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Cornell University Library 
 HA 31.T67 
 
 Steps in the solution of the correlation 
 
 3 1924 013 704 071 
 

 
 3T7!PS III mH BOLUTIOIj'^Ol^ THJ C0R:lJ}IulTI01J C0::F?ICI51IT 
 
 usiua K3Ct;iKJular corf.iilatioi^ chart r::vision,uui,ib:r 6.- 
 
 Determine which variable shall be the independent fX- variable ; 
 and which the dependent (Y-vsriable ) and ivrite their names int 
 the appropriate boxes at the top end left-hand sides of the 
 page respectively. The independent variable should be that 
 variable whic'i besrs a more or less causal relationship to the 
 other variable which is said to be dependent upon it. Thus, if we 
 are correlating intelligence and scholarship, intelligence 
 ft^e presumable causal variable) should preferably be the" Z- 
 va#lable and scholarship (the resultant variable) should be 
 the Y-variable. 
 
 Determine the class interval, Ijj-, and the classes of the X- 
 variable as follows :- 
 
 Determine the' highest score (H) and the lowest (L) of the 
 X-variable. Subtract the lowest score from the highest sccr e 
 and add 1 to the remainder; this ausntity is the inclusive 
 Range and indicates the possible number of different gross 
 scores to be taken into account in our classes. Divide the 
 inclusive range by EO and raise the quotient to the next 
 higher integer. This quotient is the class interval, I^, 
 vifhich should be entered in the ly-box at the top. of the 
 page. Now, in that compartment ox the X-classif ication 
 which is labeled "lowest X-class" enter the beginning or 
 zero-class of the X-distribution. Bggin all the classes with a 
 multiple of the class. ' interval. I . 'Write in all the other 
 classes in ascending order toward-^the right, Then repeat tie 
 X-classes in the horizontal row through the middle of the 
 chs.rt which is labeled "Repeat X-classes in this row.'' 
 '■ 'Caution : Do not omit writing in any classes just because you 
 happen to know that you have no frequencies in a given class, 
 for instance. This is a common error of beginners. 
 Illustration ; the lowest score fL) in a given intelligence 
 test is 8. The hi?rhest £ core (H) is 96. The inclusive range 
 is therefore 96 - 8-hl = 89. IJow^ -1^ = 5, when we raise the 
 quotient to the next hip-her whole number. Therefore, the 
 proper class interval is 5. Step-0 of the X-class should 
 YiOM begin with a multiple of the class interval, that is, 
 with a' multiple of £. Since the lovvest score is 8, we must 
 therefore b^-gin the 0-step with £, rather than with 10 or 15, 
 in order that the score of 9 may be included in it. Our 
 stops therefore i'un as follows: 
 
 Step-0 5-9 
 
 Stop-l 10-14 
 ""■^ep-^ i:^- 19. etc. 
 
The original of this book is in 
 the Cornell University Library. 
 
 There are no known copyright restrictions in 
 the United States on the use of the text. 
 
 http://www.archive.org/details/cu31924013704071 
 
Determine the face value^,,^ of the 0-step of the X-vtrlable 
 
 This is readily done as''f3li<ws: Decide what is the lowest 
 
 possible vslue, carried out to 4 decimal places , which can be 
 
 plotted in the 0-step; th3n decide the very highest value, 
 
 carried out to 4--decimql Pisces , which can be olotted in the 
 
 0-step » Add these two values and divide by (. . The result 
 
 is the face value of the 0-step. 
 
 Illustration: Thus if the 0-stop is, 5-9, as above, the 
 
 conventional usare of statisticians generqlly is to consider 
 
 that this includes all scores between 4.5001 and 9.4999 
 
 inclusive. Addinp- these two values e-ives 14.0000; 
 
 dividing 14.0000 by 2 .elves 7 .0000, ' which is the face 
 
 value of the X- variable, and 7.0000 should now be entered 
 
 into the ¥ -box. This technique of determining the face 
 
 value of a .Ttep should be carefully memorized as its use 
 
 will prevent the very p-rea t confusion that is common in 
 
 intei polating the true naean from a correlation plot. ^ 
 
 Caution: Sometimes the tabulated score, 5, does not mean 
 
 all those values from 4.5001 up to and inclusive of 
 
 5. 4919. If, for instance, w^ had taken the ages of children 
 
 only as on their last birthda y then the tabulated ap-e, 5. 
 
 would mean the a;-'e of children who are at least past 5 
 
 and not yet 6; consequently, in this ca^e, the lowest value 
 
 which can be plotted in the class £ is 5.0001 and the jp^ighest 
 
 value is 5.9999 and ^q ^^^ ^^^ ^ i" this cap.e is :. .'^ • 
 
 5.0001<f5.9999 ° 1:. £000?- By usins' this value for Fo , when 
 
 usinc? ^this method, one will obtain the true mean for the 
 
 children just as accurately as L f he h?:d taken the spres 
 
 of the children to their nearest birthdays . V/e could of 
 
 course have taken th3 G??es of the children according to the 
 
 a?e which they will be at their next . - bitthda y. 
 
 ■In this case Fo^ for the clas's 5 = 4 .00011~4.9199 = 4.5 
 
 In all work witn tests it is desirable to*^ assume i^ 
 that a test score, 5, extends from 4.5001 up to 5.^^999 
 inclusive, in order that we shall conform with 
 conventional statistical usap-e Generally, and in order 
 that any averat-e' obtained from '■'•rouoing the data will 
 check with the aver'-fe found by the customiary method of: 
 f<S^) adding up all the P-ross scores and ^) dividing the 
 gp-crree-a te thus obtained by the number of cases, 
 
 4. Reioeat -orocedures 2 and 3 for the deoende'nt, or Y-variable, 
 reoeatins the Y-classes in the column headed ""Repeat Y-classes 
 in this column." 
 
 * This last usage is seldom employed. 
 
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 EXTENSIONS OF X-FREQUENCIES 
 
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 "sdooj. V H'S-'aH '^ S26I 44t'!-'^<^°3 5261 'I '-'T ' o'-'^q'^'^'io^ '^4isJSAtun 940+90.40 'sdooi tf +jaqj3H Ap poyfiifeaa 
 
p 
 
 7V, ^® r-iixed meaEJuremarits as f.i^^oic mcrks in t>»o tsorniva-rtmon i-r." 
 •'nen pintt«ri -? t ic a — i^^uvxe to enter th'e comDartment tro-qikonxiisa- 
 
 is a small figure in the lower corner of each compartment. In 
 ucm,? the size 8#" by 11" chart, whore the compartments are small, 
 :; t may be desirable to enter these compartment frequencies only 
 .ynu after they have been obtained by some other method of 
 
 ^i'^ii?|l°§ilid'^LSii.|lc!i?SHl^lJ^ '^^"^^"^ ^>^^°^^^ °^ ^y tabulation 
 
 ■.Add the vertical frequencies by columns, and enter the sums at t he 
 foot of the proper columns in the column headed, fr^' . 
 C autions : Do not enter the frequencios in the column headed, 
 Cum.Fr^-' . ^Vhen adding the frequencies, fill uo any empty boxes 
 in the Fr^, SVy' and the Fr-j. columns respectively with dashes, 
 to indicate that you have considered the frequencies in the 
 particular row and have decided thet the row ' has 0- frequency- This 
 simple caution alone vifill prevent many errors in solving correlations, 
 3nter the numbers v;ith their vertical axes in a N.Iil.- S.W. direction: 
 i.e., in such a position that they may be readily read when 
 lookinp- at the 1I,„ f, g, h, i, windows. 
 
 A 
 
 7. Add the horizontal frequencies by rows, entering the sums at tte 
 right-hand ends of the proper rows in the column headed S^ry' . 
 
 8. How take the frequencies in the Fry colimn and add them downward 
 from the too of the column toward the bottom of the page; when 
 reaciiiisg the a-box take off a sub-sum and enter in the a-box; then 
 continue adding on to that sum the frequencies of those steps below 
 the a-box until the IJ-box is reached when we enter the total number' 
 of cases in the Wx-^ox. If l^x does not check with the knovn number 
 of cases one must discover the error at once and correct it; the 
 error vi?ill probably bo found to consist in a wrong addition, 
 although it" is barely possible that a frequency may not have 
 been tabulated or one may have been tabulated twice. In order to 
 determine vvhether one frequency has not been tabulated, add up 
 the Fry column in exactly similar fashion,, writing in the sub-sum 
 at a' and the total sura at Ny* If I^X ^^^ Ny^o not check with each ; 
 other or with the known number of cases, the error is usually 
 readily found and corrected. 
 
 9, Cumulate the X-freouencies from the top of the column toward 
 the footjithat iS, tovifard the IJ-^-'box. Write the first cumulative 
 frequency directly opposite tha^'t frequency which is nearest 
 the top, enfcerinsr it in the column headed "Cum. Fr^' ; then add 
 on each'' successive freouency toward the bottom of the chart until 
 all the cumulative freatienci?s are obtained. When computing 
 intercor relations one may check the cumulative frequency column 
 with th=^ standatd or "correct" cumulative frequency column and 
 note any errors. If, upon applying this check , there are any 
 errors which are not due to addition in the cumulative column, 
 it is necessary to correct the plotting and the sums of rows _ 
 and columns, and then to correct any entries previously made m 
 the a and Nv windows, and in the cumulative windows. Compare 
 the cumulative freouency in the two bold-faced cumulative irequency 
 boxes withtfee numbers previously recorded in boxes a and llx- J?, J^eso 
 chock Dlace a check mark in the two little check diamonds, with 
 the double arrows provided for that purpose- 
 
Coution: In cosg any of tha X-stoos have 0-froquency, tho cumuletiv.- 
 froquoncy of that stop is the samo as tho procodin,'? cumulativo 
 froouoncy; that is, addad on to a provious cumulative fronuency 
 dOQS not ch-3Uf^a tho cuinulativo froquoncy. AffiTin, if thoro are no 
 frequonci.Dti in tha 0-stop of tha vsriabla, it should ba noted that 
 tha cumuletivo fraouency for this box is. of course , N. 
 
 10. "lided by tha little lenas, and uslnr a paper or celluloid stencil 
 or sa2>®!tad edp-a, or praferably by drawinj? liohtly pencilled linos 
 through the diesronal lanes, add up the dia^onsl frequencies end 
 enter tho sums in the row at tha top of the page in the column 
 headed Fr-^. /i/hen adding up the long diagonal (ID) .^nter the sum 
 of tho lonff diagonal froquancies in that compartment of the 3V-p 
 row which is immadiatoly above tha label "Long Dia g,i when 
 adding up the frequencies in tha diagonal row markel "A" enter 
 tho sum in tha Fr^, row just ybcve the label "A" at the top of 
 tha pago} similarly v/hon "adding up the A'-diagonai row write tha 
 sura above A' at the top ci tho page; etc. IJotice that the S- 
 diagonal romt h^s only one compartment in it; tha R-diagonal row 
 has £ comp^rtmonts in it; the* IJ-di agonal row has S compartments, . 
 etc. 
 
 Caution : Note that tha A-diagonal row consists of 3 different sec- 
 tions m as many diffar^-^nt quarters of tha t"ble: and so similarly 
 for other diagonal rows from A-X inclusive excopT the LD-diagonal 
 row^vifhich h^^s only £ sections. Diagonal rows, J to 3, have sac- 
 tions in ona quarter only of the table. 
 
 Do not overlook any of those dicp-onal frequencies ; the safest 
 way to insure no froouoncios boin!?" overlooked is to draw a 
 lightly pencilled line '■•ith a pencil of medium hardness through 
 each diagonal row as you count its frequencies. In counting 
 the diagonals, if it should be found that any diagonal row 
 has -• 0-froquency W3 should not fail to place ^ dash in the 
 appropriate entry box in the Fr^-row in order to prevent the 
 possibility of later on shifting of the succeeding entries one 
 unit too far to tha left or to the right as tha case may be. 
 ''Me should remember that the check of the frequencies adding up 
 to IJ is a check only to the fact ihat wo have the right total 
 number of frequencies in tho column and not on the fact that 
 these frequencies srǤ in the correct boxes; that the checks 
 on tho extensions sro chacks only on v/hether the extensions have 
 been correctly done for these fr-nruenoies o1 arifid in fnn boxe s 
 in wbicb they have be'->ri n1 -ic^'d and thot those two checks alone 
 will hold if tha ari thmetical work has bean done correctly 
 irresoectiva of whether the diagonal frequencies are entered in 
 the'corract compartments of the' Fr.^.- row or not. This pencilled 
 line should not" be heavy enough to-^in any way interfere with 
 reading the frequencies of the compartments which the line Grosser 
 Take the rows in ord er, first tha A-row, then the B-row. 
 the C-row, and so on. 
 
 11. Add tha FrD- frequencies from the top oi tha page toward the 
 bottom that is, toward 1\TT^. 3nter dashes in all compartments 
 of th^'Fr -row which do not have any frequencies in them. 
 T"ko off ^ub-total sums Bt the a", f", and k"-windows. and 
 finally enter the totel sum in the II r) ^window. 
 nmitimT I-p thare are no frequonci--?s in that section of the 
 table W hich is above a^^ then the a'^antry is 0. If thare should 
 Wo no entries in that section of the table which is below the 
 k'^-window than the entry for the k^^-window is the same as I^ ^ 
 
^F^ 
 
 Clieck N-T) with Ny and N . Ii this check holds we may now enter 
 check marks in the flxit four check diamonds of the "CH:]CZS" 
 section of the shoet and enter the vslue of N in the rectansrular 
 box marked "IT" in this section. 
 
 IL. Cumulate the frequencies from the top of the p'ap-e toward the bottom, 
 that is, toward N^, takln^ account of the cautions given under 
 step-S above. If TTne cumulative frequencies in the bold-faced 
 boxes of the Cum. Fr^ -row check with tte frequencies in the a", 
 f", k", and II -windows respectively we may nov; enter check marks 
 in the four cKeck diamonds, with the double arrov.'s, provided for 
 the puroose. 
 
 Sote that there is no standard cumulative frequency with which 
 Vve can compare this column such as we have for the cumulations 
 oi th3 X and Y variables when computing in ter-co rrelations when 
 we have available standard or "correct" cumulative frequencies 
 for each variable. 
 
 17,lIow recopy tha frequencies of the Fr-. Fr^ , and Fr columns 
 
 into the columns of the same headinj?^ respectively^ to the right 
 in each of the three sections of the extensions of the table. 
 Take care to make your figures lep-ible. 
 
 I'i.Alio-n the window of the correlation stencil by placing the 
 little circular hole so as to coincide virith the little cross- 
 sectioned circle provided on the chart for that puroose, thus 
 makinp- the ^^window of the chart coincide with the -)fewindow of 
 the stencil; the a-row of the chart alien with the a-row of the 
 stencil; etc. 
 
 Caution : Do not confuse the les-ends^, a ,b, c, d, e, f ,g,h, i , j , etc. 
 here used for alip-nine the stencil with the a,b, c, etc . , a ' .b' , c' , 
 etc. windows in the '.ody end at the foot of the respective columns 
 of the e xtensions . The stencil can be used only on the 14" by 17" 
 chart. In the case of the bi-" by 11" chart the extensions cf:n 
 usually be done A'ery resdii.v h^^ mental multiplication since the 
 multipliers nevar are Isrc'-er than 10. Cn the stencil, in that 
 stencil column which corret; ^onds to the frequency of the a-fre- 
 quency-ftompsrtment of the ch'3rt, find in the a-row of the stencil 
 the Fr-r» . X' and X' (Fr / .X'J entries; the former in light-faced 
 type and the letter in"..Xy"'b old- faced type, and enter these on the i 
 chart in the proper columns. Continue thus for all the frequencies 
 of the X-variable. Do both the top and bottom halves or sections 
 of tho table.' 
 
dhart 
 
 to — 
 
 ■L5.,Mo^-e.-. tho. t-t't^ri-cil to the .,rip"ht end alip-n tho s tencil— t^^'iln by 
 maTHS of the little cros^-S'p^ctiona'i circle, winter tha^ oxten- 
 sionc Pi\^',Z" end ^vj" fX")^ in thg ?5ppropriRte olaoes. Do 
 both secr-.^.icno of th^^ tabic. On the 8-i:" by 11" chart these 
 extonsicns cannot bo done by stencil but raust be dons mentallj'-' 
 usiug as multipli--^rs tho X'', fX")^, nniltlpliors riven on the c" 
 
 16 ..Carry out procedures 14 and 1£ for the Y-varinble end slso for 
 the di^o^onal froquenoies. All of thovse can thas beat bo done 
 
 " uiuo before tho additions of the extensions arc begun, 
 at one-^ 
 
 17. Be '-■'inning at tho top of the columns and adding; toward the 
 
 botuoms, that is toward Jy, add all the columns of tho extensions 
 of the X-frequoncies . Using an.aadine machine wo can readily 
 use the machii-ie to add t^.'vo columns at onc^ by enterin.^: the 
 Fr;j£' . X' jxtensions in the left-hand half of the given board and 
 the X' . f Ft-.' ,X' ) extensions in tho right-hand half of the iioy 
 board. vVhert i"e "■et to the b and c windows in these tv/o columns 
 respectiv3ly we -irst npte the sub-sums and record then accord inrrly 
 in the b and .c "/indowe'; then continue on to the foot of the 
 page and we obtain th'^ total sum of the two colurmns respectively 
 which are entered in tho f and g windows. In li.:e maanor also wo 
 obtain all other sub-oums and total sums in the extensions of 
 the four columns of tho X- variable, tho. Y- variable and the 
 diagonal f ;.-oquencie s . 
 
 Cpution : If in the S'^^ctions of the extensions which are above 
 the, b,b' and b" windows there are no entries except 0-than the 
 sub-sums at these points will of course be 0. T^is applies to 
 all such simil'ir situetions. At this stage the person viho made 
 tho axtonsions of the fi-enuencies by upe of the correlation 
 stencil, sip-ns hir; initials in tho upper right hand corner 
 in the row "Extend od bj," 
 
 18. The ch3clcer of the extensions now bas-ins his work usin?? tha 
 typical checks erovidsd in the "QH'^CrS" section of the table. 
 Tha sum of the a 4- b windoxvs in the X-variable must check xvith 
 tho entry in the d I'^indow; cbf so a choc]: mark may be placed in 
 the first chock diam.ond found betvi'eon windows c -nd d. Than 
 tha sum of windows d■^b^-c should check with the entry in window 
 a. If this check holds a chock mark may be placed in tho second 
 check diamond found botv;eon windows c and d. If these checks 
 do not hold the error in rrekinp- t he oxt e ns ions must be found 
 and corrected. If the chec]:.s do hold all the sxtensions In 
 tho upper half of the extensions of the X variable are correct. 
 Now apply the sa'-ne checks to the IJ ,f,g,h and 1 windows, i^ince 
 in tha"^ place of the a-boxes 'wo hav^ here the IJy-'box, and in 
 place of the b-"!-^ox W'-> ha-.-o the f-box etc., it is necessary to 
 make these aperoeri^to substitutions. It is simplet to think 
 of the checks in this saction as;- 
 
-7- 
 
 Tho first t^o boxaa I'rHf must chocic with tho third box to -the 
 rior-ht nciiol^, v/ith ht than as tho Socond ch3c]:, tho throo middlo 
 cr bold-facDd boxes {if^j.h) must chock with ths fifth box, or i. 
 + ^^®f'' checks do not hold end tho nrovicus chocks h'-ovo baon n?do 
 to hold b;^^ corroctin.o' -^nj errors that v;,-oro found, then tho errors 
 Will be found in tho bottoii hoif of th3 extensions rnd o^n readily 
 be located by notin- rhich of the two checks f-ils to hold. If tho 
 second chock docs not hold then tho error will be found in tho 
 oxtonsions for the ^-or the c-windows. If both of these checks hold, 
 then rll of the work, on tho X-freauoncios, both in nokinrr the 
 e:-:ten£io-Js snd in adc'inp- them, is correct. 
 
 Caution: Do not for-^ot thot tho f? -window is the sum of the c-windovif 
 plus all tho oxtensions intervening between the c-^nd g-windows. 
 This error, if msde, would have occurred in the previous step, 
 
 L9.In simil.'i-r m'^nnor check all the extensions of the Y- frequencies and 
 all the dipgo^njal frenuencies. These all havino: been checked as 
 correct, tho^r '^ci^- 8i<?no his initials .^nd dete at the upper right 
 
 hand ccrner ofter tho v.ords "Checked by/'^- 
 
 Coution : If an error in tho extensions has occurred in the top 
 section of the ext.jnsion column, say in b, this will of necessity 
 introduce an error into the totsl sum of the column, in f . 
 In order to save tirae the total sum of the column in the f-window 
 should be corrected nccordinsrly before applying the check to the 
 bottom half of tho coli^mn. In the diap-onals an error in the b" 
 sum will be reflected also as -^n error in the g" ill^and s- windows 
 all of which will need to be corrected. ' 
 
 Bc_fce: Some people prefer to make the check at the very bottom of 
 each column. If this check holds, thon th^ checks above that point 
 do not need to be made. If this bottom check does not hold then 
 one can perform the previous checks and thereby very quickly locate 
 that section of the t^ble vih.ich contsins the errors. 
 
 ]O.At this point the computer enters into the vvindows of that section 
 of the table which ie headed "Derivation djf Const-onts 
 
 f^', D and 
 cannot be entered until computed in t?:e two previous sections. 
 Having entered all the possible values in the respective 
 parersthesos of this "CH:]CI:S"- section, the computer oroceeds to' 
 hll^d.i'Y tollow out tho computations, -' '■■"' '■ '■. "^■•; ^ •■.5. 
 
 hcte tnat tne xurnula forjg^^ irl re^xity calls for f + 10a, and 
 that in order to simplify the problem, has been placed in the 
 fa) in order to indiceto the multiplier of 10; thus, if f = 86 
 and = 2 Vi/o reed th.; o as 20 in making computations so that 
 D •. s -ac -ordi-i??lv <^(er^1D = 106 
 ■■^50: 
 
-8> 
 
 Caution; Tn cnse any^o^ the values called ifor are O's ^^!^hes 
 shou],Q be entexed iu cnos-e boxes respautively ratlier tnan l b, 
 in Older to obviate the- possible error of calling a true zero 
 in the b v/indoa' in solving for the ;^J X-^^, 200, say, instead ot 
 its tru3 value. In cf't.e b has the vnlue 0, it is even better to 
 draw the d^sh through the £ 9nd includinf? both parentheses. In the 
 entr;/ ^^ X^ = ^"- i^ ) > c ( 2-* ) -i-ic^VO) , we re-od the sum as follows. 
 f 14 f- 2(20) Y- 200 which is 754. Th'st is, the oris-inal formula for 
 which this stands, 1b ^X^ = ff-J-EOb f-lOOa. But, we must not fail- 
 to note that, in order to save time and reduce the complexity of 
 the multiplication, the of the 20 and the two Q's of the 100 
 are placed within the b and the a-parentheses respectively. 
 As scon as D has been derived twice independently in the two 
 ■oropor boxes the two values should check. If they do, we may place 
 a check mark in the D check diamond; similarly having obtained 
 the second value for A independently, if it checks with the 
 first value we may place a chock m^rk in the A check, diamond, like- 
 wase for 3 and B respectively. 
 
 The Quantity P is derived four times independently in the 
 section of the table headed "Sxtension of Diapconal Frequencies. 
 In case of certain errors it may happen that two of these F-values 
 will check but that these will fail to check with the remain- 
 ing values. If all four values of ?, derived independently, check, 
 then we are certain that the countins' of the diagonals has been 
 done correctly, and that thoy have been entered in the proper 
 windows in the Fr-,,-row. If the four F-chocks do not all check out 
 to th3 same value, then one should first reassure himself that 
 .the arithmetic in this section of the chocks has been done correc- 
 tly; then after no error is detectsd in the arithmetic, the diagonal 
 frequencies must bo r3Countod in order to note particularly whether 
 some diajronal freoucnoy hss not b-ion pieced one box too far to 
 the ri":ht or to the left. Such an error seldom occurs in case we 
 have taken the precautions undor step-10 above, of always entering 
 a dash in the Prj^-ro'^' \.honever a diagonal row has 0- frequency. 
 Once in a greet v/hile, wh-jn solving isolated correlation coeffic. 
 ients viThero there are no standard cumul.3ting frequencies for X and 
 Y as in intarcorrol'^' ti oris 'vo may have to recount the X-and Y-fre-, 
 quencles. Such an error will be vjry much more infrequent than '-" •■- 
 errors in counting thi diagonal frequencies. 
 
 The work boin? correct to this point, one proceeds to 
 determine the v':lue of C by three independent methods. If the first 
 tww values of C check the third value will clways check unless 
 an arithmetical error has been made in determinin.?? the third value 
 for C. Conseouently th9 third independent derivation of C may 
 or may not be made, dep-indiner upon one's time. 
 
 In all the above a c-'?lculati n? machine will be found to be 
 by f>.^r the most efficient method of solvino- the "CHECr"-formulao . 
 
22 
 
 -9- 
 
 21. The variou.; chaoicQ4 veluas a^ove j'c und ere ,..now entered in the 
 "Slreleton Formula" '^d the "Checi: Formula'^ and the two 
 eqaationi? sr'j c-olved to^^v. Note that -C and K in both formulae 
 are thr same value but that A' „is--felW3ys larr-'er than A; 3 
 is 'always lnrp.?T thsn B, etc- is calcula tin.;? niachme is the 
 rioE?t ready p.ethod for simplifying the two" equa tions, result- 
 in^ in a reduction down to three integral numbers. At this 
 point, L should jheck with L' . If so, n check mark may be 
 pieced in the che nk di-Tmond at the risrht of L'. Similarly 
 H' mus'c check with H, -mi Z' with K whereupon check marks 
 may be placed in the check diamonds orovided. If these checks 
 do not hold, note whether the substitutions of numbers for 
 I'ltters has been correctly made in the two formulae; then 
 recalculate that one or ones of the three quantities L,H,and 
 K, which do not check. 
 
 In case one wishes to do so, and does not have a cal- 
 culating machine available, he may make use of the "Job 
 Analysis •'^tep Solution of r llqustion" given at the top of 
 the column?' The reduction is perj ormed' merely by following 
 out in systematic order from the top downward the directions, 
 "Add, Subtract," etc. The little-h,"". andXsigns also assist 
 in roc3llin<3: the operation which is to be done at any 
 particular point. 
 
 The final solution for r, after finding the reductions L,H,K, 
 may be found by slide rule, Vifhich is the -quickest method, or 
 by perfcrminS' numerically the o|!eretions indicated on the 
 sheet, ynd in the order indicated. 
 ( a ) Solution of r by slide rulO o 
 
 Set the hair-line o\^or the numerical value of L on the D-scale 
 of the slide rule; bring the vilue of H of the B-scale under 
 the hair-line; move the hsir-lino to the index of the C-scale; 
 brine the numsric^il wlue of 11 on the 3-scale under the hair- 
 line; opposite the index of the C-scale, find on the D-scale 
 the answer sousrht. This may be checked by the follow/ing- 
 procedure .'Brinty th;; hair-line opoosita L on the D-sci^le; 
 now brine? j__ on th-^ 3-Kcele under the hair-lino; slide tha 
 hai.r-line to the inclex of the C-scale; brinp- H on scale « 
 under the h'air-line; read r opoosito index of sc-le on 
 scale D. 
 
 f b ) Solu ti on of r by lo"-qrithm s . 
 
 Or it may bn solved by loo^jrithms as follov/s: Find the logar- 
 ithm", of H and also of T; add the two and take half of the 
 sum; subtract this half-sum from the lo'^arithm of L; look 
 up the snti-log-^rithm of the tema.inder. This anti-lo«<?:arithm 
 is r. 
 
 ( c ) Solution of r on the Galcul-:3ting machine . 
 On the calculatins machine solve r in tvi/o different ways. 
 (A) in the skeleton formula, multiply H by Z end extract the 
 square root. Divide this square root into L vvhich ?ives 
 r ss the quotieno^ (B) Scusro L'; multiply H' by ![' ; 
 Divide (L')'^ by H' ?:nd 11'; and extract the square root. 
 The result is r. However the sign of l- must be q:iven the 
 sa,mo sign as L' . 
 
 Caution; Do not forget to givi3 the negative sign to r 
 in case L' has a negative sign. 
 
 ];Jo te ; In all th^ -jbove reductions, as vfell ag in the derivat- 
 ion of the Gonst-:nts of the r equ'Jtions any quantity which 
 may be negative has a semi-circle attached on its left; such 
 quantities do not come out negative in all problems but 
 occasionally may do so. 
 * Tixo vartical lines are intended to separate the successive 
 
 digits of each number and so accurately to align them in 
 
 the column. 
 
-10- 
 
 23- The solution cf the averages . Subatitute the numerical values 
 in the formula for M„ and !„ mn S'^lve. Tho result gives the 
 means directly without any Interpolation. We have previously 
 shown in step 3 above how to find Fsj ^^d. F'> . As a check 
 
 en the solutions it is desirable to i-^lve them a^ain by the 
 f.-Uovving interpolation formula: M« = £; M' = C If the 
 result of this operati'^n is, say k| V ^ ^^ F ^his is t'^ 
 be interpreted as the face value of the gth's-^ep plus tv:^- 
 tenths of a class interval '-f the X-variable,. The' face value 
 -f the eighth step of the X-variable can be readily found; to 
 this wo add I (.2). The result must check readily with M 
 s-]ved by the^skeleton f-^rmula of the chart. ^ 
 
 Sr. The soluti-^n f^f (T^ •^-^'^ <T~y' In these solutions the -quantit- 
 ies K and K have ^een foimd^above and have been checked in the 
 s'^luti'^n of the skeleton formula and the check formula for r. 
 Note that the standard deviation is always positive. As a 
 check on the solution r-f g- one may solve the formula in the 
 fi^llo^ving manner: ^ 
 
 (a) extract the square r^ot '^f H 
 
 (b) divide this square ro^t by N 
 (c). multiply the quotient by I . 
 
 A 
 
 25. The P,E,j. and ths coefficient of alienation, K, may be 
 copied into the chart fr^-m prepared tables ^r may be solved 
 by the usual formulae: 
 
 1 _ r2 r 
 
 P.E. = K = Nl - r^ 
 
 r 
 
 It iTill be n-'tod that the numerator of tha P.E. upon 
 extracti'^n of the equvare r'r^ot bec'~mes K. ^ 
 
 26. Th e solution ^f t he regression eq ua tion x^r pred ic ting_ Y , 
 Xi'riQ dependent vaiiableT^ Su.bstitute the values -"f M,^, r^y* 
 
 (3r"Y' Q'~x ^^\ ^^ '^"^'■^ I'^i-n^ula of the chari arid splv^. The 
 final equatioxa is always ^f the form Y = a+b« X. For a 
 check -n the equati'-'n, solve the following equations: 
 
 b= ^" ^Y a^FOy-hlY ( H.E-D.L ) - L.I^.Fo^ 
 
 NH 
 
 «• ^x 
 
 "• ^x 
 
 since this formula involves all .-^f the quantities used in 
 finding the corrclati-'^n coefficient, standard deviations, 
 and the means, if this for..iula is carried out to tbree dec- 
 imal pi aces and checks wi'jh the previous formula the re^'ult 
 is a check (^n the correctness ox the s^'luxion cf the five 
 quantities involved, af5 well as upon the correctness of the 
 regression equation. If th3 check does not held, and if the 
 arithmetic of tho above two frrmulae has been done correctly, 
 then it is necessary to exir^iine the standard deviations for • 
 the err'^rs involved in the.n. 
 
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 IL. If a copy of the stencil, "Correlation Extension Tables", is 
 ■not available, the small numbers placed near each row or col- 
 umn for multiplication by the frequencies, and labeled X, X^, 
 Y, Y^, fX-Y)^, may be used as multipliers ®f the frequencies 
 instead of using the stencil products. The stencil greatly 
 hastens this operation and tends to eliminate the chances for 
 error. " These small numbers take the place of the usual deYi- 
 ation steps from an assumed mean, aiid are always positive, 
 thr-« no negative products are produced. 
 
 12. Using only the bold-faced products of the stencil above, find 
 the Fr-n • fX-Y)*^ products corresponding t© the respective di- 
 agoiaal frequencies, Q, P, 0, etc. Check by multiplying 
 
 Frv; by (l-Y) multipliers ®f chert. 
 
 13. Add all the quantities in the five rows and columns from which 
 are obtained the sums which are entered respectively in the 
 
 D, A, E, B. and C windows. 
 
 14. Copy the window sums into the proper lettered places in the 
 skeleten formula and solve fer r- A calculating machine is 
 very efficient in reducing the Tormula t© the final form, in- 
 volving the reductions L, H, and K, which may be solved ef- 
 ficiently by slide rule. 
 
 15. If desired, operation 14 may be replaced by copying instead 
 th@ window entries into the lettered windows cf the "Job An- 
 alysis Step Solution ©f Equation for r". The reducti®n is 
 performed merely by f©ll®wing ®ut in systematic order from 
 the top downward the directions, "Add, Subtract", etc., as 
 found. The final solution ©f v is by slid© rule as before. 
 
 16. To find the X-average, or My, substitute the preper window 
 quantities in the skeleton lormula and solve. The answer is 
 in terms of table steps; thus if My equals 4.6, this must 
 be interpreted t® be the face value ®f the d-column, (class 
 N©. 4 in the X- variable) plus six-tenths of the class inter- 
 val, ly. The average of Y is similarly found. As a check on 
 the averages solve the follswing formulae: 
 
 Mx = ? . Ix +FOx . My = 5 . i^ ^ FOy, 
 
 where F©x ^^^ -^^Y ^^® ^^^ face values ©f the zero-step ®f 
 the X-variable and the Y-variable respectively. 
 
 17. To find the standard deviation of X, or (JT* substitute the 
 proper window quantities in the skeleton T®rmula and solve. 
 Note that H is found in the reductien of the denominator of 
 r. The standard deviation of Y is found in similar fashi©n. 
 
 18. The window sums, in order are: 
 
 2 2 
 
 N = U, number of cases A"2:x B - ZY g 
 
 DrZX E=2:Y C= I TX-Y) 
 
 Reference: Toops, H. A., - Eliminating the Pitfalls in 
 Solving Correlati®n: A Printed Correlation Foim. Journal of 
 Exp. Psych. Vol. 4, N®. 6, Dec. 1921, pp. 434-446. 
 
- 2 - 
 
 19. CHECKS . - It is desirable to have e aheck on the ecctiracy 
 of the entering of the diegonel frequencies. There ere 
 three checks: 
 
 A. The disgonsl frequencies must add up to N. 
 
 B. A cert&in quantity P must check' vdth D-E. Find th© 
 quantity, T, by three steps as follows: 
 
 (l) Multiply Ell dicgonc;l frequencies to the left of 
 the "longest Diagonal" hy the squsre-root of 
 ^X-Y) . In the reference cited sbove, Toage 435, 
 this quantity is llCl) + 18('£)+2lC3)+35('4)+14r5)-i- 
 15C6)+6^7)fO('8)+lf9)= 461. Since these Erj^ (X-Y) 
 products are negatiye, give this sum a negative 
 sign, ©r -461. 
 
 ( Z) Multiply cl'. diagoncl frequencies to the right 
 of the "Longest Diagonal" "by the square-root of 
 fX-i)^. This sum, in the reference cited, page 
 435, is 7(1)^1(2)= 9- ■ This sum is positive, 
 since it comes fr©m Fr-^^ 'X-Y) products which 
 are all positive. ^ 
 
 (3) T® obtcin F, add algebraically the two sums just 
 found. Thus in the above example, F= -461+9= 
 -452. P must now check with D-E. In the above 
 example D=607; E=1059; F=D - E , ©r -452 = 607 - 
 1059 = -452. If this check does not hold, re- 
 check entries of F^-n frequencies, and then ef 
 Frx . X and Fry • Y extensiens until the err®r 
 is found. 
 
 C. Finally in the solution ©f the formula for r, A-fB - G 
 is alv/ays an even number '"one divisible exactly by 2) 
 if the extensions have been correctly done. All of 
 the above three checks must held on any sheet correct- 
 ly computed. 
 
 PRICES: Correlation sheets, each 4</; stencils, each 10^; all 
 prepaid. Directions free. Address, Herbert A. Toops, 
 Ohio State University, Columbus, Ohio.