Clbc lilnivcrsit^ of CbicaQO ICibrarics F-e4 GIFT OF THE UNIVERSITY OF CHICAGO INDEXINa A iffil\lTAL CHAMCTERISTIC A DISSERTATION SUBMITTSI' TO THE FACULTY OF THE GRADUATE SCHOOL OF ARTS AMD LITEPJITURE IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF EDUCATION BY KARL JOHN HOLZINGER n CHICAGO, ILLINOIS SEPTEMBER, 1922 "ffW 1 r^ 1^4 TABLE OF COIITENTS Section I. Introduction a. Mental and Physical lleasurement b. A G-eneral Statistical Theory c. A Suppostitioiis Example cl. Indexing; a Mental Characteristic Section 2. Tlie uroups and Data Studied a. The Groups Studied h. Tlie Tests Used Section 3. Test Administration and Scoriiif^ a. Primary and Secondar,- Index Variables b. The Difficulty Factor Eliniinated c. Methods of Adniini -storinr* the Tests d. Authors* Plans of Scoring Section 4. Methods Srrployed '^ Part I. AlIALYSIS OF THE INDEX VARIABLES BY \^iOLE SCALES A. Coniparative Validity and Reliability of the Indexes Section 5. Relationship- s between Index Variables on the Sams Scales Section 6, Relationships between Scales by the Same Index Variable Section 7. Tlie Reliability of a Scale by Different Index Variables Section 8. Correlations with Ap^.e Section i^. Correl3.tions with School Marks Section 1^. Su^Tiary of Reliability of Indexes for "hole Scales TT B. The Discriminative Capacity of the Indexes Section 11* Capacity of the Indexes to Discriminate between Individuals Section 1^. Capacity of the Indexes to Discriminate betr/een Groups Section lo. Practice Effect with Repetition Part II. ANALYSIS OP Tli^ INDEX YABIABLE3 BY CO.'FONMT TESTS Section 14, Inter correlations of Variables for the Otis CoRiTionents Section 15. Correlations of the Otis CoT'oments with A{^e Section 16. Tlie Applica.tion of Relip.bility Formulae to the Component Tests Section 17. Safiimry of Analysis by Coiiponents Part III. SCORING FOE.iaLAE Section 16. Tlie Linear Form, S - aR -f cW a, Forrrralae with Highest Validity b. Liimjsations in the Use of the Formula, Sraf^tc\Ar G. Use of t-ie ForuTula, S ~ H + CW Section 19. Sininle Ra.tios a. O^ie Correlation between Speed and Accuracy b. The Validity of Simle Ratios as Scorin^^ Indexes c. The Reliability of Ratios as Scorinp: Indexes Section '"''. G-enerai Suiirnary Appendix A. CorrelationsTables for Reliability Coefficients Appendix B. Sieorems Relatin^r^ to Correlations Bibliorqraphy p. 1 Soction 1« Introduction , r- .(. i . . ifficultiss in mental -aeasurement are due to tlie fact that the method is necessarily indirect. This i::rclios Ta^raoness not only in the thrin- mm?•)^T(^Af but also 131 tho precision of Uio result* .jsay physical aeasure* ments on the other hand are direct^ raBkim possible a clear- ©r definitian of the traits aeaeiu'er- nn-l --reater accuracy of their dctermnation. The caso of u ^oy to :jg woip-^ied and also tested for intelligence id.ll brin-r out this contrast* If the boy in r^laced'on thR -nhyf^ical seal© and weirded, the measiireiawnt iu recorded diroctiy in objectively dofinsd units J accuracy of deteniiination depending: upon the instru- :nent and the individual ^ymkinT %hf>. -ne-nTrrrrimt, 'T'ne reaction of tho joy hiiiiself at the ti^no of ii.io -iVGigiiin^ wiil have no effect on the rosult* In case a mental measurenient of intei- lif-'QncG in required, it in neces.-ar; to -ib-Tiit, t-^ the boy a seriea of questionn to t;hich he niust re3;;oua before a score for the msintal trait may dq obtained* This very indirect- ness of Qjrrryrofxfh I'^.'^r'';' /: '■^^■nbirniit.^'' re.-^ardinft; the trait measiuod, and to uncertainty in the precision of the score. Thus in passing from iihysical to inental measurement the role of tho boy has shxftof! fro ■ nr^Wn ;., active one, thoreDy coi:npiicatin<7: t.,Q uiioie prooecurc. It nrny be pointe; ' '. ;^rocediu:e in the c^n?" : ^r!r> nnr^cj^nnts has y^v .railed the phynical methocl. Unit3 liaTo bean defined and scales con- structed. These units for the most -^art aro functions of group Yuxitiuiiiby on pjix-uicaiax^ bypw.: " v.terial, ajid con- sequently lose in accuracy and simificance when applied in iiKiividuK.! measur cnont* Tii-? "irrtillelisiTi in mathod has even been carried "' " ' ~ *■" ritteuipi bo deterniine "zero points" for certain mental abilities so that " just not any" amounts of the traits could be used as reference -joints. In the C8,se of the ineasurofnont "/' :■',-- : v. - .araruioit ther- mometer does not moan "just no heat," but noYertholoss serves as a convenient reference T5ointo Si-iilrxly a r^ood many of the forraer nolmious 'zero r*oint" '-■ lontai /aeasurenient imve dise|> -pestxedt or have noved up to the median where they belonp:. b. A Cronoral St ^^tisti c al Theory The difficulties briefly sketched in the above r)ara- f^ra-nhB su'™"^ost the f^eslrabl''.it'r ef ?;T-ie -"rere rnnoral method of approacii to tiio problea of irioiix^i .uoasui'Gmc^nt , ana indeed such a ::iethod has been irnplicitly unal in some of the later work on tost reliabilit," rx]& validity- In order to Tit ideas, tlierefo.:. ^-^rtain terns will n-r^ ha deriued so that tnoir meanin.": will jq clear thraur)iout the saboequent discussion. Tlie term characteristics rdll be used to denote the nay- sical, :;ientai, or social traits •which i iidiYidrnds oi a rygui) have in coTi>non« The ^ou-p may consist of a nuiaber of persons or ttiin^s eac : ' -rhich Trcist nosner^f: the characteristic in question before any statistical ztm.'j is possible. A man of p. 3 certain heL^-Vit, int0llieo;ence, and 7/ealtli fumislie-s an example of an individual with the three types of characteristics com- monly studied. Tlie pharjG of a characteristic ins.y be briefly described as the status of an individual rith respect to the character- istic. This conception is sn iiTiports,nt one because it is in- troduced for the pujrpose of distinf^ishinr: a particular thing from the number that niay be attached to it. Phases may be nu- merically or verbally expressed. Thus in describing the char- acteristics height and political affiliation of a certain man, the nur.ber 68 may be attached to the phase in height and the word "Republican" to the phase in politics! affiliation. It is conceivable that a nunerical scheme for the latter might be worked out, but the phases of political affiliation ver se would of course rerfiain unchanged. In order that a characteristic be n u/iGrically indexed ^ it is desirable that its phases be arranged in some order 0.O-. iii,:e the points on a line. If the linear arrangement be, m8.de, the trait may be termed a lineg-g- ordinal characteristi., . For chej-act eristics such as heif^ht, an infinite nomber of phases arc assumed, indexed hj the real nusnber system (dense set). In the case of sucli chGracteri sties as size of school class, hoYrever, the nu-nber of phases is finite, and the in- dexino^ is fcco-T-ii^iihed by assi^ing only integers. The dis- tinction is essentially that between continuous and discrete series, the continuity and discreetness appearing; in phase. Finally an. index variable will be described as a quan- tity Ti5hose values are in one-to-one correspondence mth the phases of the character istics indorecl. Before proceeding farther, the nieaninf^ of these Tarioas terras will be illus- trated by means of an artificial exa^iple. c« A Su-^opostitious Exairnle Consider a set of 9^ cubes of horaoj^eneous material. The problem is to describe these cubes by ordinary statistical procedure. Assuiiiing that the size of the cube is the char- acteristic to be studied, some mode of indexing or index va- riable must be adooted. Takin-'*. the edge as a first choice, the distribution nmy be given as fGllo7;%s; edjte frequency 3 30 4 20 The mean edge is clearly 3, ir/ith corresponding face area 9 and volume 27 i.e. a= e";^ T-e\ A second node of indexing by the area of a face gives the distributions area frequency fa 1 10 4 9 30 270 IC Z^ 3T 25 10 25^ The mean face area is nor; IO3 with n corresponding edo'Q and volune appi-ozirnc^tely 3.2 and . : i'espectively. Again indexing by volume gives: p. 5 TTolianie 1 frequency fv 10 8 ?S 160 27 30 Cl^ 64 20 1S8^ 115 10 125^ The mean TOlurae is 39 9 with the corresponding edge and area approxiamtely 3.4 and 11.5. 'Biese results "aey bo set forth in sii'T-iary forra as followss TABLE 1.* IMl^S AMD COBRESPONDriG VALUES FOR CUBES II-IDIXED 6Y EmE. ilSA, A^S YOLOLIE ^ „ „ .,M ,1, 1 , 111 „ „ TI ■ . i ui. i. ri I — i-i T iniifc i ■ , ii jt- ■.- ^ ■HJLi M. ..^.^> j *- v :;;--»«'J^:-.--^--"'^- r ''- '^ ^ "■"■' ■■ J'- ■■■■•■ — ■ : '-■ — ^ . — i i> ^Myy^iM-M^ ™ . ■■■■^■'■■■P*! i^i^ Mode of Indexinf; or Index Tariable Area Yolnne Mean end Corresponding Indexes mf'',e tJm U 3.4 Arec 10,3 11.5 Yolujiie ^7 33.2 39 InsiDection of these figures rcverls a complete incon- sistency imder the three raodes of indexing. Moreover the three distributions are quite different. Tho frequency poly- gon according to 9d.";e is syrmrietrical, r/hile the distribu- tions for area and volume are skewed toward the sraaller val- ues of the variable. It is also clear that the cubes re- mained ivt the same phases in pie characteristic size but that various modes of indexing' o-ave inconsistent results. The above example then illustrates the fact that althoup-^ a set of thinrrs be unaltered in phase, the for.n of the distribution and the statistical constants depend upon the particular in- dex variable employed. p» 6 Fip;ure !•- Distribution of Cubes by Edr^e li Zt> iS 2.S IS Figur - Distribution of Cubes by ibrea Figui'e 3." Distribution of Cubes by Tolurne It JTsay be well to point out briefly that the inconsist- encies in the aboTe example are clue to the relationships be- tween the index variables ;i*e. It mil be. sufficients to note here that simlar ineerxsisten- cies wil' -r?"!- iraienever the reiationsl>i;' between the index variables is other than linear (y=:ax+b). In the latter case the statistical constents will be merely affected by propor- tionality factors. Thus r.!hen height ir, -neaniired in inches and again in centimeters, the distributions will bo similar and all constants easily conTerted. by the sirm:;le linear re- lation s,hip, 1 iiiCli ^ 2.54+ centimeters d. Inderlnp- a Mental Characteristic In mental measureraent the method is to set a body of material before a child end elicit certain responses from him. Tliese res^Donses are then recorded and combined in ¥a- rious ways to produce livhat is Icnomi as a score. Lloreover these responses exhibit considerable variety under different modes of administering the tesbs. Two boys took the Terman (Jroup Test of Mental Ability v^iich is administered under the plan of keeping the time constant. Tlieir responses may be set forth briefly as follows? Score (Autlior) Bidit Attempted ?/rong Accuracy John ISn 1^5 175 50 y71 Henry 15*^ 14'' 15^ l'^ .93 The problem of indexing here is^ unlike that for the c^se of the cube»s. Intelliri!;enco is the characteristic to be indexed, and this ir. possible by usin^f; the Rif?:hts, '.''rongs. Attempts, or so::iG com^jination of tliese ind^:>:: •^-^iri^.blo^ in the form of a score. It ¥dll be shO'.m later that for the particular scale in question the author's score is a rather Gom"plicated f^jcie- tion of t'lr Y-]'lables Ri'-^-'t anc ^'r-yix^:, :jl.o for the --resent it is sufficient to note the possibility of such a daaracteristic p. 8 bein:<7; indexed in several 7/ays. Questions irrrnediately ailse as to the best mode of indexing". Ig it beet to ubg only one of the index var- iables available, or to co.ibiae several of tiiem,and if more than one variable is used how is the co.nbination to be det- ermined ? These questions are of most vital importance in mental 'neasui-eriientj arising; in one form or another irhenever a new test is devised ^:''-r\ ot'^ridardiEed. It may be pointed out at once that this stuay does not attem:pt any general solutions for these problems, but by analyzing a definite type of test material, aims to carry t'.:^ :'. .ivestio-ation a little farther than is possible under incidental treat- ment in the construction of a particular scale. Intellir':ence test ?imterial . a;^ uhosen for tv/o reasons. First, a lar^^e body of such data ?7as available. Dr. F.3. Breed and 'tr. E.R. Breslieh made available their escellent data for t'lree intelli-'rence tests Tiven at two parade levels in Tlie University of ChiGa{T;o Hi,p;h School, .jlr. Guy Capps also turned over two thousand copies of the Terman Oroup Test of Mental Ability, Forms A and B, -,■:.. ......listered and repeated in a number of Kansas hirh schools. A second reason for usinuc:i :;tati;^tic;al ' ility? '' ."eved, i!3 due for the most -art to tho lon-^h of the to ts and to their care- ful Gonf,t:aictr' „ Sectio-ti ^ Groups and Bats. a, Tho G-ro ^ pi StucUecI For nn rnal^rfii'cnl f,t/Uf'?y !*if this ty^e it i" de?!ir?-blD that tlie groapii stusjied siiouid possesa two i. .■; ortant properties* They should be l&ic^Q enoiit^i to insiu'-e reasomible stability in the ^tr^.fAntior)! co^^atpnts; ., :-nca"'id, they nhould be as hoiaonioiiooaii as 7303sibio# iiopro3feiioo.tiY& s£L;ipiGs froia lar'^e '^crnulations arc ©xbreniely difficult to obtain in test nciTk ''rr " • ' . -■'i<^T^^^ v.^t ai^rtnntialf to tho prosont ,::v::iu.y» ..GvorUioloG:; for UiC iarnest gi'oup described belo'- . .ith sar.'iplo of 135 out of ca:;QS wae t^iken by co-nb-^ ,. '-bri^^ ^r' rrfth r--n"To-rf.-iatGly tv> :■; nean scores as tne t>ovai i;^roi4p» iho p^Qiiv desir^eted horea" 'rt^Ae 7 couBisted of GrouTJo i iii'-^:: A oxi. inciuju-ed d^' }5Upiis re- speetivel\\fro!n the University of QhiGtir-o llim School. The Vth Q'rade . . '. " .■.■;.., c-nd ^©re pr©- pared Lj enter ^ie r6'f<:ui&.r first, yocu^ ciac -©s the follo^ria-^ year, there bein,n: n<| ei^iith grade In tlie laboratory Schools, p. 10 Group I Hi.?Tji B diffei-ed from I Hio-h only by the addition of 10 pupils in certain of the tabulations. IMle the groups C^^'^rih(-d ^re imdoubt--^;!:' ?:olect.bh6y are unusually horao{>;eneous as ref^ards social status, training, emerience with tests, and ap:e. The j-rp;est f^roup,! Eirji C, consisted of 135 pupilG from the Ro 11a , Salem, and 61, James public high schools of Kansas, selected as described above. The age distributions for these .":rom?s .are.:»iven in Table '3. The '>--■- hi-: school pupils -..ere a year older than those in The University lii.rrh School, -mile the latter were about a year and a half older than tlie 7th 3 i 4 4 4, 4 3 5 7 4 5 p. 14 Next by esceedinnily tedious coinputation a Tjeir?;hted and an unweirfitod score for each pupil was obtained. Thus a boy responding correctly to iteins i, 3, 7, 9, and 11 on the test receiTed a weighted score of 21 and an wiweipiited score of 5, , the nuiTiber rip^/it. A correlation table was then nmde for weigj-ited and unweighted score ivith a resulting coefficient of A-uvAi - -^7 2 . The standard error in sstinatinp: unv/eigrited from 7/eir^hted score ,, So-.l .The correlation between the two miT/ei.p^hted forms of the test was then obtained, giving Aa6-.S"=i7 The st.andard error Su in eotinrnting mvneirjited form A from unTjei;Q;hted B was 4.8. Thus the correlation between tf/o lor.os of the same test is much lor;er than that between weighted and unweighted scores on the same form. Also the standard error of esti late of unweip^ited from weighted is about one-fifth that from form B. The wei^ting of the items then gives a degree #f re- finement considerably beyond the roll -bility of the test it- self i.e. correlation of two forms. For 40 weighted and un- weighted items of different material the ?7riter obtained cor- relations of .9.^8, .9975 "^^S ■'^^ith the corresponding reliabil- ity coefficients between .85 and .9^. All of these results point to the conclusion tiiat for a fairly long series weight- ing of the separate items is unnecessary. A conrolete solutior. of the problem would involve experimentation with series of variouB lengths, items of vai^^ious difficulty, and populations of different size. Such exhaustive treatment is clearly be- yond the scope of this study. It may be finally pointed out that a nuTiber of recent achievement tests have appeared first with weiglited items and later with weights dropped when it p. 15 was realized Ttiat a sli^it difference thes© made in the resulting scores. C» Methods of Administerinfy the Tests With the factor of difficulty eliminated the scoring or indexing problem is greatly siiaplified. The plan may now be described as the method of Unit Eesponses, the response to each item being scored as a unit point. Furthermore if Amissions be neglected or counted as errors ^ the reiriaining primary index Tariables are reduced to Time, Attempts, JRights, and Wrongs with the relationship, A =.R+W if omissions be coimted as errors* This assur/iption will be made in a subsequent discussion. The nujiiber of omissions occuring in the tests used was negligible. It thus a pears that four' variables 9 T,A»E» arid W ?/ill have to be'studied? the last three not being independent* Also all scaring ^for- mulae or secondary index variables will be functions of tliese four primary indexes* Two plans for administering such tents are possible. One plan is to fix the number of Attempts allomng Time, Rights J and Errors to Tary, Tiiile the second method is to fix th© Time, allowing Attempts, Rights ^ and Wrongs to va- ry. In the last analysis then, only tliree index variables need to be considered, the fourth being arbitrarilly fixed by the plan of administering the tests. According to the first scheme outlined, one allows all of the children to p. 16 finish the test thus keepin,^ AtbeiTipts constant. The time is then record od by stop ivatohes or clock device and RifT-jits and Wrongs obtained from the pa.pers. By the second plan a fixed time limit is set for all the pumls, Attemnts Ri/3^t-^, and Wronr':s bein/^ then scored on uhe test papers. Th.e second method is obviously sinipler than the first, and is no?/ followed in the p^reat nmjority of tests of all kinds. Certain te^ts, it is truSB, neglect actual time but these are not considered here. The material used in this study is all adjiiinistered under the plan of fixin-^ time giving the three priiimry index variables A,Rj and \^. d. Authors' Plans of Scoring For the three intellir^once tests described above, the authors have set forth scoring foriiiulae for each test of the battery. These fornsjilae 5xe obviously expressed as functions of the primary variables A,xl, and I, while relat- ionship A - R-ff makes it possible to set down the equations in terms of any tvro of the variables. In the follomng tab- ulsjT scheme, therefore. R and W have been e.Tiployed throu";h- out» TABLE 4.- AUTHORS* SCORING FORIIUUE ON THE THREE SCALES Scale Test r^ 4 5 7 8 9 10 Otis Tearman Chica'"o R R ■ R 2R R-W R-W R R R R 2R ^R R R-W R R R R-W R R R 2R p. 17 It will be soon frora tabic 4 that ti70 of the scalcc consist of ten tests each waile the third inoludes only five. Considerable variation in the scorinp; formulae is also to bo notedo The otis Scale has 9 tests scored by R and only one by H-f ; Temanj on the other hand, scores ■ four tests by R, 3 by 2R, and 3 by B-W. For the Chicago Scale t:-o ^er fo'-^^"'^e a-^ear^ H- i W arid 2(R« ^f). All of the above foruiulae are clearly special instaaices of the general linear form where a and b are constants* The simr-licity of the Otis' scoring formulae imnedi- ately raise^ the quostion rs to the advisability of scoring the other scales by the same method. This problem is dis- cussed in the following sections. Another question concerns the for:Ui..3 R-W. and R- ^W. These f ornialae are employed for material of the True-False t^?pe and t}.irer3-choice variety, on the supposition that"' they correct for the element of guessing for 3uch tests. Tliis proble-^. "' ■! also be taken up in some detail in later discussion. Finally the doub- ling of scores for individual tests is a point that needs consideration. So far a3 the i?vTit9r it abls to determine, this doublin'-^ T?as effected because the author of the test felt 3uch tests to be worth about twice as much as others, or wanted to increase the total points • o'^'-"'JIo to soiTie convenient number. The Doint at issue is the same as that p. 18 between Fei'^j:i,tedand iinweiHited. items* A .crrs^uate student ma,de a ^y^ii..^/ of the Cliicago Scale regarding'; this problem and found that the welghtin,g of the tliree tests as indi- cated in Table 4 affected the correlation bat slijp;htly, the coefficient between ?/eip'J:ited -iind imweLo-jated scores be- ing .SB. It thus appears that Ei.f^its are the basis of most of the SGorinfr fonmlae employed on these tests and thet other forms have been used to correct Right responses for gaess- ing or to wei.^th theT±iole test because of the relative im- portance in the battery making up the scale. Section 4 Methods Employed Taci -^--oneral method of this study will first be to an- alyze the interrelationships of the index variables invol- ved, and tlien to set up certain criteria of good indexes and attairnt to evaluate the variables in terms of these. Obviously this As an indirect approach and it must neees- saril^y be so from the nature of the problan* The actual technique employed will involvo a considerable amount of correlation, for this is the best method of studying the relationships between index variables fro i tests. For the case of the cubes deserib^^''"''. «bova the index variables p. 19 were functiondly related,!, e. Tolume - (edge) Such f'uiictionality can only be approached \)}[ empirical data, the correlation coefficient f^iying the most convenient ap- proxii?iitio:;i for linear fimctions. Thus if score iaid Rigiit^:. are coi^related to the extent of .98 icLth regression lineai% there is a very close approximation to the functional re. lationship S - KS. Tiw chief adTantag.. o: correlation is that it gi7es a nuiTierieal estimate of the closenesG of such relationships or approach to linear fimctionality. Index variables mil be analyzed by batteries and by single tests, ige, school rii3xks, and otlier intelligence tests will be used as criteria against ^iiich to check the var i ou" f or': liiil ae . All of the calculations beloF have been preformed by the Tnriter and have been checked T/ith care. Correlations were obtained by the usual product L-ionient method with a specially designed correlation forra. Tliis was found to cut do?fli the labor of calculation very rmterially especially when "batteries" of coefficients 7/ere required. Blakenian's test for linearity was applied in a few cases with the re- sult that the miter believes the great i:ia,jority of the tables Gzliibited sufficient linearity so a^ not to reduce the correlation beyond the limits of probable error. PaxL I Tfill be concerned -Ith an analysis of the five index variables by ?iiole scales. By the indirect method of correlation, the relative merit of the indexes ?dll be de- termined. Part II will involve a similar analysis of the individual tests of the scales. In Part III certain form- ulas ?dll bo develoriod end their validity fmd reliability detGrmined in '^"■f'J^i"^'^.'^tical ter-^f^*. Part X iinalyais of the Index Yailabl es by Wh ole Scales A>~ ComnaratiTe Talid^ t y and RQllability of the Indaxes Tae most direct approach to the problem of indoxini^, will '^^' ,y '^■''^T''^ ^'''^ '^■'i^.'i.-Of* -^^ -'^l'^^: of p.nri~**;ir' js^ Q2« hibited in i!able 4 with the slm^'lar plans of coimtinr; meroly Atts^w^'ts- Hi'-^^t^ir'Srrorn, and determining Accuracy for ^t".', ■:-:->nHi^iirr>' -'■?i Vyr^iTA-i f^fi rslatiTS merit In reiiaDiiity of tiieso siirrf\lGr primary variables #ien soTsral teets are poolod to ,r?ive a scale score. The torf, "f'C""^'"' '^■f^'^''"^^ "J f^ hn-jTi «?i*T,-"' A',*'' ,15. '^.t-"^^'>«^'^ ^h it from test soor© i^iiiiosii will be used to aonote Uw score on on© of the eonrnononts iTkhiri up the tot^l scale* IPhus the Otis Scale i:. irndQ w^ 0^ '^^ ,-.,-,' 7-.-.r-nn^-if, ■*, - . ^ Fiv tirpoB of eomrjai'ifion will be ;:ii>jde in evalu0.ting the indos: vari-^blars for rm??ral reli"-bility» Thee© includes ft. B©latlona : ■ /iju-vr;;ai index Taria •. ^^..0 smse scale b* RelatioT-shipji; between scales by the sa.ne index variables Co Heliabillty of a scale b|r different inde:-?; variables d* Correlatic-" "'^^'^ "^-'' ©. Gorrolations ?7ith school ?narks Sec tion 5 Rel5^1onohir>n bet^oon Indo:: T-iri^-blGS on th e Uomi Scales As pointod out afoOTe the relationships between the va^ TD. 21 ibles camiot in general be functional for empirical data such as thopiG obtained froa montal tests. Noyertheless iu irill be Taluablo to discoTor tlis closenGSii of linear re* lationship as indicated by the correlation coefficient. These results are set fori- '^abl^ 5, All of the corre- lations were copjoutsd on total Aotor.rotOj, Eiglits, eto. foi" the entire scales. The agreement froui sroup to group and scale to scalo betiveen correlations for the same two var^ tables is striking- and may be viei^ed mUi pardonable sat- isfaction by one #10 has viewed many isiexnli cable differ^ ©nces for other t;/i39s of test*^- ■■■-'■/.'^ : _: . ,' .■; TABLE. §•- COEKEUTIOMS BSTWESI^I YMII^BI^S FOH THE SA'IE SCALES Scale and G-rom:- Pairs of ITariablas Correl at'i'-; S^A' S^R Otis 7 Otis I B Ter*? Tor. I B Chi. 7 Chi. I B S> +.72: "♦••99 -.52 +.67!4'.99 -.59 +.59 j-f . 96 : -.45 +»&9 '•f.96i-.67 +.45 +.9'^ -.54 +.54 +.95! -.71 S>-^ A>H -*»,73|+.73 +.81: +.37 +.06 1 +.7^ T R! A +•33 +.69 +,02 +•74 +.59 +.62 +.21 +.13 1 -.51 +.41 -*i.6> -iS*^ +*14i+.10 +.31 -.1^ +.19:+.^3 !■..- -•61 -.39 -.05 +.70 +.77 +.53 +4 73 +.59 +.7D W>| -.91 -.93 -.94 -.93 -.93' Mean ff.Ov' +.56U.58 +.76 +.C9 ori +.69 -.93 .- v^- for example the correlation between authors' i;icore and to- tal Attenrpts for Otis G-rade 7 is .72 +.05, i/hile for Otis I Hif^h B it is .67 +.05. By the forimila P.e vhen the number of attenrpts is fized and the rels^tionshix? A-= R^ VJvT still obtains. In this case the functional relationship holds strictly, and this i-^-i^/--- --^r-Poc- ;:3;r^ativ© corre- lation as is shorn raore fully in a section below. If the index variable A be used as a measure of speed a number of interesting xelationshi"''" ".r-; bro^ifrjit ou,t by the renrdning coefficients. Speed and accuracy are evident- ly uncorr elated, the highest correlation between: these two variables being --,16 ^,.»r.9 which •" " '^isi^^nifieant because it is not even tivice its probable error in amount. Moreover p. 24 the difforencGs in si^ are such as to giv.; x •isen corre- lation for the coliimn. of less than ,Ci» It will be furth- er noted that R in correlated mth A to the extent of -!-.69 on th- "T^T^^ ^ ^''^ '" •"'^;'-i A f^i^e- ' ■■■•'■•r-.i eorrel^'.t-^-'-*-^ of +«24# Ml a? 6r age. Gorr elation of +.59 between S end A was preyiously noted. These results indicate that the way to get ^ hlff^i intellifrence ^'•-■-•■■^ -'/; ■' - :-ork fast. By working fast one is likely to xuake more rrd stakes, but he Is much more likely to ,n^et more itema ri,p^t and make a hi,c^her score than 5^f ' orkdv. .ior^ oxOvdy. Intellifr--"- f,^~\tn have frequently been called "alertness" te.cts. The above find~ inf^s indicate that with considerable appropriateness they might '^■''": ''^■^ termed ''spec '•'''' ^--^"t". ^h? -.n sumption thus far in deterrnininf?; general reliability has beea that the author3*score is the best index of intelligence. If, on the other hand, Accuracy had been assumied to be the best indexs the speed factor would have been eliminated, there being no s;enoral tendency as indicated by tlie zero corre- lation for a ijupii uo get a high or low accuracy score by changiFig his speed. It iiill also be recalled tliat Accur~ acy was hi.^hly correlated I'dth Score (■J-.76) so that f- as an inuw-c o-ariable is t £ar^^u#A y coasistent uith S witn- out being miduely influenced as is the latter Tariable by i^e undesirable speed factor. The average correlation of -.49 between R and W means that the raore items the pupil gets ri^it, the fewer he is T). 25 likely to get wrong* If Attempts ara eonstojit tko above correlation becomes -i.^^ as in the case of Wrongs and Accuracy. F:'r/.lly the yrr^y^^ '>"-^r><^. T? and a -re found to correlate an the aTera^~;e ^oCj, a coefficiont viiidi becomes ♦1*00 -^ea Attempts are constant. (Theorem 3 Appendix) Section 6. Relationship's betT/eon S cale s bj- the Same "fh© relationships between index Yejriables on the same scale have been disGossed in the precedinp, 'Bection, with uJ.O :. ; ■a'',-*- ■*"h^'^ ■'■-■^ r— ,virM--^T r,--/?: ' -r^-;i^■!^ -n th score as a criterion is Rights, Accuracy, sind Attempts or Errors. 15ie variable R r^ossesses a si>Trlicity #iich considered hi oonQectit. ■ ' v' '-' -■--,^;^ ---k:' 3 suggests that it mglit well be substituted for tiiat va*- riable in inde:dnf? bj/ batteries of tests. The index x was found to be in close £^;;xuuaenu -atli S and R and to possess the advantage of being unaffected by speed. TtxG next procedure mil be to evaluate the index v-s'xiables indirectly by determining; the correlations be- tween pairs of scales indexed by the same variables* The closeness of thi? corresT^ondence id.ll f^iv; a measure of ti^ effectiveness of the particular uoc.: indexirig. The former comr/arlson was, inter ^variable, the present is inter- te-lt. Table 6 o-ive?^ these results for p-rn-i-r. I Hidi B and Grrado 7. Coin|.jarison of this table mUi Tauio 5 reveals the fact that the correlations in tlie former are rmich more ne- arly tlie same size. All of the coefficients are signifi- cant, T^ile there ar© few of the differences between any p. 26 two Ttiich may not- be attributed to saanpling by the usual method* Ths neenn for the coliirmB and rows of the table brinf^ out the stability more clearly inasmuch a.3 the de~ viationa frohi these are in ;:i03t caoes Yary sliglitt « The really surprising; results exhibited by this table are indicated in the col-oirin of means for tlie various in- dexes. Here it appears that with inter-seale correlation as a criterion, S,E,E9 and 1 are all about equally good I for purposes of indexin.^, ftiile A is someviiat poorer than the root. To discoTer ai.iiost as hif^i a correlation between two intelligence scales by merely tabulatii^ total errors as by usins; the author's score or total Rights is at first somev;hat surrprising. It is a closer a^eeimnt than migiit be expected from the areraf^e correlation of -*58 between S and ?/ from Table 5, Inter-test correlation giTOs a meas- ure of the extent to i-iiich two tests a^ee in msasuiln.?^ the sacie character is tic. From Table 6 it appears that this a^;r©em9nt is about equpjly close isi-ien any of the four' in- dex Tariables is em^oloyed^ the general ordar of merit be- inp, indicated in the table as S,E,IljW, end A« This order j I it will bs recalled 3 is in harmony with thtit found in the preceding section. It may finally be pointed out that the lack of variability ainong the coefficients stron^c^ly sug- fi!;ests that combining these index Tariables as in the case of S f/ill give but slightly better index when batteries of tests sudi as these are employed. P* 27 TABLE 6.«IIJTSR-SCALE CORRSUTIONS BY FIYE IWKi YAHIABLSS Judsy Pairs of Scale,-} ajid G-roujos Taxiable Ter;:i. a Otis f''?9nn,)cChlc. GhioicOtis Mean Order 11. H.S. El. H.S. El« H.S. Score ktiermtz Wrono; Accuracy +. 7,y +.73 +.72 +.70 + .63 +.72 +.82 +.63 +.88 +.66 +.5C +.59 +,6<^ +.61 +.76 +.46 +.71 +.62 +.77 +.78 +.78 +.66 +.79 +.53 +.83 +.65 +.74 +.76 +.56 +.74 +.69 +.73 1 5 Z 4 3 M®an +.71 +.82 +.60 +.68 +.67 +.71 +.70 Section 7 Ttid IleliabiIi^^.:/ of €. Scale u:; Mffsrent Indj Variables Another method of studying the relatiTe merit of the different index variables is to obtain the reliability coefficients for a scale under each of the indexes. Q-roup I Hi^ C was iised for this puiisose* It mil be recalled tliat tliis group consisted of 135 pupils idio took Form B of the TeiTdan Scale and Form A of \..,..w ..a...:., oest on the follo?ang day. The reliability coefficients in this case are given by the correlations between tlie vaxiables on the t!^o foruis of the scale. ThsiJ^ o.^iTGlation.j .ix^ given in Table 7 Ydth the contingency tables presented in the appen-. dix. p* 28 TABLE 7.- RELIABILITY GOEFFICISITS FOR 1?EBHA:^ T^oRrfS A AI© B Oil G-IIOIIP I HIGH C Tariables Ssors Att,Gr-Rts Accuracy EeliaMlit- Goef.rieient +.684 ?.C3;^ +.896 +.C11 +.737 TMi It is at ones apparent \2u^'^ ' ■ oeoia na^ the hicrhsst reliability mth a coefficient of .908. A glanc , t'T-? corr^B-'^-.indin.T correlation table in th© ap- pendix wiii. give tiiio ro^uit aore liiyuim^i. Here -the lin- earity of regression is at once aopai'entj and tlrie rema^rk- able a^eesient 9siiibit©d In graphical fora. Close corrs*- spondence of this typa appears to th© m^ltsr as one of th© most Bi^iifieajit acliieTements of S'tai-idardized tests. Just v irtttiat U... to./ ^-B^^ws:.. ^.,: indfx lo „rix. ainbigaouSj but to index any mental oLaraoteristia mih saoh a high degree of reliability is in itself a most not9^?orthy achievemont. It duuaiu, ■;. - o^-.Li.j in mind '&t all such reliability coefficients (and indeed all such eorralations) depend up- on the f^roup. Selection mil in general tend to reduce such correlations iMle heterogeneity due to such factors as age mil tend to increase it. Retumin.'T to th© reumining coefficients in the above table jif© iind that the order of reliability for the five variables i^^ S,H,R,Wj and A, This is precisely the order I obtained by the method of inter-t©st correlation as shOTOi in Table 6. p. 29 If the symbol /Ixx be employed to denote the re- liability coefficient for a test indexed by the variable Xj thQ differences between the correlations in Table 7 may be exhibited as followsj J^« - A.AA ~ -^.224 +.034 difference is simificant A^s _ /xk^ - 4.,012 T,oi5 difference is insipjiif leant /Lss - A».w ^ -i-All +.^29 difference is significant /i-.s -~ rt^L. ^ +,067 +.020 difference is significant fhe forniiila used for calculating the probable errors of the differences is, The sinali differences between the reliability coefficients for S and R is insignificant^ T^hile the remaining differ- ences are sufficiently large in comparison with their prob- able errors to be significant. Thus from the standpoint of reliability the Terraan Scale is indexed equally well by author's score or total riglits, and next best by accuracy. A reliability coefficient of .74 is generally considered highs and it is remarkable that total errors on the two forms should correlate to such an extent. The difference between A-sb and Ivuw ^ howeTer, is over fiTe times its probable error so that the reliability of errors as an index is significantly less than for score. Similarly Attempts furnish a niuch less reliable index than Score , Rights J or Accuracy. . In this part of the study no attempt is made to an- alyse the indiridual tests making up the scales. N®Ter- theless it is interesting to note from Table 4 that three of the ferriCT, tonts r»ro scorsd R^Wg thrn'* "^ -^ad four Ijy I aXoRO. The uiiicrGiico iu tlio roii^^i-ity oo©ffi~ cimt P-ss wider tliin plans and /I^r bjf Qoimtinr:: more* ly ri^^itn in '^'»^V.^i,-H5 aj^ nhoTm above. ' Formulae of trie type M oua iv-w, -i^uc appua^- iu ha?o no effect on tao rollaMlity of the total ;30oro, and tlasir us© for sue . scales i3 open to quQsticm* Section (^rreXaticr.^ wjtlrik^ ^10 age factor is al?®y3 of interest ihrn fitucly* in^ tost results* Table 2 vitidi (^wqb the cire distri.- butlons in half-yoax inteiralSj shoT/s raa\^]oa fox- 'Uie v..** rious ©roupEJ frc^i four to ®l|§;it' years* la IJable 8» the corr^V'^'.t'i^'^'^Ti.'"- botrro^'^}'; ■'^.'"'r fflnsl. thn four index variablnn are given for Urade * ana i iilg. ii« A consistency in tiiose coefficients is at ono© apparmt« 'Bib laosn correlation of -«4i between a'!;o and score indioatos that the younr:* or pupils ar© brif^Iiter tlian tlio oldar mios within Ui^ B&rm g^ade groups Siii3il8xXy Uib tmm. correlation^ *»389 **299-*26, and *^2^ slio?^ tlmt -^...o yourjger pupils «^t raore itonis ri/^it, are E3or$ aeourat©» aro sp©Mier,» asad mak© fouor errors tliaai tlie older ciiildren* p. 31 fmm 8.- GOEKBMTIOHS BITWSEI^ AGE MW INDEX TAHIABLES Scale and Ap;@ with Group Score Attempts Sigiits Wron^^s Accuracy Otis 7 Otis IB f erman 7 Termn IB ChiGa,«;o 7 Chicago IB -.41 -.40 -.47 -.39 -.39 **39 -.37 -.27 -.22 -.21 -.21 -.30 -.37 -.39 -.40 -.37 -.05 +.06 +.27 +.17 +.28 +.20 +.24 -.24 -.32 -.37 -.34 -.33 -.26 Mean I-.41 — •??6 -.33 +.20 -.29 The superiority of the yoianger child is eTident, therefore, no matter which indez is ©raployev ^.1,, "DesuS* When arranged according to the size of, the correla- tion ^d.th a^e the order of the index yariables a-nr^ears from the means as, S,R,B,Ag and 1. The three variable^i Score, Eights, and Accuracy retain ttie order found in the previous sections. MoreoTer the mean correlations for score and ri^ts with age ai"e vei^y nearly the same, so that the su* periority of S OTer R as an index is again slight if any. The usual sanrlJn.<^ forrraila reveals no difference in the correlations ttiat is of sta.tistical sigLiificance. Similar correlations are given in Table 9 for the lajT^onf ., coefficients, though someitiat smaller are in hanflony with those of the preceding table. The de- crease in size is probably due to the greater range in a.ge. p. 32 !?ABLE 9.- GORREUTIONS OF INDEX TJJIIABLBS WITH AGl fui TSKIAII SCALE FORM B G-rou-o Age Tfith I M# B I Higi C Score Atteaipts -•16?.<^6 -,X4;f«<^6 Eights I Wrongs Accuracy -.37t.O© i -^.28^.09 *.34??.08 By increasing the range thro-o^aseTeral grades the corre- lation becoraaa roBitiTe. Bi fact heterogeneity, or lack o: selection appears to have a ci^rious effect on correlations with a^e« For children of exactly the sain© age the coeffi- cient i.. ..: 3oi3rs0 zero; lih.fm. the i-ange is increased to several years as in the typical grade group , the eorrela* tion is negative. As the range in age increases^ the co- efficient approaches zero a^-^ain, and finally ^.icisses through. this value to positive values of considerable size if sev- eral grades are pooled to give a long range* The theoret-^ ical curve for tlie eorrelatior. .x, efficient will thus have an appearance resembling that in Figure 4. 15ie negative correlation for Uiq age interval OA has been accounted for by the appearance of older retarded children in grad© groups* %i3 explanation, while plausible, does not seem satisfac- tory for groups such as G-rade 7^ ^nich irj usually free from chiMren of tliis type, Tiie positive correlation increases from A as the ag© span is lengthened. p» 33 Fl^» 4 'Sa0o.r9lji0al curre for eorrelatloa iTitli om ■i^^^'^*-"i^-^ -r ,.,i;'.jyrC.4£:..C....3Tl3 ^771'GJI 50X00., .. Ic:r!k:o i>|pi||ii||| ii ri illl ^ iil l) wiiiio school msKB are obviously in;-ccurata est! of the ability or adiieveawiit of -upiiB,. nevertheless th©y nK!.y sorre ':',?^- "■ useful '"e'''';'o« '■>'"■-'•- *.'-►. ^.^- -■•■•■. ^ -•% tftst result.:;* yiioir rolatl?© iii«cc'ajrac^? isuisi beeii gi^ossly ©xa^- a b g@rat0(U Burt, Proctor, Kelly, mid othears have shorn tMt the predictlYO value of ....w... rnarks ...:. v..wu.i as hic p. 34 that from intellip;enc:v .„ achievement tests. MoreoTer in the prssent ooiiiriarisons tlie question of accuracy is not of great iraportaiice inasniAch as each inde;: is checkeci, ajf^ainst th^^ ^.aie sdi .ol f^rades* ^.atoTer mireliability exists in %lq rmrks, therefore, iiill affect all corre- lations alike. TABLE 10.. GORREUTIOHS BETIEEI^I IIAEKS IN M&LISH MD THE IIIDEX YMIIABLES ON THE TIIIUr,E SCALES Scale Iferks in Enr^lish xvith Score Atteiorots Ei^ts Wrong Accuracy Otis Terraa:.! Ghica^^o +.56 +.63 +.52 +.36 +.3(^ +.B7 +.57 +.56 +.53 -.32 -•36 +.46 +.46 +.48 Mean +,57 +.31 — -^"SS" +.55 , 1 -.33 +.47 TABLE 11.- CORRELATIOMS 3ETWSSN limG IN HISTOIIY MD THE mm. Y/J-ilABLES ON Tm THREE SCALSS Scale :-.ferk3 in History with Score Attempts Rii^ts Wrong Accuracy Obi-s Terman Chica^^o +.53 +.63 +.48 +.17 +.24 +.17 II II 1 1 ■ 1 ri +.54 +.6^ +.41 -.45 -.38 -.37 +.55 +.53 +.47 H©an + .D5 +.19 +.o2 -.40 +.5J2 TABLE 12,- GOEHEUTIONS BETWEEN JilAEKS IN MA^ I>JDEX TMIABLES ON TI-IE TlffiH SCALES p. 35 tlEIIATICS MD Seal© Marks in Mathematics mtli Score Atternpts Ri;^;its f'ran^ Accuracy Otis TeriTian Chicar^o +,60 +.54 +.47 +,45 +.32 +,29 +,61 +.72 ~,17 +«40 +-36 +.27 Mean +.54 +,-35 +.53 ■*,ZZ +.34 L ,n,. -l„i —- — » Correlations of the fi?e index variables i^ith Ehglish, History j,and Ifathetnatics ars r^-X^m in Tables ^,w,.^^,and IS respectively.. Inspection of the avsrago correlations for these tables shows a close ao^reerasjit for the three subjects studiedo The coefficeints for Siir__...-. :""'jestjHiE'tory next, and Lkthernatics lowest/Dut the diflerences are sli^^iti In all three tables Score has the highest correlation mth school imrks. The next higjiest corrslations in order are Eir4its,AcGijracy,WrongGjand Atternpts. This is precisely the order found in the section on reliability. Thus if school uork be measured by luarks ti^e s -„^t,tiTe merit of the T0.rious indexes for prediction is SjH.a W,and A. As in the precedin?^ sections, correlation invoiring S and B are aore nearly equal than the otiiers. Section 10. Suiimary for Reliabi lity of Index es for Ihol© Scales For \iiolG scales consisting of batteries of tests, the authors' formulae appear to be sli^tly superior to total Rights as as index. Table 13 gives the avera^^^e correlations and differences in f^vor of S (absolute values considered) p. 36 TABLE 13.- SirtiABY OP G0RPJEU7I0NS FOR SCORE AMD RIG-IITS Yai-'iablcs Correlated AYers/'te Coefficient DiiferGnce in FaYor of S Score and Riplits +.^^6 Scales Indexed by Score \ +.76 Scales Indexed by RlQ:hts i +.74 Terman Forais B and A 'by S ; -i-.i?! Termaji Perms B and A by R | +.;)0 Ai^e ?7ith Score i -.-^U A^e i^th RiF^ts I -.38 lErks ?.ith Score j ■^.•'^5 Marks mtli Kip-hts 1 +.b3 Total Difference .02 .01 «03 .08 The extreme sinr:"icity of scoring?, by Pj.r;:hts,howeTer, would seem to more than outweir';h the slic'^t advantage in favor of niorG Go^ir^icated formalae* Accuracy has been shorn to have the pecaliar e^vant- af^e of being unaffecte<^. hy speed, and at tha seae time to posser- -■-•-. •"'-'^^'-■'. . .,v, "■■'') Guismarsj: cc.-w, tions in favor of Score are shovin In Table 14. The total differ- TABLS lA," m^^^MCf OF nomyHLATIGNS FOR SCQIS AliC ACGURACI Yario.bles Correlated Averr!.'-''6 Coefficient Difference in Favor of, S Score and AccLtracy +.76 Scales indexed by Score +.76 Scales Indexes by Accuracy . 70 » . « . 03 Terrnan Fomis B raid A by S "1 » t/ *^ Terman Forms B and A by R +.84 . . - , . . .07 Age ^dth Score I Afse iPTith Accuracy 1 "^ 1 A . • » X :' .< Marks i,ith Score +.55 Marks mth Accur-acy + .44 . . f . . . .11 Total Diff'^-r ence . 00 P* 37 ^ces in favor of S are ohowk in Table 14, Tlie total differsncea in favor of S indicates that Accuracy is som - T^at less satisfactory than R accordinp^ to the criteria eniploy-30.. Moreover it is a more linvolved complicated index than R, but not so involved as S# The results for Errors are -nresented in Table 15. Th® general merit of Wrong as an index is less than that of the preceding variables. Erroifs, however j have a surpris- ingly hig^ reliability and are utilized to advantaf;?;e in formulae discussed in the following sections. TABLE 15.- SIPriAJiY OF CORPIUTIOHS FOR SCORE MD ERRORS Tariables Correlated Average Coefficient Score and WroA<^ Scales Indexed by Score Scales Indexed by Wrongs Terman Forins B and A 1^ S Terrnan Forras B and A by W Age With Score j^e With Irongs mrks With Score Marks With Wrongs 4 o « c 9 o -.58 •¥,16 +.69 +.91 +.74 . -.41 + . *j^- » « • » » a i +.55 . o » o » o Total Difference Differences in Favor of Score .17 'i o a » ^ .68 Attempts , ?Mch are frequently used as an index for tests 5 appear to have the least merit of any of the var- iables discussed. Table 16 gives the averages and differ- ences as in the above tables. The total absolute differ- ence in favor of S is greater than for any of the pre- ceding variaiiles. p. 38 TABLE 16, - SUWMEY OF GOEPvEUTIONS FOR SCORE MID ATTEMPTS Yariables Correlated Score and Attempts Scales Indexed by Score Scales Indexed by Attemnts Tei^nan Fornis B and A by S Termaji Forms B and A by A Ag© with Score Age with Attempts mrks ?ath Score Marks with Atteiirpts Coefficient +.76 +•56 +.9i ♦.68 -•41 -.26 +.55 • « Total Differsnc© Difference in Favor of Score » « p « a tt o o d G « .20 .27 Bi-rSi3 Si 3 cri?ninati T8 CaT-.-- u:.2.0 Ind©2C93 In addition to the general, reliability pf an index, another valuable propeir!" ' r" sudi " — 'i.able is the extent to which it makes possible discriniina,tion between indivi- duals ezid between groups rixon real differences exist. A test Miich reveals too narro?/ a range for a given group fails to discriminate between the individuals of that group. Such undistributed score is a defect in the test or in the' mode of indexing. Similarly a test or raode of indexing which fails to discri.-.iinate between groups is defective if the characteristic is in reality different in tjrpe from group to group, ©lus a test #iich shows all individuals in G-rade 5 to possesB the same ability, and at the same time reveals p. 39 no difference between mean scores for Grade 5 and 6 is lackin,<5 in individual and in group diseriininationf Th© fundamental assuniption is, of course, tbi?.t such indlYi- duals and gro-"nr. do Tary and that failure to detect the variations lies in the particular mode of indezinr?- tiie trait in question. Section IX Carac ity of the Indexes to Pi scriroinate between IMiYiduals Disoriraination between individuals of a group is best studied by meanss of frequency distributions. In the pres- ent study J however J such an elaborate method as this is unnecessary inasmuch as intelligence tests are ©f suffici- ent length to give a fairly good spread for all indexes, !aie distributions for S,1»A»W, and E» in the Appendix are typical of those for all tliree intelligence scales, fhe standard deviation for these variables are given in Table 17« TABLE 17.- STAJ^ARD DETIATIONS FOR TERIIAII f'OMS k Mm B mm aRoup i high c Tariabl© C^i (first) (o;^( second) c^~^ iDiff^P.S.di^ Score Attempts Ri£^lt3 Wrongs Accuracy 33. 79+1 ,39 24a 250. 99 26, 9551 til 19,73+0.81 f. 13+0,01 30.91+1,27 20*63+C,85 23.5^r^,97 19, 66+?^, 81 0.127^.01 2,88+1.68 3,49+1.30 3.41+1,47 0.07+1.15 o.oiTo.oi 1.5 2,7 2,3 1.0 1!here is sowe evidence that there is less variability in perfomaance on the second trial (Form A) than on the first ( Form B), Tliis is incidentally a bit of evidence to the effect that equal practice for a ^oup of pupils tends to bring them raore closely together about a central t.^ne, a result contrary to that held by some psychologists. The differences, however, are slight, altho^ji/y^i in one direction, and the tw) fomis of the test i!i8.y not be equivalent for this purpose. The result is then merely sugp^estive. ^e standard deviations for A,R, and W in Table 17 ad- mit of direct comparison inasziach as they are all ezxDressed in point or response units. The order of discriminative ca« pacity for these variables is then R,A, and W. The indexes S Kid. I .-^re expressed in different units and hence may not be compared mth the rest. Considered on the point basis however 5 author's score has the greatest capacity for dis- crimination between individuals on account of the 'jveigiit* ing and formulae invslved. The standard deviations -for Grade 7 and I High B are also given in Table 18. Tbe results ag- ree with those of the preceding table. TABLE iw.- STMDAI® DWIATIQNS FOE THT; TIimHE SCALES Scale and Stsndazxl Deviati :.i] L.g for "■"" - -"-'■ ■"'■'■'-- - Group Score Atternpts Ri^^its i'rong Accuracy Otis G-rade 7 19.91 18.00 19.92 13*46 0,06 Otis I I-Iigh B. 22. 4n 18.02 21,55 17.02 0,^8 . Termeji Grade 7 23.52 19.17 16.52 12,46 0,08 Terman I Hif^- B 26.87 18.39 22.34 14.87 0,09 Ciiicago Grade 7 i^.sa 6.20 6,16 ^.60 0.09 Chicar^o I Hif'-h B 12.03 5.8^ 8,34 6.50 f^.l<5 p. 41 In order to study the variability of a group by a statistical measure independent of the units employed, Pearson's Coefficient of Tariation, V - '^^^'^ , was ©nployed. ISie results for two groups appear in fable 19. It is at once apparent that #iile T is independent of the units employed it may nevertheless le$d to results which exQ confusinin;. The largest coofficients of variation are for f , an index ifnich migtit readily be supposed to furnish the least variability. The result is brou^t about by the relatively large standard deviation of 1 (Table 18) and. the relatively low mean (Table ,1!f>, below). TABLE 19.- COEFFIGIMTS OF YARIATION FOE TI-E THREE SCALES Scale and Coefficients of fariation for aroup Score Attempts Rl^ts Wrong Accuracy Otis arade 7 Otis I Einh B Texiaan Grade 7 Terman I Hi,^ B Chicago G^rade 7 Chicago I Hig^i B 14.2 14.8 18.8 18.7 19.5 21.7 1C.2 9.5 13.8 11.5 11.7 10,1 14.3 14,3 16.6 17,4 14.6 18.8 36.2 45.6 45.5 47.7 43.0 48.1 9.86 1^.35 9.85 11.37 10,71 13.18 Iloan 17.8 11.1 16,0 44.3 10.89 The coefficient of variationj depending as it does upon the position of the distribution on the scale, is likely to give a very rnisleaxiing result for distributions such as these above , and should in general bo avoided for com- parison^ of tliis type. p. 42 Section 1"^ CaTiaQity of t-ie Indexes to Discriminate Detweon G-rou-',')S Table 180 gives the means on the tliree scales for Grrade 7 and for I Iligli B. It is at once eTident that the second group has the hidier mean for nearly all of the in- dexes. Accuracy J howeTorj appears to be nearly conaistent for all three scales and for both ^'oups. From the stand- point of discrimination, therefore, this index is of littlo Talue. The correlation ta,ble3 in tho Appendix show a con- siderable spread for Accuracy liiile the constants froin Ta- ble 17,18, and 19 indicate the extent of fiis variability. TABLE 20.. !IEMS FOP. THE TKREE B'lTELLIG-MCE SCALES Scale and ilGon?^ for Group Score Attempts Ei:9;rits "ITrong Accuracy Otis arado 7 Otis I High B Termezi Grade 7 Terman I Hidi B Chicago Grade 7 Chicago I Higli B 139.8 151.3 125.0 143.7 54.0 55.4 177.*" 189.3 138.8 159.5 52.8 57.2 139.6 152.0 111.4 128.3 42.1 43.7 37.2 37.3 37. 4 31.2 10.7 13.5 0.80 0.80 0.80 0.81 0.80 0.76 Individimls within a group, then, differ considerably in accuracy. When the above inter-group comparison is made however, Accuracy is found to be relatively constant. These results indicate tliat the groiiili curve for accuracy ordered relative to a^e ¥dll be relatively flat in com- parison with ordinary score. This iriiole matter ?/ill be p. 43 fully treated by the ^^.Titer in a forthcoming article on C3-rowth Curres under Different Modes of Indexing. In order to bring out such inter-group differences more clearly they are presented in full in Table 21. The quantity p.ir.a>>|-. denotes the inter-raean difference divided by the probable error of tliis difference ealeulated in the manner explained in preceding sections. Such a quotient gives a conTenient indez of discrimination. Indexes less than 2 or 3 sho?j that the discriminatiTe capacity of the test for such variables is not sigrdfican.t. T£BLE 21.- DISCRIMIIATIYE CAPACITY OF THE INDEXES AS SHOWN BY INTSH-r;ISAI\I DIFFERENCES IN GRADE 7 AND I HIGH B Scale Otis Terms2i Chican-o Inter-Mean Difference and Probable Errors for Score Ul 1 1 .P,E, IS. 7 2,7 3.2 Attenr^ts Diff.P.E 12.3 4.4 3.4 0.8 Diff ,P. ia.4 16.9 1.6 2.7 a. 6 0.9 Vi/irong Diff .P.i; 0.1 3.8 2.8 .n 1.8 0.7 Ac cur a c:^ Diff. 0,0 ^.01 -0.C4 0.01 0.014 Ave. D 3.7 4.3 t . ij Ttie five vsxiables in ordGr of their capacity are A,E,SsWj and R. Thus the groups studied show the greatest difference f with respect to speed sjid the least with respect to accura- cy. This result is quite in agreement with coiamon teaching experience. Pupils can be easily made to hurry, but it is exceedingly difficult to train them to be accurate. p. 44 Miilo A shows the best ca,pacity for inter-group discrimination, it is not superior to tho other vaxi- ables for differentiatiA^ indiYiduals. Score and Rigiits again appear to be superior to tho other indexes for individual discriioination, a property rfhidi is raore inrpor- tant than inter-group differentiation. Section 13 Practice Effect with Repetition Form B of the Termon Scale was given to .'rroup I Hi^i C and Form A of bhe ;iajis test given ui^j r oilowing day, Assuining that these tT70 forms are equally difficult a practice effect for each of the variables may be noted as in 1!able 22 • There is a positive difference between the means for each of the variables except f. !Eliis last nex^atlve difference also means an iraprovement on second trial, so that the practice effect is imieated on all of the variables. The last coluim shov/s the sif^ificance of this gain. Tlie indezes R,A.,S5 and R, reveal gains A tliat almost certainly cannot be accounted for hy chance fluctuations, rdiile tlie change in W is in hanriony with that of tho other variables. Errors orA Accuracy show changes of less significajice for practice effect. m TABLE EArlS FOR TSEiIAI^ F0KI3 A AND 3 I'lTH OliOOP I HIGH C ?ariabl9£ M A ( second) M e, ( first) Ma ~ Me | Dlff-vP.E.db.^ Score Attempts Hint's Iron?:; Accuracy 101,74+1.79 1^2.41+1.37 58.59+1.14 ^■.63+<^.^l 84.11+1.96 146.19+1.4r^ 87.15+1.56 61.f^4^1.14 0,59+^.^1 17.63+2.35 1.3.61+1.84 15.26+2.08 -.S. '45+1. 60 o.^'4+n,r>i 6.7 7*0 7,3 V'l.S 4.4 t* 45 Part II jjgalysis of tho Index Variables by ComDahgnt Tests ^e authors* plans of scoring giYen in Table 3 show that 9 of the tests niakinr- uv. the Otis Scale are scored by the f orimila S R. Thsse nine tests ?/erG therefore cho- sen for analytical stiidy. The 3tabilit3" of correlations for whole scales has been shomi in Part I. In the folloT/- ing sections the inter correlations of the coiirponent tests show a M:^].i degree of consistency. The coefficients in general are loF/er than for wliole scales but they indicate the sarsie relationships between index Tariables. It will also be shofai that pooling tests increases both tlie val- idity and the reliability of the indexes j an effect which may be roughly forecast by certain predictive fomialae. Section 14 I ntercorrelations o f Variables for the Otis Oonroonents The correlations between Index Variables for the saiae corrqionents axe given in Table 23. In the last line of the table the coefficients for ail nine tests pooled are given for comparison. p. 46 TABLE 23.- CORRELATIONS BETl^M INDEX VARIABLES ON NTNE OF TIE CO:iPO!JSITS OF THE OTIS SCAL3 Test G-rade 7 I Hi,^h A AH A>W R^W A-^E A>f E-.? 1 2 4 5 6 7 8 9 10 +.37 +.32 +.72 +.72 +.58 + .55 +.28 +.5G +.:m +.43 +.30 +.04 +.48 +.5" +.33 -.81 -.IG -.42 -.66 -.44 -.20 -.66 -.27 +.19 +.81 +.62 +.70 +.47 +.32 +.56 +.81 +.26 + .4;^ +.55 +.05 +.12 +.48 +.65 +.16 +.18 +.29 -.77 -.05 -.75 -.63 -.55 -.52 -.86 -.43 -.85 Mean +.61 +.56 — . -j.C + .5"^ +.u5 -.CO All +.75 +.21 -.51 +.5.5 +.21 -.70 It is GiridGnt that these are hi^^^ier than the meems of the nine correlations on conrponent tests except for Atterapts with Errors, in 7.hich case the pool H.Te-j the lower value* In certain cases, therefore, pooling or len^tliening the tests has the effect of increasinr^ the correlation be* tween the indexes. Tlie exception in this instaiice is worttiy of note as a warning against applyi-ng general rules for the correlation on lerif^iened tests. The hir^i de.c^ree of con- sistency in +;.- coefficients indicates tliat pooling of such components is a justifiable procedure inasmuch as the test material is fairly homogeneous for mirnoses of indexing. Inter correlations between Rlr^its on the nine conxoon- ©nt parts of the otis scale are given in Tables 24 and 25 for Grade 7 atid I Hi,^i A respectively. Both groups consis- ted of 50 pupils. All coefficients larger than three tiraes A p. 47 their nrobablo errors are printed in heavy type. TABLE 24. ~ GORKSUTIQ-LS BS^HEEl^-I RIGHTS ON TIIE HII^ OTIS CO:!P0!-n5ITS for GPJU)E 7 Tepyt 1 9 4 5 6 7 G J iO X +.54 +.40 +.47 +.50 +.34 +.33 +.50 +.34 2 +.54 +.^Y +.53 +.39 +.49 +.26 +.27 +.27 4 -^.4^ +.S7 +,35 +.43 +.37 +.4C +.38 +.19 5 +•47 +.53 +.35 +.51 +.3C' +.35 +.14 +.16 6 4.. 50 +.39 +.43 +.51 +•56 +.3^ +.36 +.15 7 +.34 +.49 +*.'^7 +*30 +.5G +.13 +.52 +.35 8 +.33 ■ +.r36 +.40 +.35 +.3:: +.13 +.19 +.29 9 +.5f^ +.S7 +.38 +.14 +.36 +.5S +.19 +.38 10 +.^4 +.?.7 1 +.19 +.18 +.15 +.35 +.29 +.33 Mean +.41 + »--)0 +.M +,35 + .4'^ +.'.y/ +.34 +.26 A siiUTile calculation will show that this includes all co- efficients numerically greater than .^7. In Table 24 only 8 of the 36 correlations are not sif^^iificant, Miile in Table 25 the sswne mmber occur. Most of tliese low eoeffi- cients @xo foimd in the correlations Mth test 10, the mean value for whidi is lower than for any other test. This one conr)onent then appears to be out of hsxinony with the rest; i.e. to fail to measure tlie same thing as the otl> er teats of the battery. Inspection of the Otis Scale shows that test 10 is for memory^ a trait quite different from those involved in the other components. Except for this one test a fair degree of consistency is found for the coeffi- cients in both tables. The means for all 36 coefficients in each table are +.33 and +.36 respectively. ■p. 48 TABT.K ^5»- COBREUTIOIIS Bm^'mm RIGHTS d^ THE mm OTIS C0ilP(3iWS FOR I HIO-I A Tost 1 1 1 *> z. p» 6 7 8 ^ iC X +.36 +.23 +.4"> +.5.^ +.38 +.5'' +.42 +.3^1 2 •f.ac +.41 +.36 +.42 +.a5 +.56 +.47 +.21 • 4 1 'SO +•41 +.S3 +.ri4 +.36 <¥^34 +.38 + » vJii< 5 ^'•45 +•36 +.23 +*52 +.38 +.41- +.17 +.33 6 +•59 +.49 +.24 +.52 +.4G +.5': +.45 +.17 7 +.3GI+.33 +.38 +•38 +.4^ +.4i +.:i6 +.02 8 +.3v +,5e +.34 +.44 •¥•0 J +.44 +.3i^ +a^ 9 ♦.4'! +.47 +.36 +.17 +.4D +.36 +.3^ +.13 10 +,34 +.r}i +.33 +.iV +.^2 +.i»J +.13 Mean +.41 +.4^ +.33 +.37 + .'-a4> +.35 +.4^^ +.35 +.21 Tables 26 axid "7 iIidw the correlations betueen errors for the GORFionent tossts. Only 14 of the 36 coefficients in Table 26 are sif^iificant, U%<3 mm for tlio nliol© table bein?? +.S4. T©ot 1^ shows uBxt to Uie lor:€3t averacje cor- relation TTlth tlie other teste, so tlio.t it is of little significance indozed by the ri^rhts or m-on^s. In Table 27 the coefficients are soaortiat hifliorj the iiiean of the iHhole tabl© boinc^ ^•S^* Eiriit of tbe 36 coiTelations are si^ificantj rdth fi¥e of the lowest values a-^^^^earing Tdth Test 1". It is difficult to explain the difference in correlation for the two P?"aup3 Mien indexed by W« Ta- bles rJ4 and 25 showed moan values nearly identical , but the difference bet?/een the riean coefficients for W is too larije to be ascribM to chance. One explanation of tliis difference may be found in tlie fact that arour> I Hir^ A made more errors than Grade 7 ( See Table ) * The effect of this was to f!;i70 less jaminr'^ in the contingency ta* bles with a resultant hif^ier correlation. p. 49 TABLE 26.- CORRELATIOMS BST^;.?EEN SFIROES ON THE NINE OTIS aO:;ffO !E^ITS FOR ; ]iUDE 7 Test ' i 2 4 5 6 j 7 8 9 10 1 •^.16 +.171 1-.^ + .11; +.29 +.3^"^ -.^1 +.15 2 +.16 +*24 +.15 +.18 +.35 +.16 +.17 +.13 4 +.17 +.24 +.3?. +.30|+.44- +.37i+.33 +.28 5 1 +.49 +.15 +.32 +.18^ +.46 +.35 +»09 +.19 6 1 +.11 +.18 +.30! +.10' +.40 +.24 +.18 +.22 7 i +.29 +.35 +.44 +.45. +.4^ +.43 +.21 +.23 8 i +.3O1 +.16 +.37 +.35 +.24: +.43 +.21 +.14 9 ^ ".n^ +.17 +.33 +,09 +.18i+.21 +.21 +.12 10 +.15; +.i;-i +.19 +.^': +.^3 +.14! +.12 Mean +.'51 +.19 +.31 +.^^8 +.23 +•35 +.27 +.16 +.18 TABLE CORREUTIOnS COIPONEfWS ITEEiy EimOES on THE NINE OTIS FOR I Hian A Test 1 ' 2 * 1 4 5 6 7 8 9 10 1 +.44 +.42 +.28 +.36 +.58 +.4i/ +.46 +.39 •J +.44 +.44 +.36 +.54 +.57 +.51 +.18 4 +.42 +•44 +.3^ +.37 +.54 +.5i +.39 +.22 5 +.2G +.36 +.301 +*4e +.52 +.46 + .12 +.29 6 +.38 +.37 : +.46 + .-4*0 +.49 +.26 +.16 7 + . .JO +.57 +.r34!+.52 +..13 +.54 +.50 +.31 8 +.49 +.36 +.51 +.46 +.49 +.54 +.21 +.24 9 +.46 +.51 +.39 +.12 +.26 +.5C +.11 +.10 10 +.39 +.18 +.22 +.29 +.16 +.31 +. 24 +.10 Mean 4.4:j +.42 +.4^ +.35 +.4^^ +.51 +.41 + .32 +.24 Conroarison of the four tables abore slio^s that the inter correlation of errors on the 9 component tests is about as hi.idi as for Blfrhtz. It may be noted also that none of the correlations are as hir^h as .6 uhile the means in all the tables are less than .4. Such coefficients are not considered hif^. The correlations corresponding for ■Biiole scales as f^iven in Table 6 are +.74 for Ri,'^its and +.69 for Errors. Clearly then the cwjalating of tests to p.50 form ?jhat haa hem called a scale score has the effect of raismo; the correlation or^ in other words, lerif^thening a test increases its reliability for these indexes. A more detailed discussion of this point rail ari-ear later. Correlations bet?/8en atterrr^ts on the nine corar)onents have been worked out for one group and are given in Table 28. Of the 36 coefficients, 27 are significant, the low- est averaf^e again occurring for Test i«^ with each of the others. By tliree modes of indexing, then, -this test shows up as distinct in type from the rest. The ;Tiean correlation for the whole table is -^.32 which may be coiTipared with the mean coefficient of ,56 in Table 6. Lengthening the test also increases correlation ',ihen Atteirrrts g^o employed as the index vazlable. The coinparison iis only a roug^ one, hoT^^ever,; for soriiewhat different scales and groups are em- ployed in the two cases. TABW. ^!8.- UORRELATIONS BST^EM ATTEMPTd 0>I THE Nimo OTIS COAlPOim^ITS FOR I : HlOIi , A Test 1 2 ■ 4 5 6 1 7 1 8 9 ir 1 +.41 +.^-e;+.r;8 +.35 +.4^ +.21 +.31 2 ■j-.S^' +.3^1 +.13 +.38 +.41 +.3y +.:^9 +.S6 4 +.4li+.3'l +.41 +.S9 -^-.29 +.32 +.35 +.20 5 +.48 +.13 +.41 +.31 -i-.^g +.34 +.37 +.18 6 7 -{-.ac +.3a +.?.9 +.31 +.52 +.27 +.4<^ +.26 +•35 +.41 +.29 +.52 +.37 +.22 +.33 8 +.4C +.39 +.32 +.34 +.27 +.37 +.40 +.26 9 +.21 +.29 +.35 +.37 +.4^^ +.32 +.4^ +.C8 i^ +.3i +.26 +.20 +.18 +.26 +.33 +.25 +.C8i Mean +.34 +.31 +.32 +.31 +.34 +.35 +.34 +.29 +.24 In Part I, Table 5, B and W shoiTOd a moan correla- tion of -.49 for 'fstiole sco^les. The oorrQ3;'-ondii%^ coef- fioiento for the Otis eonr'^^^^n--^ are giTcn in Table 29 with a nieaii of -•'^1. The Inoirease in correlation b^ pool- ing is a'?pin STidont. The oorrGlG,tions in tha principcil diagonal aro betweon Rights snd Errors on the sains test and are therefore lar.<^r than the rest, Uis mean bein^ -♦48, The re!TiainfIf?r of the table r^iverj correlations for all poseible co.nbinations of Hi^^its anci Wrongs on the nine tests two at a timo. All biit tliree of tlie 81 coefficients are ner^itiiro Miile just one third of tJiea axe si;<5.iif leant aoccrdin/^ to tiie usual rale. TABLE 2^.- UOBBEUTIOHS BIGHTS AHD SRBQHS ON THE DTI3 OG . . FQIi OIL^DS 7 Alffi I HIGH A Ei'-'hts Irongr^ •* ^ .■■, 4 ;) 6 7 8 J 10 ilean 1 -•8Jl — *t5 -.'M -.^4 —33 -.41 -..T/ -.rJ8 -.31 -.35 -.18 -.17 -.15 -,r -.r -.25 -.17 4 -.11 -.•l*^ — . *- -.11 -.^-0 1 r^ -.f^9 -.OS —13 5 -.^M- •.27 *.4^ -,66 -•44 ".29 -•Dl -.^7 -.la • .36 6 -.U -.^a -.^9 -.4'4 -.'^■^ -.S^ -.11 -.Id 7 -.36 -'.i^ (1 1 '■■J 'T r. . 8 -.22 -.42 -.17 -.Cc *.<^7 -/^6 -.28 'J .,<^4 -.C?^ ^."'^ — • -.1-^ -.^7 ^.n -.^y i^ 4..'^7 +.^4 -.^9 -.^7 -.If) -.17 -.13 -.08 -.54 -.13 !iean -.•:;) ' ft -•^1 •»•■■- -.23 -.19 — . :.0 -.14 -a^ -.21 ThiF5 au^T^Gnts tiir.t the criterion of ihroQ times the -irob* able error i"; too ntringsnt for tests of this kind. If tiTice th© rrobablo error viere edovted in Uw present case all ooofficients over.lO vjonld be sL^ificent including p. 52 five ninths of the total nui'nber, while all of those great- er than one probable error or over .l*^ ?;ill include 56 out of 81. For the last case 25 coefficients are less than one probable error, yet 22 of them are negative in sign. Thus a coefficient less than one probable error appears to give assurance of negative correlation beyond the expectation from the usual rule of even chance; i.e. the probability of significance from the data appears to be greater than by theory. In any case, highly consistent negative corre- lation is exhibited by the whole array. Section 15 Correlations of the Otis Components with Age The correlations of the age factor with each of the Otis corrrponents are similar to those for whole scales. Ta- ble 30 shows higlier correlations for the nine tests pooled than for the mean of the tests. Here again the effect of adding tests is to increase correlation. The formula for estimating the correlation of the pool of "n" tests with a criterion may be written in the form: Considering l,u^~ - r,o as Rights for Otis G-rade 7 and age as a criterion, the constants in Tables 24 and 30 give for this coefficient, p a value identical with that obtained by pooling the nine components. p. 63 TiJLE 3^,- COERSLATIOHS BmiEm ASE AM) THE imU YAHI- ABLSS ON TilS 0?I3 SCAIJi3 FQH QBME 7 Ai^D I IlIG J A .itt"!!..'!:.',, ;i! Grade 7 «_ TE » Jl, u,^...^ I liir ii A Teat Ajt50^A Af;e*R AgGM',' An;G>A Ap:BkR An;e. ' ••» Ji> •?-,lw 4., ^2 ^,29 -•»43 +»I6 -r.} •.•3i 4", 26 4 -3" -•24 -r^7 -• 3. J -..29 +•21 5 ■"•♦CfU -•42 "h.S3 -•3;^ -47 +.23 6 —23 •^•'^5 -•f^l -.6?! +.33 7 -..^u -^G 4»^i «•• ■ .;, ~.S6 +.19 8 -.^« —'^6 **^6 ••aii^ -.*4i +.33 9 i ■■-; -.::.: ^.«^ ••S3 -.-^14 +•24 1^ -I-.C3 "..35 —« •-»',; ■f*''"l •.04 Lfean -•?6 ^•'^a •••f2v> -.3-^. +.2^ All «»♦ 3'.^ +•07 -..j2 "f.34 Siriiilju" coefficients are piroxk in Table 31 • The differences between -predicted raid actual Talues are lii no case sif^iifi* cant* Tlie above forrmla, then, appears to be & useful on© in preuiotiiif, \ho Yaiidity (correlation mUi a criterion) of tests by poollrig oouiponeiits* ^^ :l-: U uo noted also that the formula will hsT© hi#i values for large values of J^* axid snF.ll values of Ax> • -y.e- mSLS i^l.- PBEDICTED AH) ACTUAL CORKEIATIOIIS BETWEEN AGE A!'^ I!'1DM YARIABIJ] (Jroup Variables Predicted Yalue Actual Value Grade 7 I Mr^a A Crada .7 I Mmi k 1 Higi A RiHits AfT.o Hl^iits Ar^o Iro'nrs A^ Wropif-n Age AttQ';?p^ts A-^e -•39 Ai "s +,--'i +.34 p. 64 To obtain a scale of hir^ Tf.lidity, therefore, ca^^onent tests should be f^elected T^hich haTs hif^i correlation id-th the oritarion but loi? corrslation a^nenrr thorasolves, Thom- dike Justified tlie use of tests Tdth 1o?j inter- correla- tions on tlie ,-nli--.?:tiDn of Hel iabilit:: roiTiulae to In the proeedinr^ section it Tms sho'sm iheX pooling coniponent tectr. has the effect of increasing; the validity of the tots-l Goale; i.e. to the extent to viiich it corre- lates Tith a eiiterion, Tho TK)olinp; of tests ?Jill next be shoim to have a siimla^' effect tt?ion ih€ roliability of a scale; i.e. the coixelation bstT/een te/o for^'os of tos same tost. , It irlll bo iecallec. f*b,ai tv'o foiri^- of tlie Terwn Scale viQTo {rlTGn to Group I El^i G on successiiTo days. IU»- liability' coofficionts have been calculatcxl for ©ach of the l^ co'rrir^onsnt tests and fox* all oombinod. to r;;ivo the total score. The roBuJis sxe given in Table 32. lieniKjirs of the national Acsslei^y of Seisnces^ Yoi.X?? p. 316 p» 55 TABLE as*- RELIABILITY COSFPICIISITS ?0H TIIS TERLIAM SCALE BY C ; Aim TOTAL SGOBB f«Bt Porriala Correlation bet?feen Rank """"'"■' '" For!;is A and B 1 11 +•630 7 z m ^*Q^9 3 3 II-W •f.G32 6 4 B W-J^^ 1 5 aa +*a52 S 6 H-W +•482 iO 7 R ^••683 3 8 R-f +.53^ 8 9 H +.5X4 9 10 :":R 4-. 7^2 4 Memi 4.. 679 All ^•9ir Certain of the indlTidual tents reveal a hi^p. decree of reliability» eopocially test 4 (lor;icrl selection) with a coefficient of .u'-^. It will also be observed *toat soae of the loT78st corroletions occur ?/itti tests scored R-?/» This point will be dealt TTith inoro folly in a follomrig section. ThQ moan of the reliability coofficionts on the tests is 'i-#67^, while ttie oorrole.tioa for total Score on the t~ra forms is -f.^lC, so timt pooiincj tho tests has tiie effect of increasim reliability* A •nrediotive foi'miila givon by Bro?aa, am also implied " b in Spear>;Kiri's G-eneral Tlieoreta may bo given in the foiia, irtierc n.. iz the correlation betwoon t7;o tsr;ts or the avor- a. fillia-.! Broimj, Essentials of Ilental 'leasurement, 6am- bridn-e diversity Preso, London, 1911 b. C. Spearman, Correlation of Sims m&. Bifforancon, Brit, Jour, of Psy*j ?ol-5, ,419«-^i36 p. 66 age of several J and N tho niinber of tests thus amalgama- ted. Ill tii8 present exa^imle the average correlation frorn the firat three tests is, IJ,, In order to predict the re- liability coefficient for 10 such tests, it is only neces- sary to substitute these values in the above formula giv- The value from actual amalganiation is .92. Simlarlyj a calculation based upon the average of all i" tests also gives The use of the forniula in these cases, tJaen, gives con- siderable over prediction. In order to test the applicability of Broivn's For- mula more fully and to analyze more fjilly the effect of pooling tests on reliability^ a inore detailed procedure is next eraployed. Reliability coefficients on emulated tests -are obtained in t?;o ways: Ta.Q scores for tests 1 and 2 on each forra of the Ternan Scale are ad.ded, and. the correlation determined; next tests 1,2, and 3 are pooled and the t'TO forms correlated, and so on until all 1^ tests have been cuimlated in tiiis fashion. The ^^econd pro- cedure is to befjin mth tests 1^ and 9 and araalgamate in the reverse d-iroction. Tliese em>3irical results are then coiirpared mth theoretical values obtained by substitut- in,r^ r ^.68 and N from 1 to 1^ in Brorai's Formula. Table 35 gives the results of this lengthy calculation. p. 57 TABLE ..^.-.TimOKETICAL A!1D ACTUAL HELIABILITY COEFFIGIMTS OBTAIIM) FRO;.! BROM'S PQK'UHJl Ml) BY SUCCESSIVE GHULATION OF THi;: Tm TEiriAIJ COlIPO^^EnTS Number of Teots Cianuilated Theorotical Value Order of CirJjilation I to i^ 10 to i I Z 3 4 5 6 7 8 9 +,68 +#62 +•37 +•9^ +•93 +.i)4 +.94 +»*?5 +•96 +•^4 +•81 +.87 +.i?'^ +.80 +.03 +»i3y +.9X +.9^ +.70 +♦79 +.83 +•86 +.64 +.86 +.07 +•07 +«r +.9a Inspection of the table shows a roiii^i agroemmt in the tliroe series, Pirpro 5 iMch ii3 based on -tho table brings out tli© oor.imrxsons niore olssrly. Tlie thr©© curves show a I'apid initial ris© wp to 'i Ctifua-c-.tsd tests and then a moro .Tpracluai incrsaso to ^i© nnxintuLi valu©. The more ra^ pid rise, of the cuxie cumlated from testr. I to 1^ t^on the i-everso one, is.m doubt due to t^w ^'j-;-lei reliabil- ity of tlie first fer- tests as indicated in Table 33* Willie tliG ^onorr.l s.'^G^-ient in thb tuiree cunres is videiit 9 no verthel c , : :.: \..l'.,^^v.i. O i 3 & very gIoot tcaridcmoy for the thoorstical cunre calculated for i ~ »C>Q to give an over T'.rediction beyond 4 or 5 cw^iilated tosts. This laay b® partly duo. to the unoqijai value>s of UiQ individual reliability coefficients given in Table 32, but itiatever the oaas©9 the use of Broim's Fortmila for prediction in a case of this kind is open to question* The equation re jo^ju^^^ ' ^ 5"7 p. 58 and corresponding theoretical curve indica-te that to get any desired degree of reliability with +1.00 as an upper llm3.t. it is only necessary to amalgaiiiate tests indefin- itely. Tliiro is, of course, absurd. The fortniia giTes an over prediction fairly esxly in the series of cuinulated tests. From the afooYe tables it appears that foui' or five bjrpical tests of the battery will ^ive almost as reliable an index as the pool of all ton components. This result would accoijnt in part for the hir;.i reliability of such tests as tiia Chicago Scale Gonsistin/;^ of only fiTe components. Txiis problem is one of great i.u-.ortoiioe in test con- struction. If intelligence can be indexed with almost as great accuracy by a short scale as by one twice as long, the savinp; in time alone is enonnous. Moreover if the short series can be shofm to be as valid as the longer one by correlation ?dth criteria^ the abbreviated method is fortiisr Justified. Inasnmch as no suitable criterion other than age was available for the present data, the check camiot bo rio^idly applied. Tlie p,r5 correlations by half and by 77holo scales, however, agree almost exactly (-.37, -.39), so that with age as a criterion, the five test "battery is as valuable as tb.e ten test scale. p. 59 Section 1? Summary of Malysio of Condon ents The relationships found betiYeen index Yariables in Part I arc yerified for component tests. These coefficients are in general lower than by ?/hole tests ^ so that pooling has the effect of increasing the eorrslation between indexes. Inter-eorrelations betT^een components for R,f , and A reveal a hi^cph degree of consistency for such short tests but are less stable than for similar coefficients by whole tests. Furthermore the consistent batteries of correlations qybxi for R and 1 on different tests indicate a Mgii degree of homo* geneity in the tost material iltli the possible exception of Test 1^. In addition to raisinf;^ the correlation between index variables J pooling tests also has the general effect of in- creasing^ the validity and reliability of a scale within cer- tain lirdtB. Predictive formulae are useful in this connection but are likely to give an over-estiinate of the correlation to be expected by pooling. Moreover the physical endursnos of the children deterrriinep. the ma-ximora len^h of the tests at a sitting, s!o that the formulae are limited in application. The gain in validity and reliability is rapid on pooling the first few tests, but the point is soon reachefl i^tiere the ad« dition of similar material affects the correlations but sligiit- ly. The roBultn indicate that a battery of foiir or five care- fully selected conrjoonents tall give an inde:: mth substantial- ly the same reliability as a scale tmce that length. p. 6C Part III Scoring; Formulae Section IS The Linear Forra ^ S^a. ( _Jjt K' ) a. Formalae y/ith Kirdi est Validity In Parts I and II it has been Bhoim that the scoring fornmlae employed by the authors of the scales have little effect ir:ion the resultant scores -Khen a nuraber of coiiroo- nents are pooled. The Terman Scale T;ith the coinioonents scored by the tliree formulae , S-E, S^2R, aad 3-R-Wh« a corre- lation of ■^,J& with the score obtained by using 3-H on all ten cofiiponents (Tables 4 and 5). The amalgamated score then, is not Tery sensitive to such chan.'^Gs in the comrionent scor- ing f orralae and simple fonns are recorniiiended on these groujids. The 3in;<2le conrponent, however, is rduch more violent- ly affected by chan-es in the formulae enwloyed to index it . Chan,Q;es in iveigiits ?;hich affect the pooled score but slight- ly, id.ll be found to have a pronounced effect upon the in- dividttal cormonents. Table 4 which gives the various co.nnonent scoring for- roulae used by the authors of the scales , includes only for- mulae of the linear type; i.e. equations of the first de- gree in the variables employed. Tnese variables are K and W, so that the most general formula used may be T?ritten, (1) S ^ o-^R + ^w) = o^R.-t-c\N" ■a^ere a, b, and c, are constants. It Tvas also noted in Section 3 that tlie relatiomship, A-^'^ + vT p. 61 makes it ^-ossible to e3r-,ress this formula in terras of R and A or W and A. For.Tiala (1) , ho^^ever, has been so gen- erally erii^loyed that it r/ill bo ado'oted here for further analysis. Foriiiuilae expressed in terms of the other vari- ables may be obtained by substitution if they are required, The question irn^^iediately arises as to the best val- ues to assi,Qyi the constants a and c in equationl. A gen- eral solution of this problem raay be obtained by the rnetk- od of least squares. Values for R and ?; rro obtained for each of the N individuals of a given population. Assuming that a criterion, K, is the best measure of such determin- ation^, " set of M equations may be for.ned, K , r= CX, R,-+ C,\J\)| K N - uLn Rm + Cm \1\Jh T/here the K's, R's, and W's are knom, end a's and c's are to be determined so as to minimize tliG inconsistency in the equations T^T^aich is asswned to be due to iniperfect measui'ement* Nezt Vi.V^^ V,., -ill be la^ittsn for the differ- ences bet7?een K.^K. -- K ^ and the values obtained from the best determinations for the a's and c's; These differences or "residuals'^ are assumed, to be p. 6Z nonnally dir-tributsd. YMIg the assui-option is open to question for data of this type, it is nevertheless thw best that can bo made. The most ^.rote.blo Yalues for a and c next require that the sum Tlie re.iainder of the procedui'e consists in setting up the "nornal equations" in the usual rray. Trans f err in^^ the variables to their respective means, and setting up these Gquations givec, Since, ^n - — ^ , and ;c>^- 77^^^ ? these equations may be m"itten in the form, a- H^vij S;;: -*- C v^oo - A Kvxr <^ Solving these equations for a and c rj;XTQS, (31 o~ = .1? ^, (UKvj[\a\Al -/vkyvj <-£P ^-' (AVv^-0 The value c rnay then be written a This last result has been obtained by Thurstone as a aT'L.L. Thurstono, A Scoring' ilethod for~liat?J Tests, Psy. Bull., vol. }:YI, 110.7, July, i0l9. p. 63 value for C in the for.-.iil^ S^R+C^^' such that the correla- tion Aks io :■ jaajcinuii? i.e, C or c is determined, in a such a Tjay as to r^ossosn the hif^ie.^t rnlidity Tdth a cri- terion. The foriiEila for thifj coiroli-li-ju is, a Thurstone also niakes use of Yulo's ©quation f or l-u1» tinle correlation to obtain an ezpreosion for the hi^thsst correl^.tion rxth tlic linenr fon-rala S = B -^ Cf/* This re- sult nr.y bs -.^xittGn [ .R'^ha -^ Akvm - 2 A-kw Rica jlyvv^f (7) RkCuJxi-cw) =\J i —^ Ar\wr Thn -^.ctual procGciiiTG inYolvecI in detcrmininft the con- stants a end e rail then be as foiiov;ss X. GlTe the tost to a p^om md obtain alr.o the criterion of validity ri.rf8j.n3t which the formla is to be checkeri* 2« Score the test for R anc' '^ •'^'^<-'' no"r-n.t~ the constants RkA, Akvm, n^v^, c^r^^ a>--i G^vw ; ^K la iioii required if For:uala (5) is arj:ployed). 3. Substitute these l^.st resiiltr? in equations (3)^(4)5 and (5) and obtain the forfiiolae Sc cx^l+c vjO ^ S^R+%:V\r ( dif faring only by factor of pro-nortionr.lityj L). 4. To riralict the hi^ihest eoiTrelc'.tlon obtriinable ivith these foroulac, substitute the conr'Uted conPtsjits in equa- tion sO- a. a,U.Yu}.G, Introduction to Statistics, C»G-riffin,. London 1919, "48. p. 64 b» Limitations in the Use of the Forriiala S -(x R-t-eVAT. In section 3, tv'o "?lans for ^'•''/^linisterinr; tests i^ere described. Accordin,f^ to the first, the ti:ae is fixed and Att0;:-!pts, Ri^^Jits, and Wronp,s allowed to vary; accord- inp;to the second, the mmber of Attev^^tc iz fixed, while Time, Fiip-hts, and Wongs axe recorded. T?:o of the index variables are thus alternately controlled by the method of adziiinisterin^, the test. The fonnula S- c^R-*- e\j\/ ^ ^--ith constants determined by hiHiest validity with the criterion, serves very well for tests given according to the plan of fixing the time. All of the tests eiTiployed in the tlii-ee intelligence scales are of this t^^nie and hencs no difficulty is enco^jntered. For tests adrainistered by fixing the Attempts, how- ever, the above fornola is inadequate. This arises from the fact th;;<.b S is no longer a firf-cti on of t?/o indepen- dent vaiuablGSj but of only one. If Atbeiiipts are constant and A-'^ir^ , then \/VJ - - R -t cH where d is ^constant. SubstitutiH;?: the value for \V in the eiqpression S^^P-+cW gives S^o^F-.t-c(d-f^') or S- Ca-^:)R -b cj^, Thus the seorin^^ fonuala is indeponcent of , and no fnatter what value is assigned to c (with the exception of c^o- , for which value tlie correlation is zero) the correlation /'-i^-s given by fornula (6) is equalfe/lkA. This last result fol- lows fro:n the fact that /Uco^^i^j- fU^ where a and b are constants (Theorem 1, Appendix) p. 65 In other words if Attempts are constant, the soor- iiiirr foruiala S- ol R has the same validity with a criter- ion as aiiy linear fujiction of ^ and V\^ (?rith the ezeep- tion RrW for ^hieh Hks^o ), A further liinitation in the use of the formula S- Ri-CW lies in its sensitiTeness. The Yalue of G as determined by Fornula id) de-oends u-oon -^)^v\r '-OR *0 \N '^- 1 (^.R R^VvT (J^/VaT 1 «.38i -•8^3 ■^a?.6 2.6^1 ?.,r^H -.r;7 -.767 +.^17 Z •».176 ^•183 S.S13 1.794 ,1 " •• -.3<^6 -.047 +.m2 A •X •♦•!..>■,.' i -•^66 ,1 1 0'-> •■'!'•• v/i/'r. -.752 4-.S12 5 -.417 -.66,^ ^.2AB 3.47V 4^.16^ -.468 -.627 -►•:530 '"•'^WC •?-.v-4.-ir 2.^Y6 •f^.o ^* -.052 +.3'"^ 7 -•261 «.r:76 •^.214 2»758 S.C59 -.6S4 -.^■56 -.516 +.169 G •* ♦ J "^ */ -•GVD -^.'^V^^ .'■i.a^^r r;,5j^7 •f .4I-1> -.4^0 -.857 *.3r:7 % ^» Xv^.^> -.r;74 +.rrc 5.116 .';.4r'7 +.5V0 — •'^i^iG -.4oy •f.S44 X'> ^.f^3C -.540 -.M7 s.9as g*5V9 1.376^ >.0C6 -.t^l'-.04^ !i^SLE 3^. . COI:TI Test I Kir-i A C5^ kf^vfj Cz. C, " Ct^ 1 i.^ie ^.le^ -J-C.C5': ^- ■:- p; 2 n.0^i4. -1.''69 ^.875 4 '^. 665 3.665 .r.C32 5 3.372 Z.ATi *f^-.272 -.112 6 %514 ^.^mc -^.l^^j ^,^^'-^ 7 2,871 3.57?? *^.267 —347 8 3.83^ ';.655 ^.im +.2:?5 9 4.593 2.715 -C'.P.VC- ^#948 1^ 3.^:36. 3.f>79 ^1.^36 •i-1.14^ c. Tho Use of the For:isla S - (R t C ^ In tsst roaterial ^^mrQ t-vro altematiYos are given for each item aiid c^aessiiK tlierefore posaibl© the forrnola S- (R-W (i^.«X--~k) has been frequently adopted. Simil- ar f or:milae are used -mmi tli© nmmQT of choices is greater. These O2:prop.3iom5 are asBmiod to oorrect for guessing ele- ment Ira'-ol^od. According to tlio r.boro or;ur.tion a person f^essiiir; blir/dy on all of the itoms will get Ii^^-lf of thea ri^t by ch^^Gej antl hence & zero score v/hieh he deserves* imla penalizes justly; but it also penalizes for errors which are not due to guessiix^^, and hence unjustly. As ii result, for very difficult material, nearly all of the scores may be ne.niative. A noj'iif^oi^ of oxperimental atteiiiiybs haye been imde to deteriiiine the amount of fj^uessing in tests of this sort. After admni staring a set of Trae-False tests of the syl- logystic reasoning type , the v^iter asked the pupils on which items they had guessed. The number guessed ^vas about %-^jo per cent of tho total number of errors :aade, and of those items guessed only 8^ per cent T-ere LTong. For such groups and tests, the penalty attached to errors by the formula R-Wis enoniiously too rreat. It may be noted, how~ eyer, that the children did not always Iqiow whether or not they had j^uessed on any item. Reasonin^^ and guessing are often iiidistinguishable, and who has not credited himself mth reasonoiig when he has only made a lucky guess. Instead of assurninp* that a penalty should be attached for guessing, Thurstone proposes to use the forinula with the value of C to be deteiinined according to validity as above. This method at-ynears to be ^referable to that of a priori ^.a:ror.aixic.vio-i, wi-C'i os::- . _. . ministered wita the tine fixed. If Attoinpts are fixed » however, the foraa- la beco.aes independent of 1 as has just been shoivn^ and all values of ,C (ez-^cept G~+i) givea the same correlation p. 68 This point is of great practical irrrr^ortance because most tests of the True-False type are administered so that all imy finish; i.e. Atterjpts constant. According to the above repjultsj, all fomiala,e of the type"^-^-:^ vv/ give the same correlation with a criterion as is obtained by using Eiglit'i alone? i.e. S^J^ , Inasrmch as the expression R-kW does not correct adequately for gu^ssingj and has the same validity as S ^ ^ for Attempt,^ constant, the loiter be- lieves it should be abandoned in favor of the simpler form. The following exairrnle illustrates the foregoing dis- cussion. < K \N i^-W K /^ VA/ A-vaA 1^ 2 8 -6 -.?c 9 **.'-j 2 -4 20 2 8 -6 ^ -4 3^ 5 5 C 1 -a i 4^ 4 6 -2 1^ 5<^ 7 3 4 on .^ -3 6 ^5S ::io 3C 4 6 -2 ^" (^~^^^- ^ l<-(A->^) 4^0 4 16 4<^ 8C IOC 4 16 '}■'* 4^ 1 4 4 « ft o 1 nr f*. J.-- ■ • • « • 400 9 36 er^ 120 7^ -2^ HT^ VK-, \JlS<^^ ^ I -T-t^o ^_ AHa~w)-= ^p^^^;^;^ - »r5 p. 69 Section 1;? 31.:rile Is^tlos ■?.. The Correl'^.tion Bet?;een . Sr>eM «'^'^^- Accm^aoy WW*a— M— >— ilMI W iiB n i l ■ U MlJlll llilia m iilii ■ » i ■ > ' il» IM| IW H lllll ln a Tli8 ror-Tolae S^- -% ond "S - ^ "r.;; ')c conveniently ©riTployoc; to irdex Speod aad Accuracy respsctiTely, The latter form has been xised Girtenclirely in the first two parts of tliis study and haa been fouBi' .■ave Taluable properties Tjfiich other inde:xor; do not ^^oasess. Witii Time constimt r.s in the intellirenc© scalGB diacusoed, the for- uaia Ijr ^poed roaaoeg to o c c A ; i.e. tiie Atte-iipts give the moasure of Speed directly T;7h0n Ti'ia is fi3ced. fhen AtterT/tf. ^xq constant, the Speed ie r^iYsn by the re- ciproail of the Tinie (in suitable imits). Tlio fTtenoral expression for tlie correlation between two ratios -^ and ^ may be Ts/ritten in the fonn "^ore the Vs are coefficlGnts of variation ^J.YQn by the for^rala V - '-^^7^' (Theore-^ " Appendix). For t!i9 two ra« tios -^ and -^ this expression becoaes For T- const^iiiij Vr - constant, and all correlations vdth T are zero, therefore, which QXfression reduces to p, 70 Equation (9) thus p;iveg tlie correlation between Speed and Accuracy for tests T*iere Time is fixed. The Ila- xitnajn Yalue, or AA|-+i-(Joi3 giiren for TIar^+i- . Thus if the pupils get every iDrobleiTi that they attempt right, Speed and Accuracy mil be r^erfectly co2Tel€i,tGd. For zero corre- lations between Atteraptrj and Hic^itSj Speed and Accuracy are nG.p^-atiirely correlated, the value ap;":roxlmatins - -5; • Finally, if the ratio ^ is equal to th.e value of/lAR , the correlation between Speed and Accuracy will be zero. Tables 5 and 19 indicate that these l?.st relationships will hold very closely for intelli/^^ence test data. 'Tlic ratios of the Vfi and the corresponding correlations are approximately .7 For.iTdla (9) is very uaeful for obtaining the correla- tion betv/een Speed and Accuracy from the Dingle correlation table for Attempts and Eights. This tabic gives the values • A^a,^^.^5a, Mr^ca.,^^ (Ma i7hich are all that are required to obtain H-aJ . The correlationsiTith Accuracy in Part I were co.TTiuted froiii ratios -^ obtained for each variate by division. About a third of the coefficients were then check- ed by the above foriTiula. Substitution in equation (9) re- quires but a few tiioments, liiile the division for ratios a- lone takes about an hour for 5^ esses. A great serving of time is ^ therefore, effected by the use of the above for- mula, especially if the constants in the foriTula axe needed for other purposes. If Atterupts are fixed, Speed is raeasui-ed by the recip- p. 71 rocal of the Time, and Accijiracy by Rights. Equation (8) then, reduces to For reasonong test material of the Trae-False type, ad- ministered mth Attempts constsnt^ Iott ne,?:r:,tiYe correlations were olDtainod for Ri?3;ht3 and Time, indicating that the cor- relation between Speed and Accuracy given by the last for- mula is positive f-nd low. For 15 .groups of about Z^ impils each, the aTerage correlation jIar,- .:i^± ,o5 . For both I A t^poen of test achiini strati on, then, Speev.1 and Accuracy ex- hibit correlation that is zero or ba^rely large enough to be sifmif leant. \.y b. Thg Yalidity of Sin?:le Ratios a?. Scoring: Indexes It has Just heen shoim that the ratios giTing Speed and Accuracy are relatively independent measures of intel- ligence. The choice of the proper index mil therefor de- pend upon criteria such as the purpose for ffliich the meas- urements isjere mrde, the Taliditv of the index, and its re- •■ft' 7 liability. The question of validity will be taken up first. The use of certain linear foruralae has been justified on the basis of their validity or correlation witli a cri- terion. Tliis same ^-^rinciple may be applied to ratios. If a criterion J K, be substituted in Forr.iula (8) in place of -^ an exprension for the validity of -^ may be i^^ritten in the form (1^) Hk^ ^ /lK>cVx-/lKxyY \]v^2JLnVxV^,^VY'- !Hie correlation tables for H-k^ and. ni -, T c> The HeliaMlity of 3 i;!rlo ....:. vi;..- .w 3oorinp: Indexes Equation ;3) is arsln useful in --irocliGtin,^ the reli- ability of p. ro;tio mtliout direct calculation on the ra- tios ths:nse?.ires. If ^' pud ^ denote tiie ratios in qucotion on succesoiTe trials of -S^lie nar,i© test or b;: pai-allei forros, the reliability forrmlc, nk^y be ^Trltton in the forn In order to ealciuj^te thin qiientity, foiu" correlation ta- bles are rsquireds X/i., X,Y^^ X,X, ^ o— ^ Xi.Ya_ p. 73 If thoy cx^: -^rf-j-naxed all at once, the rn.'^.r.n'i n?>l frequen- cies give ezceilent checks on the distributions. As in th© case of validity if J^if^u is significantly greater than Aa,>^ 0^ /l^,x^^ , it is to be preferred, as the more reliable index. This method was applied to the Terrnan Scale Forms A and B. The results of the direct cal- culation appea.r in Table 7. Predicbion oj Fooiiala (li) Accuracy end Rif!;hts are si?«iificently i;aore reliable tlian Attempts, but are not essentially different from one another. Tlie intelligence quotient, and similar ratios, are es- sentially a score divided by a chi'onolo-^ieal age, fihen the score is expressed in af^e units and taken from a suitable or- igin. The choiee of the age unit is important cliiefly because the resulting; ratio is then a pure number end easily inter- preted. As fai' as th.e validity aiid reliability of the ratio are concemiKij the dioice of the unit for score is of no con- sequence , inasmuch as correlation is a measure independent of the units eiiiployed for the two variates. The origin from Tufeich the scoro is taken is alway" of importance since any shift obviously changes the ratio e..g_ Furthomioro if j (s.o^ - jU^^e^-p- tiiis expres^3ion re- duces to Forriula (13) ?;ill have the value +1**^^ for fUs^tMroj and will be zero for I -t-KsvSi, ^sJU-^^^^^ , For a civen positive value of AsvSx_ , t^e reliability ,^iven by the fonmla rail increase as A^-^^-s^ decreases fro^ii the val- ue ^ — • As an illustration, let As.s^ -+. ^ . Tho funotion [1 s, s^ jmy then be t^ibled for the ar^pnent A' folloiTSS Ih 1,0 ,i vr'^^ ^ •i-,75 +.83 +.88 +.9 +.91 +.93 +.94 +.94 +.95 A*^g-- ,s ■ t +.9 .T +.8 +.6 +,4 ^ +.2 . o.n • -.2 .:^ -.4 ^i p. 75 -.6 -.8 ■' ' -.9 ^ ^^__^_ Figure 6 T7hich is based, on the table sIiotts that the high- est reliability of the ratio -51^ ocouxs with the hi^- est ne'-'-ative Talues of /L^-*^ and decreases as /*-s^>-^ increases (to the right). The value for 7Mch j W Ji^ -^^ .^ is found by solvinr^ the equation giving x^ . 5 5 so that the ratio has a greater reliabil- ity than RsiSa, up to the value A^^^ - . 5 , Inas- much as .3 is a good reliability coefficient for score, and fls-*^.^ is seldom as high as +.5, the suppostitious exai'Tple above indicates that ratios mth age have in gen- eral a greater reliability than that between crude scores. The ne,«;ative correlation usually found betn^een score and age for a given grade group (See Tables 8 and 9) also in- dicates that ratios of the above tjoQ are most reliable ?7hen the group is thus selected T^ith respect to age. A final exarrrle will be given to illustrate the effect of transferring the origin to eliminate the . Assuming the apr,roxiraate values \lf\- ^^ -o^, / (ar - . ^ ^\) -o-^ ^ ? c3u^ fli-o^^-.v , formula (11) ?/ill give by sirnple substitu- tion /X,*o The raiorsl -oropertios of the ratio as compjared with other Tarisbles e.g. the intellir^ence quotiorit facilitates coniparisons with norrml adiieireiaGnts* 3. The validity of tlie ratio to bo detesniiined by. Foniiiila 4. The reliability of tho ratio as deter-iined by formula (11). Sectio:; 2^ Ceiioral Conclu sions 1» The various types of response to test material have been treated as index variables for the traits in question* An analysis of theoe variables for intelli'-enoe tost data p. 77 reTsaled fairly definite relationships bet??een them as indicated by the coefficient of correlation. S. By eliminating^ the difficulty factor, the -primary index Yariables were reduced to AjH,! , and T, one of these oQin^ fixed by the method of test adniini strati on. 3. in analysis of whole scales indicated that all of the primary variables have Yalua.ble prooerties as indexes. The introduction of the sir.orple ratio ms,de possible a com- parison of the indexes revealing; them in order of general reliability as SjR,RjW, and Aj mth Time fixed. I 4. According to the criteria employed, the co'T-licated formulae used by the authors of the tests, are not signi- ficantly better than Rights alone. Acc-uracy comes next as a 5;enerally reliable index, and has -oro":erties ?;hich the other variables do not possess. For batteries of tests, then, scorinr!; by Rights alone is justified by reason of greater simplicity and practically equal reliability as cojiroared with more com"lieated formulae. 5. In discriminative capacity, Attem^pts proved to be highest and Acciu-acy lowest; i.e. individuals and groups differ more widely in Speed than in Accuracy. Lack of discrimination between groups is of less consequence in en index thsii failure to differentiate bcti^een individu- als. Accuracy, therefore, retains its high place as an p. 78 index regardless of the sli/rht inter-group differences shoim. 6. Analysis of the scales by conrponent tests furnished a check upon the results obtained for whole batteries. The coefficients in p-eneral are lower and less consistent than by whole scales. 7. The validity and reliability of tests are increased by pooling coiTtponents, Estiimtions of these correlations are furnished by certain predictive formulae , which in p;eneral, tend to give over estimation very early in the emulated series. 8. Both theoretical prediction snd actual results in- dicate that pooling tests soon ceases to increase valid- ity and reliability materially. A battery of five well- selected tests is about as satisfactory as one tv.dce as long. 9. The formula S-&-R+c:\>7 i-^^c^ haen slioYin to be the most general linear form. Foniiulae for validity have been worked out by the method of least squares. 1^. The linear forTiula above is open to - question because of its sensitiveness for values of C. This implies that a new deteiraination is necessary for each neis; group dealt i^rith. 11. The gcorin<7 forraila^-v^/ is criticized becauso of its p. Id failuTQ to correct for r^eosinf^. It has boon shoim that for Attempts constant (the asual method ?rl th True-False tests) corrective formilae of the tyr-e R-^W have ex- actly tli8 saiae validity as Hirjits alone, 12, Special f orTulae and niothods have boen r/orked out for detcrmininr^ the validity and reliubiliby of siraplo ratios, i3. The valiiahle propertien; possessed by siraple ratios indicate that they are hi^dily dosirablo and useful scoriiv^ devices in nite of the labor of division. Special tables give such quotionts directly. 14« More conTolicated for-mlae ha,ve not been dealt v?ith be- 'u cause 1J,i0 labor involved in tlieir use would be m-ohibitve no '-T^'/'^or iTiiat virtues they r-iir^h'^ '}^' ^:-nd to -obsess* The sensitivoness of such forirulae is additional reason for a- voiding them. 15, The results as a whole rioint to the concluaion that for batteries, and for sinf?le tents as well, tiie snost de- sirable and useful indexes are K and B, I Bibliography Memoirs of the National Acadeiny of Sciences, Yol. X?, Psycholop;ical Testing in the United States Array. CSpoannan, Correlation of Sum-'? and Differences, British Journal of Psyoholoi^, Vol. ?, vv* 41-^-^1^6. A.S.Otioj An Absolute Point Scale for the Group Measure- ment of Intelligence. Journal of Educational Psychology'', Yol. IX, Nos.S-and 6, Hay-Jane, 1918 L.L.ThuTGtone, A Scoring Uethod for Mental Tests, Psy. Bull., Yol.XYI, No. 7, July i9i9. ^ T.L.Kelley, The Reliability of Test Scorer. Jl.Ed.Research Yol. Ill, IIo.6,p.379,May 1921 A.S.Otis GJid H.E.l^ollin, The Reliability of the Binet Scale HY^ p6da{50gioal Scalec, Jl.Sd. Hptsoarch., Yol. IY» No. 2 121, Sept. I9n • Biblior^&r^y — Terts ^MJ.'oil • ■ , ■'• " .r ^ ;tatisticB,. Chaxlss Griffin and Comnany, London, 'Ui-J* Sons, Hew York, 1920 D.G.JoneSj A First Course in Steti sties, a.Bell imd Sons London. 1921 W.Bro"^ piv! C.H.ThoriTOBon, 'Hie Ks^^entials of Mental Aleas'xro.teuu, GaTibrid^^e univeraioy Pre;^;^, London, i92i APPENDII A Correlation Tables for Reliability Coefficients COHRELATIOI'I TA3LS FOR SCORE ON TEB2IAII FC: TEE!.li^J FOEl 3 VflTH aEOUP I HIC!!! C "./ITil SCORS ON SCO Oil TEmiM FOK^ B ( FIRST TRIAL) ?j <5- co &- ::}- 1 <5~ 0-. o 1 C^ fvl CO 1 T <>- 3-^ 0^ ) 5 H 1 ; en 1 o In o o 1 o 6o o 1 o 5 £2 o O I-' tS a ^or..2?^9 . • • • O • « e « « » « 9 » 9 9*9 9 » 9 « 1 > 9 9 tt e> 9 ft « 9 • e • « • • ...1. a.l . aa. 1 ) lo -IGv:' • • to « 9 « a • • 9 9 9 9 ft 9 » 9 ft 1 O 9 ft 9 ft » b • » O > • • {a s • a u • a a • 'ij 170-17- . « • « • • « 9 9 9 O » • f> • 49 t 9 « ) 9 9 1 • • • 9 9 I . a ; a - • a • (/J 16^-16;? . « a « « • « • » o O 9 • 9 9 n 9 Q 1 9 9 ft B e b • • t • « .i. kl* ..^.. ..1 « « .1. ,2. o » >'U 9 .1. . . !> a ' a. 6 14^-149 . • « • « 6 « ft • « 9ft* 9 9 9 .2. 3. ' ft 9 > » . » 'J ♦ a. » a • a ( ..8 S. 13^-130 . O <» • 9 « O • ft 9 9 9 .1. kl. .1. .3. .2. 1 « • I , . 0,8 l?^-!-^.^ , .1. • <* « 9a 1 .3. 1 .i 9 » 9 e .1, 1 9 9 > • i> • 1 .10 tL li^-l.U . • Ck A .1. ft-'^. .5. »'-i9 » • • 1 a • 1 W > • i> a < .16 -7 i^^'-i^i? , « O 4 • « • .7. .7. -? k e f< • o © i c » ■ • 9 .17 « r. , « o 1 a b > t> a a « .13 S 80- 8Q . ^1 •-> .6. 9,^9 9 9 9 » .1 a I- * » i> -•. } 4 4 1 i» a .16 oi 70- ?•;} . .4. n .3. .1. 9 a » « 9 1 a ft > o « . r. o r c e > « a e f .16 60- e'i? . .3. .1- .1. .le e • a « • i 1 B a 4 ' e 1 a A 1 O a r o a a < ..8 50- 5';? . .2. « ft A o • « • 99 • « a 9 9*' * V » 9 9 1 ff 9 V <3 e > 9 > . . i> o a ..ia. ..4 2 40- 49 . .1. • X. ,1. .1. o ..4 O - • • 1 • 9 i • 1 ,.1 n « ft 6 o « • • • « o e 9 4 « * 9 • • • . . . • 9 « ft 9 « t * V » « o . » 9 1 • '1 > a f a ' ..1 o 1/0 f 3 10 10 ir^- 13 14 30 VI 9 9 3 4 4 4 3 - 1 135 PL- -h. ^o-gi. O [ o Ma-A^B^ n,43i2..4,5" grdn is significant diff.may be significant (r TABLE FOR A5?Tn',!PTS Oil TEKlAIs FOHLI A WITH ON TEK-IAN FOR!] B ITITII GRO.'? I Iliail C -1. \J^u. a B :first t. ■T '■r \ < - o- a- o- o- o- Or- o- C3- c>3 1 i o T l 1 S2 1 ( 1 1 1 f u. s g o 5 o CO O o o § < ,» - ,\ -i -> «» ,. n «1 2<> 1 » • 1 » * • ' « » 1 I • • o « • • • • • 1 u II « on 16f^«iCv? > o a 1 « « ' A a 1 ..4 . .;J. ..4. L«6. ► .4. ,..,.. • a 31 l- 15^-1').' .U, • o 1 « • 1 • vy » • ■> » >«•.-• • • %' « 1 to ft/* » » « ft « ' 9 « O O 2 14^.^;-/ » O » 1 tt p 1 • • >. > • • • < 1 A ^« • • • «i « 1 • '' » » <1> o o ^ « tr * 9 • ■Ui^ o a.3^-.i... h • ci> ' « « » * <> .A > • :^ • • « • > • V » >.x* h O A o » «> n tt ,.,ft.,l.,.D ir-i:. ► • o • ♦ la ..i L.i 1 a O « • ftiiv ^ o o « L... » • o ^ ...4 t- un-i.ij » « « ' ,u ,.1 ..i 1 • > ' • I 9 * « » « « » » • » ^ ► *•**» ..» .| «««0 e. ic^-i-.^ k s c < « • » o e » « a a ft 1 • 9 > » « o ..1, » C O 17 ? w • » • • * « 1 • • •!> ^ * « • .1 1 • • ' « • • • • 1 « * • • « « O f a «. o * O O » ? # (» » > • « W « & « i& H 1- 8^» .Jy e o t * u ' m • a 9 a 1 « • W • * 9 ! 9 a o E> <> • tr ft » fc to s « » » 1 • 9 ail < f 3 4 l^ li 18 17 sx gr Ivt- » 135 jl^ +,t -^^ i .032_ CSi~^^^ 3.<^q i '-^^ r • ^in it: sifpificsnt /.diff. riiay Ij& sigaificaat COimSLATIOlI TABLE FOR KIQIFTS ON Tmm ON TMIAN FOR-' B ^-'IT^' ffflO^F o Z. o Co f- 5 HIGil C ixiui..j.i* tj.; i'^jiuiAi, -i ii vi'liUji' li' ilAL) 1 1 1 <3- 1 1 \ >- O 1 (3- T C3^ f T 'J * 1 r o % o ^ o § o ^ CO 2 '2 o 17^-172 • • • ft « » « o U 4 3 • « « 1 « * • a • f* w » o » » i O 9 » * » « » ft ft » ft o d ft! ft • it • 16^'-169 • ft » • « a 4 • » • * • • < »•« ■» t '3 » • » » « « ... ft *» ft a a « ■ ft A ■* "i » ..a i4^-:.-w • 00 a 9 w « » « • a * » O « 4 e « o » • » » • A n> • a ft ft » • « v^ » ft '■■* ft ft • ..§ l^^-lo^ • * • • • s « i> a • » » 9 rt o -. » » » O ft .a ..3 •Ct 1 • a^ ft ft I ,ac i.r-.i':p A « » tt n o a A • « e «X « « ft » ..6 ..;«i ft ft ft.X > • ft ' ft c ft .14 ii^-iiQ » o fr « « V a « • '* » 4 *^ fl t • '-. .•4.0.4 • a ft • o • » ft ft ft «■ « ..15 ir'O-i^^tj • • • • • • • « s • « • ..a a: ..4 l.X « • • ft » • > • « « ft • • 26 y^- dd O O r» » » » p t. -^ > • •• > 6 * ' »* ■■'"* ft M A ><•»(>•• » • •? » ft 1 ft « ft ,.r/ 3^' Q9 ft O o ..r^ • •u » •=.-» K •'^V« • '-a- » » « » O « > o ft « » • « • » > (1 ft • ft o ft .a^ r. 79 .u ,a • •6 1 • * « « a » k « » • 9 •> • « > • • 1 • ft a >a4 6^« G9 .•1 ...-s ..3 1 M 9 |t • • 4 » » « » • o »«9I0««<»«« m » '.■» S A ft a a a .,.6 13^- ;)^ ^ 4 •> 1^ » • .9 » r» '> 4 ■> « © >'••<' 1 <>«^a«t»»* I* ^« > • « 1 » .» ft ft, 3 ^- 4^ » ■ « » • • » • • » » • B • « Q • e « * a * 1 » « » j> • • >* • .•• > • • • • a 1 • «• 3^- 3:' » • • .a » « 9 ) e a k O <3 4 » • » ► ■» * » ft • > ft • t • • r 9 * • •« ' ft ft « ..1 f L_ -f^ Cj 16 -t '1 X3 ll»>f^ m 12 / ,d eft Z 135 /1_^ -H, S^ t ii , O I \ (Ma ^ (oz.4-( dt (. i7 GX = -^3. 5^-± 0,17 .^^-<5;:= 3.4-1 =L /.M-7 /, 5^J.n is slnnificsnt ,', difft my be si^ificant d coBEEyiTioii TABLE FOR micms ON tseia:i foe.1 a with ^OHSS m FOM B WITII OSOIJP I IHG^I C t3lOI^ 3 0' ] TEEMAH FOE' B (FIRST TRIAL) <: ^3— :^ '3~ .:3^ "T~ 'J^ tj~ a— a— o '~j~ % 1 1 1 1 1 1 f-^ ■>3 1 1 T f o -7 o ^ o o o '1 o .S ^ & o i<^0*lC9 » « » » > • « > e ( O « 4 > o « e < oi&a a e • » » « « tt o « 9 ,« ft . 1 « « «<* « < 90- A' »^'' > c > • «' a 4 > a ■1 '7 > • . ,; • < 9 « >> • > A A » 9 « • • > 9 • • / s. sr- 0»' » » « • « ft ft 9 a ..2. » • « 1 » ;-.( » d o-^» o " • » > • a b « tt 1 a e . a. ' 9 f » < .9,. 9l9> • o ,1.9 .699. ,..ZZ A0„ > • « s * » » * «i 1 a<4^ « . .„. » • •.>•••...'.«« .«« .1.9 «.9.»,' .,..18 o 50^ 52 »« « ,a 1 • • ,•?. 1 ^ <&.'-•* .•5«< • 4« a • o*0»cio« s a 9 * .9 •....2.9 4/>« » » o ;» (t ,«4 ,.9. ,1 1 • ..i • 4 » « » ft « » 9 » 9 9 B «.» » , ...31 30- m » • • .»! ,5 -^ >•...» 1 1 S vV « « » « • » » 9 O c » c c? u « 9 » « • >. 9.9 9 »,9,.,..»,» i8 2. o i^- 19 ,,i .a 1 V 9 > « • ' « « « 1 « • » « • ♦ • « > « * e « « 9 > 9 • • 9 • 9, ■ .9.9 .^li? f z 7 13 ■25 •zc •2 4 '.I . 139 7L- -h. 13-7 ±,,0 3.7 ,\ loss is ins i{?!niif leant /, diff. i-^ in«i??iific^t Ji- COBRELATION TABLE FOR ACCURACY Oil TEK5AN FORLI A WITH ACCUHACY GiiOIiP I HIGH C AccuEACY m 'iimm Fom b (fibst trial) < MM* o- — |. ■3- t ^ j- o- J~ ^a~ 4- o— d~ "i^ > H 1 ('<^ 1 i-.j In jj ^ r- 1 — ^>0 <^ C5- )- i.n O In ■o "^ o \y^ O ^ •o ^ S lo Q Qd c^f c^ 1^ ^ -j- ^ ^ >o ^o ^■. Oo yt o 9^-96 • • s • • o « . < • • 1 • e » • 9 » • • » « e 1 e > « • 1 • 9 • 1 « a « I 9 » « • A* 9 9 ....i z. 85-39 e » 6 « « iS a « u. » « U » « > 9 « 1 • * » ft • 1 « * « ' • ?;« 9 > * • a • 9 * » • * • 9 9* t: 8<^«evi • Q « » « ■ » e M e ► • a k «4>i»*«'t** 1 • e '- ".J ^^ a ^ > • ''^' • < 1 • .e« • .,•10 ^ 75-79 © tt s « < • 9 ■ o * . e 1 ^ «, ^ O -» W » -A >••..' » 9 -! - ,4. 1 • •:^ 4 p a <> * • » « a .9,13 <£ 7^-74 « k» a a e « a • 1 « 9 » • « » « • A .-I » • O 1 •'■21 > • vi ;■■■ < O « t » « (& > » « 9 .9.15 'iJ 1- Q«.> ••(."►• « • ' a 1 • « 1 S XI « » ^^ i « 9 3 1 « 9 • » e • » 9 ...20 60.64 • e !• a » • > • « < a ..cue 1 0.^ '1 1 .Ilk 1 a « o 1 a » ft 1 • 9 9 < I 9 9 i • 9 ...20 ^ 55-55 » « o |a • . a » « **> ..5 1 •• . > » o ► a » « 1 • * » 19 9 9 .9.24 O 5^^-54 » • • u. .4- ,.4 .*4 >a ..i ^.1 » • o 1 • • > 9 • « > O « 9 k • 9 9 > i 9 4 « 9 ...15 V- 45-4i^ » « « L*. aC* l.l 1 ••;rf ) • « ) » • 1 s • 1 « » > » ft 3 > o c » 19 9* 9. .12 u 4^-i4 » • • L «i 4 0^ ..I • • 1 «ra > « 3 » « « 1 e • 1 t > S tl V 1 J e » 1 9 • 9 k 9 9 9 6 9 ,9 ..4 < 35-39 > • • » .i 4 • 1' ' a !* t o « U » « > • • > * • k « • > « O 1 t * * o J * o « » O » > • 9 A 9 9 « a « V^ 3 o 3<^-34 i > S 4 • 9 • 1 ' • « « • ) o a > • » .,. ► • • 1 ft e 1 e 9 a > o a « I 9 « • 9.9.1 'i: f i 13 a i^~ •' ,-. •■^r j'.. ■J i.v.> A ■ <-, I 1 135 /L - f.'^ifi i- -on [V\/v-_ 0. b3.3 ±0, (>07 r^A~i^a- o,of^± 0,0 II ,', gain is cigjiificent , ; dif f . its not significant APPENDn B "■Dieorens Relnjdn£f to Oorrelr.tion Notations Xy ^ ^ 2 - - irririablQs fron arbitraiv oririns >; ^^^ variables from respectiYG metyas l^>, risfinB of t?io variables X, G^ st:ndaxd deTiationn of X^ . V,.^ PearBon'n coofficient of Tra-iaMlitrr, -^^^ JU^ - - - - product monont correlation Z - sum of Buch (iuantiti©3 as . . N frequGncjr of the popiiJ.ati on o~,h-,c, — constf ?heoro:!i. I* Tho correlr.tion bct-r/asn trio 7?.ri able3 is the sa^. a3 that bntrcfm any two linear faietionc of eg.ch of them i.G. Tra-isferrini-^ the variables to their respective momiBf /l(aX + jt)^c^ -hdl) = jTl^jU"^) TlieorOiTi 2, The correlatio n bo two en tiro ratios -^ eM. — - " — "^ — ^ — *~~~ — ^ — — -^ — ^ — ~~- — ^ -~-~ saT is ^iveii by th e for.mila.. ^ w Ihe nieRns mid ntandarcl deviations of -^ are given by b. a equations (9) and (i^^) in Yule, The renuired correlation J Ix^ mil then be given by Expsuidinp;, ne^clecting terms higher then the second degree and substituting forai (a) and (b) gires (2). Formula (2) gi^es satisfactory results for fairly long series, but for very short ones, considerable error occurs due no doubt to ner:l0cting tlie higher po?/ers in the expansions of the binomals. Theorem 3, If X - tj^"^ + -b- Z. , the -partial correlations between the variables may be ?Titten » J (.y:^- > - - I. as is evident by inspection. IbcaiTiple: Since A -i^t^ » J ^(^w --' s if A-G&m/i^, a. S.li.Mo, xntroctiction tv tHo"'TC;'ary'"or;Jtati sties, Theore'n 4<, Tho correlation betijoon ?. variable and. the Siom a of n othe rs is p;iven by the forniiia , (this orpression is a special ca.se o.f .jioarrrrji's General ForTdla) Since ^^ xi+ ^^ -r - - - ^) — \p?^- ^.^^ - ^,^-«-:L(nx,^,t>x, b-^^ ^ J^>:fi7^ and ^x-j - A//l/tY'^^ J tlie ri.n;hu hand ;neiaber above roduces to the ejr^srosaion in (4) Corollar;/. If -t^i? staidard deviationo Oo^ arc all Qgual Theorem 5« The /3Qi^rela.tion bet?^eeii tlie 3it:i..ofn variables aiid n otiior vari ubiss is p:ivon...by.,.-bi:ie fQrr;Mla Wae proof is similar to (4) Coroll?^y X, If tho str';t¥'iaKl devia,tioii3 g^ ixe all equal /•>.* n _ / u,x,' H-/(-A.xj. -t - - - ^T*^^ xia^^a^ --h-n. where A>i>v ■lonot--^ th^ loft h-vi^l ■■lo-ibor of (5l « a. G.3i^'eaiii^a, British Joixmal of Psycholofry. X9i3p Yol.?, b. G»Spearm?aas, loc. ^t» Corollary 2. If the oorrelationg flry. ire all equal , equation (6) may be TTritteii (7) /I... 7^, This iz Brovii's Theorem, but an sho'sm aboTe it is merely a special case of Speaiinaii's General Formula. * :'<^