +Gy 3 )f(x) for /(a), before averaging, we have [p}[q}[r {(i + 2a + 2b + 2c)-(ay 1 +by 2 +cy 3 )\f(x) {/(*)- (0+46 + 9 c)/'(a:)) pqr _ [p][q][r. pqr h2\ n 2\„2. »/(*) + { £ ±£ ~ 3 -(a+4&+9c)}r(«). If then a, 6, and c are so determined that p 2 +q 2 +r 2 — 3 «+4&+oc = 24 a series of the third degree will be exactly reproduced. We have here three unknowns and only one condition, so that the equation is an indeterminate one and the various formulas result from the use of different values of p, q, and r and of a, b, and c, the condition in some cases not being exactly satisfied. 146. The first summation formula correct to third differences was that devised by Woolhouse. It was not originally con- structed as a summation formula, but it was afterwards found to take that form. In this formula p = q = r = $, so that jfr 2 +ff 2 +7- 2 -3 _ 24 3 ' Also a = 3, and b=c = o, so that the formula takes the form M3 3 i7 — 371 }/(*)• Woolhouse's formula when expanded as a function of the terms of the series becomes T-k( 2 S + 2 47l + 2I72 + 7T3 + 3T4- 27 6 -37 7 )/(x). 147. J. A. Higham, who first showed that Woolhouse's formula might be applied by the summation method, suggested an alternative compensating adjustment which was equiv- 84 MORTALITY LAWS AND STATISTICS alent to putting a= — i, b = i and c = o, thus still keeping a+4& + ac = 3. His formula therefore took the form ^(i+7l-Y2)/(*)=^([ 3 ]-7 2 )/(*), 5 3 5 which expands into ih( 2 5 + 2471 + 1872 + 1073 +374 -276- 277 ~ys)f(x). 148. Karup's graduation formula uses summations in fives with the compensating adjustment a=— f, b=o, c = £, so that the formula becomes, ^-(!+l7l -l73)/(x) =^f(3[ 3 ] - 273)/(x). 149. G. F. Hardy used the same compensating adjustment as Higham, but used successive summations in fours, fives, and p2J t _ ( j2J rr 2_i _ _ sixes, so that for his formula = 3 T x -2 , and it is 24 not fully compensated, the remaining second difference error being -^f"(x). This is, however, small and is approximately counterbalanced for some functions such as q x and m x by the fourth difference error. A lack of compensation to this extent is therefore considered admissible. This formula becomes [ -«%]-7 2 )/(*). 120 150. The most powerful summation formula which has been put to practical use is probably that of Spencer, which includes when expanded 21 terms of the original series. In this formula two summations in fives and one in sevens are used, so that i—Ll - = 4. Also a=— i, b=o, and c = \, so that the 24 formula reduces to [5]2[7 - ] (i+l7 1 -^73)/(x)=t5H7](2 +Tl _ T3 ) / ( x ) i7S 35° JsPM (i+b]-73)/(x). 35° 151. A still more powerful formula would be given by putting £ = 5, q^J, and r = n, so that * — 2 — r _ — 3 =g) and 24 GRADUATION OF MORTALITY TABLES 85 in the compensating adjustment we have a=~i, b=o, and c = i, so that the formula becomes, ES][7|M / T i w s [s][7]["]/r 1 W/ N -^— (i + 71 - 73 )/(x) = t^i-J([ 3 ] _ 73 )/( x ). 152. The weight of a graduation formula is found by expand- ing the formula, adding together the squares of the coefficients and taking the reciprocal, that being the average proportion in which the mean square of the errors is reduced by the gradua- tion. The smoothing coefficient is found by similarly expanding the third difference of the graduated terms, adding together the squares of the coefficients, dividing by twenty and extracting the square root. It measures the effect of the graduation on the mean absolute value of the third differences. 153. The following table shows a number of summation formulas and their weights and smoothing coefficients : Author 'No. of Terms. Formula. Weight. Smoothing Coefficient 8.92 I 125 5-SO I 15 5-4° I 60 5-87 I 56 6.07 1 95 6-73 1 85 6.14 1 i°5 6.98 1 160 6.70 1 141 9. 11 1 326 Error. Finlaison. . . . Woolhouse. . . J. Spencer . . Higham G. F. Hardy. J. Spencer . . . Karup J. Spencer (a) J. Spencer (6) Henderson. . . 13 IS IS 17 17 19 19 12s giio- 3 [ 3 ]| [5iur 320 [5] 14+371- 372! [4][S][6] 27 120 [5JI7] 175 [sP 625 [sPM 350 MtsPM 600 [5M11: lbl-72! {3— 72 1 3b]— 2 73l t + [3l-73i 13— Til 38S fbl-i 3/"(*) +£/"(*) ~5-4p v (x) - 3 .86f™(x) -6. 4 f™(x) -J"{x)-6. S f^{x) - to.6f"(x) -7-8/ IT W — I2.6f"(x) ±-j"{x)~io. 2 fr>{x) 44-8/ IT W 154. One difficulty in connection with the application of summation formulas to the graduation of tables is that in 86 MORTALITY LAWS AND STATISTICS order to determine any graduated value of the function it is necessary to know a number of ungraduated values above and below it and, unless the function is such that it disappears and the ungraduated values beyond a certain limit may be assumed to be zero,, there will be a portion at either end of the table for which values will not be obtained, and some sup- plementary means must be adopted of completing the table if it is necessary to have it complete. In the case also of insurance companies' experiences, the values of the functions near the limits are usually derived from very limited data and are consequently very irregular. Both of these difficulties may be overcome by taking the three last graduated values of the function that are considered as determined with sufficient accuracy as a basis and determining a function of x of the third degree which will reproduce these three values and will also make the total expected deaths for ages beyond equal to the expected. The general expression for a function of the third degree in x such that /(a) =u a , f(b) =u b , and /(c) =u c is fM- ( x - b )( x - c ). t , (x-a)(x-c) __ , (x~a)(x-b ) (a-b){a-c) (b-a)(b-c) (c — a)(c-b) +k(x — a)(x — b)(x — c), where k is an arbitrary constant. 155. Taking up next the application of a general law with arbitrary constants to the graduation of tables the law which is most frequently used is that of Makeham, according to which n x =A +Bc x , or l x = ks x f x . The constant k is merely in the nature of a radix and does not affect the rates of mortality. This formula may be applied in two general ways: first, to construct a graduated mortality table from the original data of the exposed to risk (or population) and deaths without the explicit deter- mination of the ungraduated rates of mortality; and second, to graduate a rough table without direct reference to the original data. 156. Probably the simplest method of determining the Makeham constants from the original data is to group the exposures (or population) and the deaths into quinquennial GRADUATION OF MORTALITY TABLES 87 age groups and then, by the process already described in con- nection with interpolation methods, determine the values of q x , m x or n x at quinquennial intervals. Where q x or m x is determined, we can proceed immediately to colog p x from the known relations. Now we have colog p x = \og 4 -log 4+i, = (log k+x log s+c x log g) -{log^ + (x+i) log s+c x+1 log g\, = -l0g5 + (l-c) C x l0gg, = a+ji<7. It is therefore of the same form as p, x . If, then, we neglect the values at young ages and at extreme old ages as derived from insufficient data and start with some age y in the neigh- borhood of age 30, we have colog p v +s colog p v+ s + 5 colog py+io + S col °g Pv+li +3 colog ^ y+ 2o+ colog p y+2b = Si = i8aH-/3c"(i+3C 5 + sc 10 + 5c 15 +3c 20 +c 25 ) = i8a+/3c I '(i+c 5 )(i+c s +c 10 ) 2 . Similarly colog A+15+3 col °g A/+20 + 5 col °g Pv+k + S colog p v+30 +3 colog />„ +3E -r-colog p v+i0 =S2 = i8a+/3c* +I5 (i +c 5 ) (1 +c 5 +c 10 ) 2 , and colog py+zo+2, colog py+zi + S colog p v+i o + 5 colog p y+m +3 colog /v+so+colog py +S5 =S3 = i8a+^ +30 (i +c 5 ) (1 + C 5+c 10 ) 2 . From, these equations we see that 15 _5V-52 MORTALITY LAWS AND STATISTICS C™Si-S 2 S1S3-S2 2 l8« = - /3c t '(l+C 5 )(l+C s + C 10 ) 2 =5l-l8a «-I 53-252+5!' 5 2 -5i_ (5 2 -5i) 2 c 15 -i (5 3 -25 2 +5i)' Thus, c, a, and (3 are determined. Similarly, the values of c, A, and B may be determined from the values of p x and from either a and /3 or ^4 and 5 we may proceed to the values of s and g. 157. If a further refinement is required, we may assume that the values determined as above are only approximate and that colog p x = (a+Mh) + (f3+Mk)(?( 1 +- where ft, k, and I are small quantities and M is the modulus of common logarithms, or logio e. Then approximately q x = q' x +hp' x +kc x p' x +lxc x p' x , where q' x and p' x are derived from the constants a, fi, and c. We have, then, three unknowns, h, k, and /, to determine and three equations are required. These equations may be ob- tained by making the total number of expected deaths and the first and second moments of the expected deaths equal to those of the actual deaths. The equations so obtained are : h2E x p' x +ki:c x E x p' x +nxc x E x p' x = l{e x -E x q' x ), Ji2xE x p' x +k?,xc x E x p' x +lZx 2 c x E x p' x = Zx(6 x -E x q' x ), hXx 2 E x p' x +kXx 2 c x E x p' x +nx s c x E x p' x = Zx 2 (8 x - E x q' x ) . These equations are seen to be equivalent to h2E x p' x + (k+ar)2c x E x p' x +l2{x-a)c x E x p' x = ?(9 x -E x q' x ), k2(x-a)E x p' x + (k+al)2(x-a)c x E x p' x +l2(x-a) 2 c x E x p' x = 2(x-a)(8 x -E x q' x ), hX(x-a) 2 E x p' x + (k+al)-2(x-a) 2 c x E x p' x +lS(x-a)h x E x p' x = 2(x-a) 2 (6 x -E x q' x \ GRADUATION OF MORTALITY TABLES 89 where a is any suitable quantity used for the purpose of re- ducing the numbers involved. From these three equations the values of h, k, and / are determined, and the values of a, (3, and c are corrected accordingly. 158. A shorter process is to assume that the value of c is accurate, and consequently Z = o, and to determine h and k from the first two equations or to determine A and B directly from two equations depending on the relation m x = Mz+s = A +Bc x+i . The two equations are A 1xL x +B2xc x+i L x = 2x8 x . 159. Where a mortality table has been constructed and it is desired to graduate it by Makeham's formula, the simplest method is to determine the constants from the values of log l x at four equidistant ages by the method described in Chapter III. Unless, however, the ungraduated table is already com- paratively smooth the constants so determined will depend to too great an extent on the particular ages selected. To minimize the irregularity we may take, instead of individual values of log l x , the sums of a number of consecutive values. Then we have : f n(n—i)) c x (c"—i) Si = if +n_1 log l x = n log k + \ nx+ log s-\ — y — - logg, S 2 = 2^ +n-1 log l x = n log k + \n(x+t)+-^- — '- log s 2 + C - n(n—i) S 3 = x x+2t+n ~ 1 logl x = n\og k + \n(x + 2t)-\ — -[ log 5 2 + c-z bgg ' i ftift I ) I Si = lf +3 ' +n '^ogl x = n\ogk + \n(x+3t)+-±— — logs $-$-ot I 2 1 C x+3«( C „_ I ) logj 90 MORTALITY LAWS AND STATISTICS c c ,1 .c x {c l -i)(c n -i) . S 2 -Si=ntlogs-\ logg, c—i S 3 ~S 2 =ntlogs-\ logg, c — I 5 4 - 5 3 = nt log s +-. i '- log g, ■< c — I 5 3 - 2S2 +5i = — — log g, c — I 5 4 - 2S3 +S 2 = — log g, c—i Si — 253 +S2 , = c . S3 — 2S2+S1 160. This method does not, however, entirely eliminate the objection that special importance is given to special points of division. To obviate this it has been suggested to use all the values of log l x except those at extreme old ages and at young ages and to so determine the constants that the sum of the values and the first, second, and third moments will be reproduced. This can best be expressed in terms of sum- mations as follows : Suppose a is the youngest age to be included and let n be the total number of ages to be included. Then \ogl a+x = \ogk + (a+x) log s+c a+x logg, = (log k +a log s) -\-x log s+(fc" log g, = log k'+x log s+c x log g'. Where log k' = log k +a log 5 and log g' = c a log g. Also 15;= 2JT 1 log a+I =xlog/e'+~ logs-] — —-logg', *Aj 1 7/1 " 1 t I ^ ^- ^ ,S X = 2S-S5, = ■- log k' + i- log 5 + ? i— ^ - -^- log g', 2 6 [(c—i) 2 c — 1 J ] , + i( C -l) 4 ( C -l)3 2 ( C -!)2 6 ( C -l)j l0gg - Where x table at the extreme old ages is due to the fact that the values of l x and d x were tabulated to integers only and the values of q x recalculated from them instead of being calculated directly from the constants in the formula. CHAPTER VIII NORTHEASTERN STATES MORTALITY TABLE 1 66. Some of the methods described in the preceding chap- ters will be illustrated by the construction of a mortality table for the Northeastern States. The data used will be the death returns for the five calendar years 1908 to 1912 inclusive and the census returns as of June 1, 1900, and April 15, 1910, for the New England States and the three Middle Atlantic States, New York, New Jersey, and Pennsylvania. Table I shows the total deaths in these states for the five-year interval arranged by age groups and the total population at the two dates arranged in the same way. The average population by age groups for the five-year interval is also shown. 167. In this table the average population in each group is determined by means of Eqs. (-5) and (9) of Chapter VI, only the population for which the ages are stated being taken into account. The cases where the ; age is returned as unknown are an extremely small percentage of the total and the effect of their omission is negligible. June 1, 1900, is 7 T \ years before the beginning of the observations, which cover five years, there- fore, *i = — B- April 15, 1910, is 2^ years after the beginning therefore, fe=|i- Also P 3 = 2i 004 724 and P 4 = 25 836o88, so that we have log r = . 0455239 and the two factors entering into the determination of the average population are -.03154363 and 1 .030481 7 for June 1, 1900, and April 15, 1910, respectively. The total years of life are then obtained by multiplying the average population by five. 168. We then apply Eq. (3) of Chapter VII and obtain the values of L 9i , L m , L Wi , etc., and of 6 m , 6 m , 8 m , etc. The value of 6 9i is determined by leaving out of account the deaths 95 96 MORTALITY LAWS AND STATISTICS TABLE I Deaths and Population— Northeastern States. 1908-1912 Population. Total Age Last Birthday. Deaths, Years of 1908-1912. April 15, 1910 June I, 1900. Average 1908-1912. Life. 397 98S 574 480 476 810 576 951 2 884 755 I 84 939 505 632 426 773 5°7 583 2S37 9I5 2 35 757 553 698 448 816 556 419 2 782 095 3 22 116 54i 178 449 855 543 484 2 717 42O 4 15 694 515 976 442 067 517 760 2 588 800 0-4 556 49i 2 690 964 2 244321 2 702 197 13 510 985 5-9 42 475 2 400 180 2 no 213 2 406 778 I 2 O33 89O 10-14 25885 2 285 642 1 908 183 2 295 121 11 475 605 15-19 42677 2 385 256 1 888 668 2 398 388 n 991 940 20-24 64 604 2 554 686 2 024 318 2 568 703 12 843 515 25-20 71 700 2 4°5 723 1977342 2 416 681 12 083 405 3°-34 75 273 2 121 420 1 738 577 2 131 243 10 656 215 35-39 86752 1 984 723 1 562 115 1 995 946 9 979 730 40-44 86342 1 671 571 1 305 952 1 681 329 8 406 645 45-49 90895 1 399 363 1 053 884 1 408 775 7 043 875 50-54 99 493 1 174 250 899 808 1 181 660 5 908 300 55-59 103 030 840 368 697 132 843 994 4219970 60-64 118 590 686 755 575 880 689 523 3 447 615 65-69 128 504 5i4 97o 418332 517 47i 2 587 355 70-74 128 074 355 427 292 946 357020 1 785 100 75-79 113 343 210 122 177 814 210 918 1 054 590 80-84 82 412 102 741 88019 103 097 515 485 85-89 45 174 39617 31 5°4 39831 199 155 90-94 15 993 10 198 7 9 2 3 10 259 5i 295 95-99 3 497 1851 1 523 1859 9 295 100 and over 678 261 270 260 1 300 All known 1 981 882 25 836 088 21 004 724 25 961 053 129 805 265 Unknown 1038 32485 41 971 All 1 982 920 25 868 573 21 046 695 at ages o to 4 and assuming that 8 2 w 7 is equal to 5 2 wi2 in the formula. The exposed to risk at each of these ages is then determined by the formula E x = L x -\-\d z . The same formula also applies at ages 1 to 4 inclusive, but for age zero it is found that the average age at death of those dying within one year of birth is only three-tenths of a year, so that we use instead E = L +^d . F rom the values of log 6 X and NORTHEASTERN STATES MORTALITY TABLE 97 log E x the values of log q x are then determined. These figures are shown in Table II. TABLE II Calculation of Values of log q x for Infantile and Quinquennial Ages X ioL x Ioe x ioF. x 1 °SQ X o 28 847 550 3 979 850 31 633 445 I 09972 I 2S379 J 5° 849 390 25 803 845 2 SI742 2 27 820 950 357 57° 27 999 735 10621 3 27 174 200 221 160 27 284 780 3 90879 4 25 888 000 156 94° 25 966 470 78134 9i 23 180 579 57 344 23 209 251 39283 i4i 23 234918 62 207 23 266 021 42712 i9i 25 046 068 109 046 25 100 591 63793 24i 25 302 916 138 167 25 372 °°° 736o5 292 22 725 822 145 08s 22 798 364 80372 34i 20 660 018 162 848 20 741 442 89494 394 18 499 612 174 237 18 586 731 97194 44i 15 378 331 175 75i 15466206 2 05551 49i 13 005 892 190 556 13 101 170 16271 54i 10 068 338 201 374 10 169 025 29672 59i 7 530 953 220 568 7 641 237 46038 64i 6 039 903 249 733 6 164 770 60756 69i 4 351 °46 260 643 4 481 368 76464 74s 2 796 270 246 45° 2 919 495 92642 79i 1 5°i 735 199 302 1 601 386 1 09501 8 4 J 650 085 127 131 713 651 25077 89i 205 286 57o8s 233 828 38763 94* 37 512 15 140 45082 52612 99* 3879 2 225 4992 64906 169. The value of log q ih to serve as an initial term in a systematic interpolation is calculated from those of log qz, log qi, and log §91, on the assumption that for the interval in question log q x may be considered as a rational algebraic function of the second degree in x, the resulting value being 3.72417. An additional term is supplied at the end by assuming ?io« = i or log q im = .00000. Eq. (8) of Chapter VII is then applied to the interpolation. In this interpolation t is given the value 5 and h takes successively the values, \, f, f , f , and f . From these values of log q x , the values of l x and d x are then derived. The following table for the first five ages shows the working process: NORTHEASTERN STATES NORTALITY TABLE 99 (I) (») (3) =10g (2) (4) (5) =(3) +(4) (6)=antilog (5) Age. h log (3 lo S 1x log rfz d x O IOO ooo 5 . 00000 I .09972 4.09972 12 581 I 87419 4.94161 2.51742 3-45903 2878 2 84541 .92707 .10621 •03328 I 080 3 83461 .92418 3.90879 2 .83027 677 4 82 784 •91795 ■78134 .69929 5°° 5 82 284 170. In accordance with the usual custom the values of q x shown in the table have been adjusted to agree exactly with the values of l x and d x and do not agree with the values of log q x used in constructing the table. In the accompanying Diagram, No. 3, the values of log (i + iooq x ) are plotted in comparison with the similar functions according to three tables representing mortality among American insured lives. APPENDIX Fundamental Columns and Other Data from Various Mortality Tables x = age ; l x = number living at age x ; d x = number dying at age x last birthday. NORTHAMPTON TABLE * lx d x * lx d x % lx dx o ii 650 3 000 3S 4 010 75 70 1 232 80 I 8650 1 367 36 3 935 75 71 1 152 80 2 7283 502 37 3860 75 72 1 072 80 3 6781 335 38 3 785 75 73 992 80 4 6 446 197 39 3 7io 75 74 912 80 5 6249 184 40 3635 76 75 832 80 6 6065 140 41 3 559 77 76 752 77 7 S92S no 42 3 482 78 77 675 73 8 S«i5 80 43 3404 78 78 602 68 9 5 735 60 44 3326 78 79 534 65 IO 5 675 52 45 3248 78 80 469 63 n 5 623 5° 46 3 170 78 81 406 60 12 5 573 5° 47 3 °9 2 78 82 346 57 13 5 523 5° 48 3014 78 83 289 55 14 5 473 5° 49 2936 79 84 234 48 IS 5 423 5° 5o 2857 81 85 186 41 16 5 373 53 51 2 776 82 86 145 34 17 5320 58 52 2 694 82 87 in 28 18 5 262 63 53 2 612 82 88 83 21 19 5 199 67 54 2 530 82 89 62 16 20 5 132 72 55 2448 82 90 46 12 21 5060 75 56 2 366 82 9i 34 10 22 4 985 75 57 2 284 82 92 24 8 23 4910 75 58 2 202 82 93 16 7 24 4835 75 59 2 120 82 94 9 5 25 4 760 75 60 2038 82 95 4 3 26 4685 75 61 1956 82 96 1 1 27 4 610 75 62 1874 81 28 4 535 75 63 1 793 81 29 4 460 75 64 1 712 80 3° 4 385 75 65 1 632 80 31 4 31° 75 66 1 552 80 32 4 235 75 67 1 472 80 33 4 160 75 68 1 392 80 34 4085 75 69 1 312 80 100 APPENDIX 101 CARLISLE TABLE X h d x * lx dx X lx dx o IO ooo * 539 35 5362 55 70 2 4OI 124 I 8461 682 36 5307 56 7i 2 277 134 2 7 779 5°5 37 5251 57 72 2 I43 146 3 7274 276 38 5 194 58 73 1 997 156 4 6998 201 39 5136 61 74 1 841 166 "5 6 797 121 40 S°75 66 75 1 675 160 6 6676 82 41 5009 69 76 1 515 156 7 6 594 58 42 4940 71 77 1 359 146 8 6536 43 43 4869 71 78 1 213 132 9 6 493 33 44 4798 7i 79 1 081 128 IO 6 460 29 45 4727 70 80 953 116 ii 6431 3i 46 4 657 69 81 837 112 12 6 400 32 47 4 588 67 82 725 102 13 6368 33 48 4521 63 83 623 94 14 6 335 35 49 4 458 61 84 529 84 IS 6 300 39 5° 4 397 59 85 445 78 16 6 261 42 51 4 338 62 86 367 71 17 6 219 43 52 4276 6S 87 296 64 18 6 176 43 53 4 211 68 88 232 51 19 6i33 43 54 4 143 70 89 181 39 20 6 090 43 55 4073 73 90 142 37 21 6047 42 56 4000 76 9i i°5 3° 22 6005 42 57 3 924 82 92 75 21 23 S963 42 58 3842 93 93 54 14 24 S92i 42 59 3 749 106 94 40 10 25 5 379 43 60 3643 122 95 3° 7 26 S836 43 61 3 521 126 96 23 5 27 5 793 45 62 3 395 127 97 18 4. 28 5 748 5° 63 3268 125 98 14 3 20 5698 56 64 3 143 125 99 11 2 3° 5642 57 65 3018 124 100 9 2 3i 5 58s 57 66 2894 123 IOI 7 2 32 5 528 56 67 2 77i 123 102 5 2 33 5 472 55 68 2648 123 103 3 2 34 5417 55 69 2525 124 104 1 1 102 MORTALITY LAWS AND STATISTICS ACTUARIES', OR COMBINED EXPERIENCE, TABLE X lx d x X lx dx X lx d x 10 100 000 676 40 78653 815 70 35 837 2 327 II 99 324 674 41 77838 826 7i 33 5io 2 35i 12 98650 672 42 77 012 839 72 31 159 2 362 13 97978 671 43 76 173 857 73 28797 2358 14 97307 671 44 75 3i6 881 74 26 439 2 339 IS 96 636 671 45 74 435 909 75 24 100 2 303 16 95 96s 672 46 73 526 944 76 21 797 2 249 17 95 293 673 47 72 582 981 77 19 548 2 179 18 94 620 675 48 71 601 1 021 78 17369 2 092 19 93 945 677 49 70580 1 063 79 15 277 1987 20 93 268 ■ 680 So 69517 I 108 80 13 290 1 866 21 92588 683 5i 68 409 1 156 81 11 424 1 730 22 91905 686 52 67253 1 207 82 9 694 1 582 23 91 219 690 53 66 046 1 261 83 8 112 1 427 24 90529 694 54 64785 1 316 84 6685 1 268 25 89835 698 55 63 469 1 375 85 5 417 1 in 26 89137 7°3 56 62 094 1 436 86 4306 958 27 88434 708 57 60658 1 497 87 3 348 811 28 87 726 714 58 59 161 1 561 88 2 537 673 29 87 012 720 59 57 600 1 627 89 1 864 545 30 86 292 727 60 55 973 1 698 90 1 319 427 31 85S65 734 61 54 275 1 770 91 892 322 32 84831 742 62 52505 1 844 92 57o 231 33 84089 75o 63 50 661 1 917 93 339 155 34 83 339 758 64 48744 1 990 94 184 95 35 82581 767 65 46 754 2 061 95 89 52 36 81 814 776 66 44693 2 128 96 37 24 37 81 038 78s 67 4256S 2 191 97 13 9 38 80253 795 68 40 374 2 246 98 4 3 39 79 458 805 69 38128 2 291 99 1 1 APPENDIX 103 AMERICAN EXPERIENCE TABLE X lx Ax X lx d x X lx d x IO 100 000 749 40 78 106 765 70 38569 2 39i II 99 251 746 41 77 341 774 7i 36178 2 448 12 98 5°S 743 42 76567 78s 72 33 73° 2 487 13 97 762 740 43 75 782 797 73 3i 243 2 505 14 97 022 737 44 74 985 812 74 28738 2 501 IS 96 285 735 45 74 173 828 75 26 237 2 476 16 95 55° 732 46 73 345 848 76 23 761 2 431 17 94818 729 47 72497 870 77 21 33° 2 369 18 94 089 727 48 71 627 896 78 18 961 2 291 19 93 362 725 49 70731 927 79 16 670 2 196 20 92 637 723 5o 69 804 962 80 14 474 2 091 21 91 914 722 Si 68842 1 001 81 12383 1 964 22 91 192 721 52 67 841 1 044 82 10 419 1 816 23 90471 720 S3 66 797 1 091 83 8603 1 648 24 89 75i 719 54 65 706 1 143 84 6 955 1 47° 25 89032 7i8 55 64563 1 199 85 5 485 1 292 26 88314 718 56 63 364 1 260 86 4 193 1 114 27 87 596 718 57 62 104 1 325 87 3°79 933 28 86878 718 58 60 779 1 394 88 2 146 744 29 86 160 719 59 59 385 I 468 89 1 402 555 3° 85441 720 60 57 917 1 546 90 847 385 31 84 721 721 61 56371 1 628 91 462 246 32 84 000 723 62 54 743 1 713 92 216 137 33 83 277 726 63 53 030 1 800 93 79 58 34 82551 729 64 51 230 1 889 94 21 18 35 81 822 732 65 49 341 1 980 95 3 3 36 81 090 737 66 47 36i 2 070 37 80353 742 67 45 291 2 158 38 79 611 749 68 43 133 2 243 39 78862 756 69 40 890 2 321 104 MORTALITY LAWS AND STATISTICS INSTITUTE OF ACTUARIES HEALTHY MALE (H M ) TABLE X l x d x X lx d x X lx dx IO 100 000 490 40 82 284 848 70 38 124 2371 II 99 5io 397 41 81436 854 7i 35 753 2 433 12 99 "3 329 42 80582 865 72 33320 2 497 13 98784 288 43 79 717 887 73 30823 2 554 14 98 496 272 44 78 830 911 74 28 269 2 578 15 98 224 282 45 77919 95o 75 25691 2527 16 97 942 3i8 46 76 969 996 76 23 164 2 464 17 97624 379 47 75 973 1 041 77 20 700 2 374 18 97 245 466 48 74 932 1 082 78 18 326 2258 19 96 779 556 49 73850 1 124 79 16068 2138 20 96 223 609 5o 72 726 1 160 80 13 930 2015 21 95614 643 51 71 566 1 193 81 11 915 1883 22 94 971 650 52 70 373 1 235 82 10032 1 719 23 94 321 638 53 69138 1 286 83 8313 1545 24 93 683 622 54 67852 1339 84 6 76S 1346 25 93 °6i 617 55 66513 1399 85 5422 1 138 26 92 444 618 56 65 114 1 462 86 4 284 94i 27 91 826 634 57 63652 1 527 87 3 343 773 28 91 192 654 58 62 125 1592 88 2 570 615 29 90538 673 59 60533 1 667 89 1955 495 3° 89865 694 60 58866 1 747 90 1 460 408 31 89 171 706 61 57 "9 1830 91 1 052 329 32 88465 717 62 55 289 1 915 92 723 254 33 87748 727 63 53 374 2 001 93 469 195 34 S7 021 740 64 Si 373 2 076 94 274 139 35 86 281 757 65 49 297 2 141 95 135 86 36 85524 779 66 47 156 2 196 96 49 40 37 84 745 802 67 44 960 2243 97 9 9 38 83 943 821 68 42 717 2274 39 83 122 838 69 40 443 2319 APPENDIX 105 BRITISH OFFICES' M <» TABLE X z* d x X lx d x X lx d x IO 107324 658 45 82 OIO 984 80 15 531 2 151 ii 106 666 658 46 81 026 1 018 81 13380 2 007 12 106 00S 656 47 80008 1056 82 11 373 1847 13 IOS3S2 655 48 78 952 1 096 83 9526 1674 14 104 697 654 49 77 856 1 139 84 7 852 1493 IS 104 043 654 5° 76717 1 185 85 6 359 1308 16 103 389 654 5i 75 532 1 234 86 5051 1 122 17 102 735 655 52 74 298 1 286 87 3 929 943 18 102 080 655 53 73 OI2 1343 88 2986 773 19 101 425 655 54 71 669 1 402 89 2 213 617 20 100 770 657 55 70 267 1464 • 90 1 596 480 21 100 113 660 56 68803 1 529 91 1 116 360 22 99 453 661 57 67 274 1598 92 756 263 23 98 792 664 .58 65676 1 669 93 493 183 24 98 128 667 59 64 007 1 742 94 310 124 25 97 46i 672 60 62 265 1 819 95 186 79 26 96 789 676 61 60 446 1897 96 107 49 27 96 113 681 62 58 549 1975 97 58 28 28 95 432 688 63 56 574 2055 98 30 15 29 94 744 694 64 54 519 2 133 99 15 8 30 94050 703 65 52386 2 211 100 7 4 31 93 347 711 66 5oi75 2285 IOI 3 2 32 92 636 720 67 47890 2 355 102 1 1 33 91 916 732 68 45 535 2 421 34 91 184 744 69 43 114 2478 ■ 35 90 440 757 70 40636 2 527 36 89683 771 7i 38 109 2565 37 88912 788 72 35 544 2 59i 38 88 124 806 73 32 953 2 602 39 87318 825 74 30351 2 596 40 86493 846 75 27 755 2 572 41 85647 869 76 25183 2 529 42 84 778 895 77 22 654 2 466 43 83883 922 78 20188 2381 44 82961 951 79 17 807 2 276 106 MORTALITY LAWS AND STATISTICS NATIONAL FRATERNAL CONGRESS TABLE * h d x X lx dx X lx dx 20 100 000 500 5° 81 702 935 80 20 270 2 799 21 99 5°° 5°i Si 80767 981 81 17 471 2659 22 98999 502 52 79 786 1 029 82 14 812 2485 23 98497 503 S3 78 757 1083 83 12 327 2 280 24 97 994 5°5 54 77 674 1 140 84 IO 047 2050 25 97 489 5°7 55 76 534 1 202 85 7 997 1 800 26 96 982 510 56 75 332 1 270 86 6197 1539 27 96472 5i3 57 74 062 1 342 87 4658 1 277 28 95 959 5i7 58 72 720 1 418 88 3 38i 1023 29 95 442 522 59 71 302 1 501 89 2 358 788 3° 94920 527 60 69 801 1588 90 1 57° 579 31 94 393 533 61 68 213 1 681 91 991 404 32 93 860 54° 62 66 532 1778 92 587 264 33 93 320 548 63 64 754 1 880 93 323 161 34 92 772 557 64 62 874 1985 94 162 89 35 92 215 567 65 60889 2 094 95 73 44 36 91 648 578 66 58 795 2 206 96 29 19 37 91 070 59i 67 56589 2318 97 10 7 38 90479 606 68 54 271 2430 98 3 3 39 89873 622 69 51 841 2 539 40 89 251 640 70 49302 2645 4i 88 611 660 71 46657 2 744 42 87 951 683 72 43 913 2832 43 87 268 708 73 41 081 2 909 44 86 560 734 74 38 172 2 969 45 85826 761 75 35 203 3009 46 85065 790 76 32 194 3 026 47 8427S 822 77 29 168 3016 48 83 453 857 78 26 152 2 977 49 82 596 894 79 23 175 2905 APPENDIX 107 NORTHEASTERN STATES MORTALITY TABLE 1908-12) Age. Number Living. Number Dying. Rate of Mortality per Thousand. Expectation of Life. X h dz 1000 q x e x o IOO 000 12 581 125.81 50-4I I 87 419 2878 32.92 56.59 2 84 541 I 080 12.77 57 -5o 3 83461 677 8. 11 57-24 4 82 784 5°° 6.04 56.70 5 82 284 386 4.69 56.04 6 81 898 3" 3.80 55-31 7 81587 261 3.20 54-51 8 Si 326 228 2.80 53-69 9 81 098 207 2-55 52.84 IO 80891 195 2.41 51-97 II 80696 190 2-35 51.10 12 80 506 190 2.36 50.21 13 80316 196 2.44 49-33 14 80 120 207 2.58 48.45 15 79 9*3 222 2.78 47-58 16 79 69 1 244 3.06 46.71 17 79 447 270 3-4° 45-85 18 79 177 299 3-78 45.00 I 9 78878 329 4.17 44-17 43.36 20 78 549 354 4-5i 21 78195 374 4.78 42.55 22 77 821 39° 5 01 41-75 2 3 77 431 402 5- x 9 40.96 24 77 029 413 S-3& 40.17 39.38 25 76 616 424 5-53 26 76 192 435 S-7I 38.60 27 75 757 445 5-87 37.82 28 75 3 12 456 6.05 3705 36. 26 29 74856 468 6.25 3° 74 388 482 6.48 35-49 31 73 9° 6 499 6.75 34.72 3 2 73 407 SI8 7.06 33-95 33 72 889 537 7-37 33 19 34 72 35 2 S57 7.70 32-43 35 71 795 S75 8.01 31.68 36 37 71 220 S9i 8.30 3° -93 70 629 607 8.59 30.18 38 39 40 70 022 623 8.90 29.44 69399 68760 639 656 9.21 9-S4 28.70 27.96 41 68 104 674 9.90 27.23 42 43 44 45 46 6743° 66737 66 024 65 290 64 532 693 713 734 75'8 785 10.28 10.68 11 . 12 11. 61 12.16 26.49 25-76 25.04 24-31 23 -59 22.88 47 48 49 5° 5i 63 747 62934 62 089 61 210 813 84 S 879 9i5 12. 75 13-43 14.16 14-95 22.17 21 .46 20.76 60 295 955 15-84 20.07 108 MORTALITY LAWS AND STATISTICS NORTHEASTERN STATES MORTALITY TABLE (1908-12)— Continued Age. Number Living. Number Dying. Rate of Mortality per Thousand. Expectation of Life. X lx d x IOOOJz °e x 52 59 340 998 16.82 19.38 53 58 342 1045 17.91 18.71 54 57 297 I 096 I9-I3 18.04 55 56 201 I 153 20.52 17.38 56 55 048 I 217 22. 11 16.73 57 S3 831 1285 23.87 16.10 58 52 546 1355 25-79 I5-48 59 5i 191 1424 27.82 14.88 60 49 767 1489 29.92 14.29 61 48 278 1547 32.04 13.72 62 46 73i I 601 34.26 13 15 63 45 130 1652 36.61 12.60 64 43 478 I 702 39 15 12.06 65 41 776 1752 41.94 H-54 66 40 024 I 802 45-02 11.02 67 38 222 1850 48.40 10.51 68 36372 1894 52.07 10.02 69 34 478 1933 56.06 9-55 70 32 545 1964 60.35 9.08 71 30581 2986 64.94 8.64 72 28 595 2 OOO 69.94 8.20 73 26595 2 OO4 75-35 7.78 74 24 591 I998 81.25 7-37 75 22 593 I 982 87-73 6.98 76 20 611 1955 94 85 6.60 77 18656 1 914 102.59 6.24 78 16 742 1857 110.92 5-9° 79 14885 1783 119.78 5-57 80 13 102 1693 129. 22 5.26 81 11 409 1588 I39-I9 4-97 82 9 821 1470 149 . 68 4.69 83 8351 1342 160.70 4-43 84 7 009 1 207 172.21 4.18 85 5 802 1068 184.07 3-95 86 4 734 929 196. 24 3-73 87 3805 795 208 . 94 3-52 88 3010 669 222. 26 3-32 89 2341 554 236 . 65 3.12 90 1787 45i 252.38 2 -93 91 1 336 360 269.46 2-75 92 976 282 288.93 2-59 93 694 214 308.35 2-44 94 480 157 327.08 2.29 95 323 in 343-65 2. 16 96 212 77 363.20 2.04 97 135 5i 377-78 1. 91 98 84 34 404 • 76 1-77 99 5o 21 420.00 1.64 100 29 13 448.27 i-47 101 16 8 500.00 T - 2 5 102 8 S 625.00 1. 00 103 3 2 666.67 ■83 104 1 1 1000.00 ■So APPENDIX 109 RATES OF MORTALITY PER THOUSAND ACCORDING TO VARIOUS TABLES (iooojx) Northamp- English Life Healthy English Life North- Age. Carlisle. No. 3. Districts, No. 6, eastern Mixed. Mixed. Mixed. States. o 257.51 153-9° 149.49 102.95 156.53 125.81 5 29 44 17.80 13 43 10 27 7-43 4 69 IO 9 ib 4-49 5 73 4 36 2.50 2 41 IS 9 22 6.19 S 36 4 87 3-ii 2 78 20 14 °3 7.06 8 42 7 3° 4-36 4 Si 25 15 76 7-31 9 38 8 08 5-36 S 53 3° 17 10 10.10 10 32 8 57 6.48 6 48 35 18 70 10.26 11 42 9 °3 8.36 8 01 40 20 91 13.00 12 87 9 69 10.84 9 54 45 24 02 14.81 14 84 10 82 13-17 11 61 5° 28 35 I3-42 17 53 12 62 17.12 14 95 55 33 5° 17.92 22 76 IS 35 22.77 20 52 60 40 24 33-49 3° 66 22 99 32-32 29 92 65 49 02 41.09 43 40 35 35 45-44 4i 94 70 64 94 Si-64 63 80 53 24 67.20 60 35 75 96 15 95 -S 2 93 94 80 S4 95-59 87 73 80 134 33 121 . 72 135 Si 120 43 146.00 129 22 85 220 43 175.28 189 29 174 95 215.41 184 07 9° 260 87 260.56 254 84 245 °3 290.56 252 38 95 750 00 233-33 331 17 325 11 396.91 343 65 222.22 56 413 04 600 . 00 448 27 Age. Actuaries'. American Experience. Healthy Male. British Offices, OM(5). National Fraternal Congress. Medico- Actuarial. 5 10 6.76 7-49 4.90 6.13 15 6 94 7 63 2 87 6 29 20 7 29 7 80 6 33 6 52 S.OO 25 7 77 8 06 6 63 6 89 S 20 4-7 3° 8 42 8 43 7 72 7 47 S SS 4 9 35 9 29 8 95 8 77 8 37 6 IS S 1 40 10 36 9 79 10 3i 9 78 7 17 S 7 45 12 21 11 16 12 19 12 00 8 87 7 5 5° IS 94 13 78 15 95 15 45 11 44 10 6 55 21 66 18 57 21 °3 20 83 15 7i 15 8 60 3° 34 26 69 29 68 29 21 22 75 24 65 44 08 40 13 43 43 42 21 34 39 39 70 64 93 61 99 62 19 62 19 S3 65 61 7 75 95 56 94 37 98 36 92 67 85 48 91 9 80 140 41 144 47 144 65 138 5° 138 09 137 2 85 205 10 235 SS 209 89 205 b 9 225 08 203 7 90 323 73 454 SS 279 45 300 75 368 79 297 8 95 584 27 1000 00 637 04 424 57i 73 43 602 74 423 576 3 4 110 MORTALITY LAWS AND STATISTICS DEATH RATES PER THOUSAND ACCORDING TO VARIOUS TABLES Table Mixed Population Tables: Northampton Carlisle English Lite No. 3 Healthy Districts English Life No. 6 Northeastern States Male Population Tables: English Life No. 3 Healthy Districts English Life No. 6 Female Population Tables: English Life No. 3 Healthy Districts English Life No. 6 Insurance Experience: Actuaries' Healthy Male H M Healthy Male H M(5) British Offices' M British Offices' M(5) American Experience National Fraternal Congress. Infancy. Ages 0-2 249.31 I30-33 118. 19 73-48 116. 71 87-25 128.24 80.28 128.07 i°7-95 66.51 105.30 Child- hood. Ages 2-10. ■49 11 . .21 6. .07 6. -93 5- •57 3- ■56 3- 16.10 10.85 8.60 16.04 II .00 8.54 Youth. Ages 10-25. 6.62 5-43 3-74 6.97 6-37 3-56 7 14 4 77 5 92 3 84 6 42 7 99 Maturity Ages 25-65. Old Age. Ages over 65. 24 23 15 30 16 47 12 66 14 96 13 74 17 03 12 76 16 24 15 90 12 57 13 74 14 86 14 44 15 70 13 36 14 14 13 65 IO 92 91 87 89 45 37 35 69 92-35 8331 96.79 86.86 79-47 90 INDEX Actuaries' Table, 8 Age Year Method, 61 American Experience Table, n Analyzed Mortality Table, Construc- tion of, 65 Breslau Table, 2 British Offices' Life Annuity Tables, 1893, 16 British Offices' Life Tables, 1893, 10 Calendar Year Method, 64 Carlisle Table, 6 Carlisle Table, Method of Construct- ing, 56 Census Returns, Mortality Tables from, S3 Central Death Rate, 46 Combined Experience Table, 8 Construction of Mortality Tables, 51 Death Rate, Central, 46 Death Rate for Communities, Cor- rected, 47 DeMoivre's Formula, 26 English Life Tables, 7 Expectation of Life, 2 1 Force of Mortality, 19 Gompertz's Formula, 27 Graduation of Mortality Tables, 68 Graphic Method of Graduation, 71 Hardy's Graduation Formula, 84 Healthy Male Table, 9 Higham's Graduation Formula, 83 Insurance Experience, Mortality Tables from, 60 Interpolation Formulas, 76 Interpolation Method of Graduation, 72 Joint Survival, Probabilities of, 34 Karup's Graduation Formula, 84 Karup's Interpolation Formula, 76 Makeham's Formula, 27 Makeham's Formula, Graduation by, 86 M. A. Table, 13 McClintock's Annuitants' Mortality Tables, 15 Medico-Actuarial Mortality Investi- gation, 13 Mortality, Force of, 19 Mortality Tables, 1 Mortality Tables, Construction of, 51 Mortality Tables, Meaning of, 17 National Fraternal Congress Table, 12 Northampton Table, 4 Northeastern States Mortality Table. 95 Pearson's Analysis of Mortality Table. 32 Policy Year Methods, 63 Population Statistics, 52 Seventeen Offices Table, 8 Spencer's Graduation Formula, 84 Stationary Population, 45 Statistical Applications, 45 Summation Methods of Graduation, 73 Survivorship, Probabilities of, 41 Tests of a Good Graduation, 70 Uniform Seniority under Makeham's Law, 34 Wittstein's Formula, 31 Woolhouse's Graduation Formula, 83 111