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 .UNITED STATES DEPARTMENT OF AGRICULTURE 
 BUREAU OF AGBK LL ECONOMICS 
 
 THE USE OF PUNCHED CARD TABU - EQUIPMENT 
 
 IN MULTIPLE CO: ION PFT'BL: 
 
 Collected and prepared for the use of 
 Statisticians of the Bureau. 
 
 
 Bradford B. Sknith, In Charge 
 Machine Tabulation and Computing Section 
 
 .ington, D. C 
 October, 1923 . 
 
- 1 - 
 
 The use of punched c.r- tabulating equipment is not new: It 
 was used a nunber of years ago in tne Bureau of the Census for 
 •ration of Correlation tables. More recently it hae been ui 
 
 It two of the bureaus of the Lepartment of Agriculture m 
 %j. The writer in no sense claims to be the sole author 
 the methods herein set forth except insofar as the -1 tech- 
 
 .ue is devised to suit the equipment and problems of tne Bureau of 
 tltural Economics, 
 
 Especial credit is due to Messrs. Tolley and Esekiel of this 
 Bureau for devising the least square method (page 1? et sea.. ) of 
 apj relation problems. The writer also washes to express 
 
 appreciation to them for going over the following text, their 
 
 . ns and helpful criticism. 
 
 The coefficient of Multiple Correlation, R, is a measure of _ 
 the degree of agreement between a given series and the estimated 
 (generally forecasted) values of the same series, <<hen these esti- 
 mated values are determined mathematically from two or more difier- 
 ent (independent) series.* The following pages describe the forms 
 i arithmetic used in computing the value of R and in deriving the 
 on equation" or forecasting formula. A five variable tfour 
 independent) example is given. 
 
 •Tne coefficient of Multiple Correlation (R) may be defined as the 
 coefficient of correlation (Pearsonian) between the dependent van- 
 
 ie and corresponding estimates thereof, computed from the inde- 
 pendent variables, i. e. if X' , the estimated value of X, equals 
 
 b]A plus b?B and 03C plus a constant, where A, B, and C, are 
 
 - —-mrarkent varrccle^and bi, b 2 , 03, are the net regression 
 
 efficients, B equals the mean product of the deviations of X ai 
 X' fr lr respective averages divided by the product o: 
 
 standard deviations of X and X' . R is also equal to the r 1 
 
 deviation of X' divided by the standard deviation of X. Also, U 
 standard deviation of X' squared is equal to the mean proauct of 
 th- -tinns of X and X' from their respective averages. 
 
Die data *r. • coded so that the variation I 
 
 is somewhere be- depending upon the d. ion 
 
 ■ e per:, . estigu- The smaller S- 
 
 the simpler process of performing the calculations, for 
 
 the whole success of using Holl- nth Machines* lies in the groupi. 
 
 '• land, tod a grouping-- i 
 
 duction of v xn reduced accuracy. The c .1 
 
 be accomplished either oy subtraction or by ^s 
 
 ii the caries under consideration varies between 120 and 173, SUD_ 
 ~t 120 from all items, ar.u the differences oy p thus re- 
 
 ducing the variation I from to 11. This ceding Of 1 in 
 
 way | -ates the calculation of R. Coding oy division -y 
 
 reduces accuracy, but subtracting in no way affects accuracy. 
 
 On the following pages are given the cod<. In the 
 
 example. A ( 3 ( C, & D are the independent variables. X is the de- 
 pendent variable (the values of which it is desired to estimate, or 
 "predict".) Tne series should be so arranged that the dependent 
 1 ible is placed in a column to the right of the independents. A: 
 . series have been listed, cross-add th Lu B, C t 
 
 CHFCK an- r each ODsan uod list the sum in tne n 6he< - ^a" 
 
 ~~' v CO] the right, Col. 8 Ua the tabulation. This Ch< - 
 
 is not essential to the solution out is of considerable di- ir. cht 
 • ic. 
 
 1 typos of pun. sorting 
 
 1 • ^s*» used In t -ureau are for r n 
 
 -. 
 
 ** to be made .. L, figure a that 
 
 can b<- rocess. 
 
IV FOR EXAMPLE 
 
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 15 12 D 
 
 lb 12 4 
 
 17 6 3 
 
 10 12 
 
 19 3 
 
 20 io 3 
 
 21 10 3 
 
 22 10 3 
 
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 25 l 
 
 26 io 3 
 
 27 10 4 
 25 10 b 
 
 29 12 3 
 
 30 io 5 
 
 31 12 9 
 
 32 io 3 
 
 33 . 7 6 
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 37 3 5 
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 28 
 
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 11 
 
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 32 
 
 1-6 
 
 10 
 
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 31 
 
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 6 
 
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 23 
 
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 38 
 
 150 
 
 3 
 
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 10 
 
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 19 
 
 131 
 
 10 
 
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 8 
 
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 132 
 
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 10 
 
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 10 
 
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- 6 - 
 
 Record No. 
 
 .)- 
 
 B^V 
 
 '35" 
 
 — i«r- 
 
 l 
 
 - 
 
 
 
 
 
 
 
 l< 
 
 
 
 
 k 
 
 6 
 
 14 
 
 It 
 
 2b 
 
 
 10 
 
 lj 
 
 
 
 2 
 
 27 
 
 
 10 
 
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 C 
 
 
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 23 
 
 
 I 
 
 u 
 
 11 
 
 
 
 
 2o 
 
 170 
 
 3 
 
 1 
 
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 11 
 
 22 
 
 171 
 
 7 
 
 
 
 
 
 11 
 
 5 
 
 23 
 
 (*) 0r ' • mus 33 and remainder divided by 3 
 
 [3) " " "120 " " » ■ o 
 
 " " 233 «i it 11 H 5 
 
 (3) " " " " " ti n z 
 
 
- 7 - 
 
 After the data has been coded and lifted as on the preceding 
 page* the values ari than punched on punch cards, the recor «. r 
 
 also being punched for the purpose of Identification, There are 
 
 1 columns on a punch card. In nur example the record number 
 was punched in columns 31-2-3, A in columns 34-3, B in 3b-7, C in }6-3', 
 D in UO-1, X in 42-3, and 3 in 44-p. One card is used for each line-- 
 for each observation, that is. 
 
 ter the cards have Deen punched, tney should be sunned: 
 The sum of A, of 3, of C, of D, of X and of S should be obtained, also 
 the number of cards should bo counted. The results should be recorded 
 in seme such form as follows: 
 
 Form 1 
 Sums and Means of the Variables. 
 
 w Items 
 
 A 
 
 B 
 
 C 
 
 D 
 
 X 
 
 s 
 
 171 
 
 1302 
 
 703 
 
 1149 
 
 o30 
 
 . 774 
 
 4b 10 
 
 Means: 
 
 7.6140 
 
 4.1223 
 
 ' 6.7193 
 
 " 3.976b 
 
 4.32b3 
 
 26.9591 
 
 The use of the Check Sum first becomes apparent here: Evidently 
 the sun of the sums cf <., E, C, D ( & X, should equal the Sum of S; which 
 is the case. Ir.e sa~e is true of the means (averages). This checks 
 the first additions used in building up the checic sum itself, it also 
 checks the accuracy of the puncing; and also of the division in securing 
 the averages. The values filled into the form above are for our example. 
 
 •It is sometimes feasible to do the coaing by punching the orignal values 
 upon the punch cards. Then sort tne cards on the variable to be coded; 
 group the arrayed values into the determined upon classes and gang punch 
 each group in a new column .vith the assigned class value --such as 0, 3, I c 
 The check sum for the individual record then can be prepared by showing 
 each card separately in the tabulator, aduing across, and subsequent!., 
 ..ng upon the card, after which the procedure is as given above. 
 
 y 
 
- 
 
 The n i to sort the cards upon the : iable, 
 
 iing the IS g iV en below: 
 
 
 r. 
 
 
 No. 
 
 
 I5i_ 
 
 - 
 
 ill 
 
 
 
 (10) 
 
 S-jm 
 
 (12) 
 
 iiii 
 
 • 
 
 iiil 
 
 
 
 
- 
 
 The cards being MM tht first group t.. 
 
 into packs--all the c.iris >f t i.. : I 
 
 the next value of A in the second pack &c . &c . . List in c lumn 
 (l)--Frr the value of A in the first pack. Tabulate this p.* 
 
 On the first line in coIutji ( 2) write the number of c^ris in the 
 pack; in colUBO (3) write the sun: of the values of A in this first 
 pack, in column (7) write the sum of the values of B in this iirst 
 pack; in column (9) the sum of the values of C in this pack; in 
 c lumn (ll) , D, in column (13), X, in column (15) S. Take the 
 second pack, list the value of A in this pack on the second line 
 the , and list the corresponding sum values as for the first 
 
 pack. Repeat until all packs have been so treated. When this is 
 completed make the extensions for columns 6, S, 10, 12, 1^, & l6 as 
 follows: 
 
 Multiply the values listed respectively in columns >i 7, 9, 
 11, 13 & 13 On any line each times the value listed on the same line 
 in column 1. List the products so obtained in columns 6, 3, 10, 12, 
 & Id respectively. Do this for all lines. 
 -a columns 2, b, 8, 10, 12, Ik &■ 16. 
 
 Take the cards and. s<--rt them again, this time on the sec 
 variable, 3. Take a second sheet (Form 2, Sh.2); divide the sorted 
 cards into packs, according to the values of B and list these values 
 : successive pac/cs in column 1. Tabulate, list and extend in a 
 nnfl>r exactly si.:: to that • a cards were sorted or. x- 
 
 cept igures need appear Is tir.« - ^ & 6. 
 

 roceed as for A & B on a new she< ., Sn 3) . 
 
 No figures need appear in the B colv . j, b, ], 4 8. 
 
 Sort r n : . • (Form 2; Sh 4): No f igure 1 la the a, B, 
 
 Linns: Column t LQ Lnc Loslve. 
 
 Sort on X and repeat (Form 2, Sh 5). He figures in the A, . 
 
 C, rr D columns; Columns 3 to 12 inclusive. 
 
 The reason tnat an increasing numcer of columns 
 oe ccai- ::ake the extensions and sun thi lid give fig- 
 
 ures already computed: Thus if we sort on C ana extend its values 
 •s D, adding the extensions, we arrive at the sane figure as i: 
 
 i onD and extended its values tixes C.) In case difficulty 
 is encountered in making the figures check to the check ran -- ex- 
 rtsd later in connect) a with Form 3--it nay be advisable to ma) 
 extensions here directed to be omitted, for the sake of compar 
 help locate the errors.) 
 
 Following are the tabulations of the five sortings made in per- 
 ing tr.p above steps for our example. Note that in each case a 
 check is ai 1 by adding up column 2. This should add. to the total 
 
 IS problem as shown by the data on form 1. 
 further check may oe afforded by adding the sum columns for each vari- 
 able— columns j, 7, 9, 11, 13 &I5. These should on every sheet a 
 the same corresponding figures given on form 1. 
 

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 — 
 
 
 ^ 
 
 J3 10 I s * <o c> o — < C 
 
 
 
 o po 
 
 
 
 
 iflOD H , . — 1| O 
 
 I 
 
 o v o eg to o 
 
 r-t - • :■ 
 
 HH«rn^rtnHvroH 
 
 
 0^10(DOO<V(J»OCOC 
 
 r-iintoor-r-^r t> co 
 
 
 0»-<CN3tO%- IfJiDt^CDOO^ 
 
- 13 - 
 
 Hie next step is to transcribe f: I tx ' fel (* to 
 
 5) of Tom 2, to another 6heet o: fona shown on the follow: 
 
 ;. (For:-. .>) . 
 
 On lii 3) coluj-n a list thv. figure taken fron Fc • 
 
 :h. 1 Col. 6. L) lir.c. ?k . other fj on li. - arc 
 
 taken frou th<- it ihoet (Ibxi 2; Sh. 1) last line colui.xs ~ , 10, 
 
 16. The fifures filled into forr. 3 apply to our lo, 80 
 
 through the various fori.-.s. 
 Th< • for line B-l cones from For:., 2, Sh. 2, last line, 
 nana , 10, 12, 14, <1 16. ('H-.is was the sheet used when the cards 
 ( sorted on B. ) 
 
 Qjio data for line C-l cones fron 7c Sh. 3, last line, 
 
 :.-nns 10, 12, 14, & 16. (This was the sheet used when the cards 
 were sor bed on C. ) 
 
 [ . data for line D-l cones fro:: Forr. 2, Sh. ■ , L st line, 
 colunns 12, 14, & 16. (This was the sheet used wh-. carde 
 
 on D. ) 
 
 B iron F r:. 2, Sh. 5, last lir . 
 , '■ 16. (This was the sheet used when the cards were sorted 
 have . to go on the 
 
 (i.O.: -c) on F m • 
 
 obtai: I igoree to go on tl "2 B o 
 
 3, we ■ oapatatj \o lata oi 1. Ji of the vari- 
 
 able eoiuputi il- 
 
 tip". - pf B, of C 4c. 
 
- 1U - 
 
 
 i 
 
 
 Form 3 
 on of Suras 
 
 of Ejtteni. 
 
 
 f 
 
 A 
 
 B 
 
 C 
 
 1 
 
 X 
 
 S 
 
 i-1 
 
 A-3 
 
 9913 -^ 
 
 2302.6 
 
 )5.0 
 
 ?3b7.3 
 27.1 
 
 9077.0 
 
 87~; 
 323-3 
 
 -4.0 
 3177.9 
 -1330 
 
 ul33.0 
 
 3393.2 
 
 2593 
 
 374 
 
 35100.5 
 27 
 
 B-l 
 B-2 
 B-3 
 
 
 2906. b 
 
 4874.0 
 
 ^737-1 
 
 l3o.9 
 
 2719.0 
 2303.3 
 
 -54.5 
 
 3100.0 
 
 3191.0 
 
 -91.0 
 
 197 - 
 1900b . 1 
 73b. 9 
 
 C-l 
 
 C-3 
 
 
 
 10099.0 
 
 7720.5 
 
 237i.; 
 
 1*051.0 
 
 -^qO.i 
 -515.1 
 
 333b. 
 
 5200.7 
 
 135.3 
 
 33-37.0 
 
 3097?. 9 
 
 2»*bl . 1 
 
 D-l 
 D-2 
 
 E-3 
 
 
 
 
 4730.0 
 
 27C- . 1 
 2081.9 
 
 3094.0 
 
 5077.9 
 
 16.1 
 
 19b9U.O 
 
 18332-1 
 
 1361.9 
 
 X-l 
 X-2 
 
 X-3 
 
 
 
 
 
 9S90.O 
 
 3503.^ 
 238o. d 
 
 23?73. c 
 
 208ob.2 
 270D.8 
 
 V 
 
roducts so obtained are list ctivttly in Columns A, B, C 
 
 on Forr. 3, line A-2. 
 
 Next the sum of tho second Variable, (3 in our example; or 705) 
 is put into the computing machine and multiplied successively by the 
 mean of B, of C, of D &c. &e«. The products so obtained are listed 
 respectively in Columns 3, C, D, >xc. of Form 3. line B-2. 
 
 Next the sum of the third Variable, (C in our example; or 11^9) 
 is put into the comtxiting machine and multiplied successively by the 
 mean of C, of D <£c. 6cc . . The products so obtained are listed respec- 
 tively in Tolur.ns C. D, &C. of Form 3t l ine C-2. 
 
 In a similar manner the computations are made for the other 
 
 lir.9s ending in "2",- Form 3. 
 
 "ote;- In practice it is most convenient to prepare Form 3 on 
 a sheet of paper vtfiich also carries Form 1 at the top. The figures 
 for marring the extension for lines of designation ending in "2" are 
 then before the operator. 
 
 Every item on a line ending in "2" is nor; subtracted from the 
 figure directly above it on a line ending in "1." (Note: naturally 
 -Id the minuend be greater than the subtrahend the difference ;7ill 
 be a negative value.) These differences are listed on the lines of 
 designation ending in "3 n . These lines are then transcribed to Form U. 
 The differences vhich have just been secured are the product 
 nts and squared standard deviations (times 1 T ) ; and are the r.eces- 
 • data for ;olution of multiple, and partial correlation coef- 
 ficients, or gross and net regression coefficients. The usual solution 
 be found in Yule: "Int: ion to Statistical Method." The solu- 
 D given in the following, however, is a "Least Square" method, fi. 
 conceived of and developed by :olley and Ezekiel of the Bureau 
 
- if - -1«- 
 
 . . -ultural Economic • ng th« 
 
 method is published ay them in the Journal of the An St 
 
 for December, 19?5. 
 
 ■ n - In case it ii jno-ica to ma*e the extensions d. 
 ner than to use . cards ana tabulating machine s--fre que; 
 
 the case when short series, such as time series, are oeing 
 a multi -columnar form should be used. In the six left-.'.. 
 List th . • i Luna headings would i< B f C, D t X, 4 8. 
 
 -lining columns should be headed: ir t AB, AC, AD «X, A|; 
 BD, EX, BS, C 5 , CD, CX, C3; D 2 , DX, D3; X 2 , XS. In the A 2 t 
 lt« the squares of the values in the A Column. In the Ab 
 write the products of the A items times their corresponding B ii- 
 
 When all columns have been extended, add them, li I 
 totals beio. jn their respective columns. 
 
 Find the means 'averages) of the- A, B, C, D, X, &. S colis- 
 hlltiply the sum of the A column times each of the means cf tne 
 ^x C , D, X, & S columns and write the products below the ra 
 
 AD, AX, & AS columns respectively. Multiply the i 
 B column times the means of the B, C ( D, X, & S columns and 
 the products below the sums of the B?] BC, BD, BX, ' & BS, colur-.-- 
 a similar manner extend the sum of the C column times the meani 
 C, D, X, & S, and inscribe the products in the C^ ( CD, CX, & 
 
 iumns. Also the sums of the D, & X columns. It is not cece_.~_ 
 to multiply the sum of the S column times anything. 
 
 subtract the last values listed in the A* 2 column anu 
 columns to the right thereof from the figures just above them. H tne 
 • - should be greater than the subtrahend the different 
 
 f a negative value. These differences are now to b* trail 
 rr«4 to a new sheet of the arrangement shown in form 4. The ij 
 :ences in the columns commencing a i th an "A" in their desi^n^t; 
 
 ferred to the first line-of form U, designated as line A- I 
 The differences in the columns commencing with a "B" in the) Jo- 
 
 nation (this of course includes the B-? col.) are transferred t 
 
 I of form The remaining differences are transfc. '-*- 
 
 BDner. Tne so arranged differences constitute the Normal kqua- 
 
 • square solution for the value of the net re 
 B.C. & D. on X. 
 
- 17 - 
 thil point the use of the Check Sun (Col. S) as a I ^ 
 
 the • to tr.is point may be shown: On Line A-l (Form !>) the sum of 
 the iters in columns A, B, C, D&X should equal the figure in Coluan 
 
 Line A-l), thus checking the extension and addition of all t 
 figures used in connection frith delving these values. 
 
 Sum of the following* 
 
 Should check to- 
 The Sum of the following: 
 
 Should check to: 
 
 Line 
 
 Column 
 
 A-l 
 
 B 
 
 B-l 
 
 B 
 
 ii 
 
 C 
 
 ■1 
 
 D 
 
 ii 
 
 X 
 
 B-l 
 
 A-l 
 
 C 
 
 B-l 
 
 c 
 
 C-l 
 
 c 
 
 ii 
 
 D 
 
 ii 
 
 X 
 
 C-l 
 
 Son of the following: 
 
 A-l 
 
 D 
 
 B-l 
 
 D 
 
 C-l 
 
 D 
 
 D-l 
 
 D 
 
 ii 
 
 X 
 
 Should check to: 
 
 sum of the following: 
 
 D-l 
 
 A-l 
 
 X 
 
 B-l 
 
 X 
 
 C-l 
 
 X 
 
 D-l 
 
 X 
 
 X_l 
 
 X 
 
 ild ck to: 
 
 X-l 
 
 and also "5" for "1" in the a 
 check may be secured It is est 
 
 oefon ftrriod to a : 
 

 
 
 
 
 (Form U) 
 
 
 
 
 
 
 
 
 NORMAL EQUATIONS 
 
 
 
 
 
 
 
 D 
 
 C 
 
 E 
 
 X 
 
 a 
 
 PM 
 
 
 • 
 
 23C 
 
 •1 
 
 J. 5 
 
 5.5 
 
 >9-8 
 
 273- 
 
 
 
 
 
 7M6.U 
 
 i.9 
 
 -.5 
 
 -91.0 
 
 730.9 
 
 
 
 
 
 
 W 
 
 -51 
 
 135.3 
 
 1.1 
 
 
 
 
 
 
 
 2081. 9 
 
 >.l 
 
 1361.9 
 
 
 
 > 
 
 
 
 
 
 2386.6 
 
 2706.3 
 
 
 
 
 
 
 SOtOTI 1 
 
 
 
 
 
 
 1 
 
 J02.6 
 
 .7.1 
 
 528.5 
 
 -133-5 
 
 259 • 3 
 
 
 
 
 3 
 
 -1.0000 
 
 -.O: 
 
 -.1427 
 
 .0530 
 
 -.1123 
 
 -1.2093 
 
 
 
 
 743.4 
 
 136.9 
 
 -34.5 
 
 -9I.O 
 
 73o.9 
 
 
 
 M 
 
 
 " -3 
 
 -3-9 
 
 l.o 
 
 -3.1 
 
 -32.8 
 
 
 
 5 
 
 
 748.1 
 
 133.0 
 
 12.9 
 
 -94.1 
 
 7C4.1 
 
 
 
 b 
 
 7 
 
 ■ 
 
 1.0000 
 
 -.1778 
 
 .1103 
 
 .1258 
 
 -.9412 
 
 
 
 
 2373.5 
 
 -513.1 
 
 135.3 
 
 24bl.l 
 
 
 
 8 
 
 
 
 -46.9 
 
 13.0 
 
 -37-1 
 
 -397.2 
 
 
 
 9 
 
 
 
 -23-7 
 
 14. S 
 
 Id. 7 
 
 -12 
 
 
 
 10 
 
 
 
 2307.9 
 
 -484.3 
 
 114. 9 
 
 1933.7 
 
 
 
 n 
 
 
 
 -1.0000 
 
 .2098 
 
 -.0498 
 
 .'400 
 
 
 
 
 2051.9 
 
 16.1 
 
 13ol.9 
 
 
 
 13 
 
 
 
 
 -7.7 
 
 l?.l 
 
 loi .- 
 
 
 
 
 
 
 
 -9-2 
 
 -10.4 
 
 73.0 
 
 
 
 15 
 
 
 
 
 -101. b 
 
 24.1 
 
 406.8 
 
 
 
 
 
 
 
 1963.4 
 
 I 9 
 
 2C08 . 1 
 
 
 
 17 
 18 
 
 
 
 
 -1.0000 
 
 -.0229 
 
 -1.0228 
 
 
 
 D 
 
 : .0229 
 
 x lo.l 
 
 
 
 19 
 
 1 
 
 /-.1258 
 
 .0U9i 
 
 .0C4d 
 
 : .054o 
 
 x 135.3 
 
 7.- 
 
 
 20 
 
 B: S 
 
 -.0097 
 
 .0025 
 
 :--1330 
 
 x -91.0 
 
 12.1 
 
 
 21 
 
 .1". 
 
 .001 6 
 
 -.0078 
 
 .0013 
 
 : .1079 
 
 x 259-8 
 
 
 
 22 
 
 
 
 
 
 
 P.M. 
 
 
 
 23 
 
 
 
 
 
 
 Sq. Root 
 
 6.94 
 
 
 24 
 
 
 3.uc 
 
 17 
 
 -3.05 
 
 : 250.79 
 
 
 
 
 
 Squart 
 
 J root of 
 
 233b. 
 
 (See Line 
 
 X-3, Col.jy 
 
 1*6 
 
 
 
 
 
 R e 
 
 qua Is 6 . 
 
 94 -k U 
 
 S or 
 
 .142 
 
On Form U Lines A-3, B-3, C- , ire th» 
 
 ing the normal I to be solved. The : lu- 
 
 tion is given on lines 1 to 23. as described below ' 
 
 On Line 1 wr irst normal Ion, i.e., copy Line A- s . 
 
 divide every item on Line 1 by * .st item of Line 1, r- • 
 
 the algeoraic signs end list the quotients on Line 2. In our example, 
 we divide by o. The algebraic svm of the items on Line 2 Columnu 
 
 B, C ( L t ani X, should equal the quotient appearing in Column S on 
 the same line. This will not always cneck to the last digit, owing I 
 the dropping of places in the division. Nov.*, draw a line un-ier the 
 figures just written in. On Line 3 t;opy in the second normal equation, 
 that is, copy Line B-3. Now put the figure on Line 2, Column B (i.e. 
 into the multiplying machine and multiply it consecutively Dy 
 the items en Line 1 in Columns E, C, D, X, and S, listing the products 
 in the respective columns on Line U. In our example, we mult 
 by 27.1 by 323.3 by .133.3 by 239.9 end by Zlib.j, giving as quotients 
 appearing on Line k t i.e., giving -.3, -3-9; l.o; -3 1; ~3- ■ 
 Now add the items on Line h to th* 3 ite.-Ls immediately above on Line 3, 
 giving Line 3. Careful attention must be given to tne algebraic signs. 
 Now, divide the figures on Line 3 by the first fie^ure on Line , 
 reverse the algebraic signs, listing tne quotients on Line 6 . In our 
 exar :de by The algebraic sum of irst four it^ems 
 
 -;uld check to the last item of the line, 70^.1, in I 
 -■^n. In liiCe manner, the ilgebraic sum of the i on 
 
 •This is the "Doolittle Method Bee Oscar S. Adams - "Geodesy - Ap- 
 
 ■ on of f Least ~ -> to the Adj. 
 
 tion." - I?!"). Special n #28, Geodetic Survey. 
 
D - 
 
 Line o should check to the last item or. Line u, or -.9^12. Now copy 
 
 D the third normal equation on Line 7; i.e., -one C-3. Put the 
 
 number la 1 )3 -mn on tne secona lias, -.1-27 lato tne mull ng 
 
 machine ana multiply it consecutively cy the items la the C, D, X, n 
 S columns of Line 1, listing the products in the corresponding columns 
 Line 3, giving careful attention to the algebraic signs. Next, put 
 the item in the C ^ lumn of Line o, -.1778, i-'ito the multiplying machine 
 ■ad multiply it consecutively oy the C ( D ( X, and S column figures on 
 Ljne ), g the products in their respective columns on Line 9, 
 
 giving careful attention to algebraic signs. Now, add together for 
 each column the values in Lines 7, 6, and 9, giving Line 10. The ii" 
 items of this line should check to tne. luot, similar to the case : 
 Lines 5 a ™i 1. Divide each of the items of this line by the first i* 
 in the line, that is, divide by 2507.9, reverse the signs and list the 
 quotients on II. In a manner similar to Line o, the first ite-^ 
 
 en this line should check to the last when uaaea together. Draw anotr.cr 
 line. jt\ Line 12, ".rite the fourth normal equation, that is copy Line 
 D-3. Put into the multiplying machine the value on Line 2, Column 
 
 I multiply it consecutively by the D, X, and S column values of Line 
 1, listing the prouuets on Line 13 in their respective columns, giving 
 
 -oful attention to the algebraic signs. Next, place the value on 
 Line o column D into the multiplying machine , .1103, and multiply con- 
 sec . • y by the values on Line 3, columns D, X, and S, listing the 
 
 ir re. re columns on Line 14, giving careful atten- 
 
 tion to tne alg» Next, put tne figure on Line 11, column 
 
 multiplying machine and multiply consecutively oy 
 the values on Line 10 columns i), X, ana S, luting the products in tneir 
 
- 21 - 
 
 respective cr>luuis on Line 15, giving ciirtfal attention to a] nL/ns. 
 
 t, r • I - 12, - ; . !■'• • L6i 
 
 -lie ■* Lren 
 
 Lc su of the fii 
 
 .1. 
 Lret ite:. on " l e , 19? '. . I 
 
 . . list the 17. ' i of 
 
 17 should chock t 3 " -. 1". 1 1 w for 
 
 s 11, 6, and 2. Ue have now finished the ■forward* solution for tho 
 
 on8i • - tho 
 
 tho i Lutioo to r 05 vri- 
 
 -" 1. - will t Wq are now rcidy for V 
 
 bf»c Lution, lii.es 1 . 
 
 - I to tK ".".-"" 
 the ■l£n. T.i~ valni i; the net r ."f icier. * ic 
 
 D X. Next in Golunn g lines IB to ?1, Inclusive, list i - rse 
 order the values in colUED X, Lircs JL-3, B-3, D-3, D-3. U Xt [ :■:. lino 19 
 Do In c . write 1 revors J Ob 
 
 :.- ^luc on Line G, colucc X, revere:- 
 
 ai.- . - . .-.-.., ri1 •. c h :. Xt rev 
 
 the si ■ 18, c . - oltipl; 
 
 " - 
 
 ict 1 01 
 
 c- 
 , • . • . 
 
 val 
 
Next, add toget.vr tne \uiues on Line 19 in columns C and L 
 
 sun luran X, Line 19. ~ ' - last sum 1. I ion coefficient 
 
 of thf C on X. 
 
 Put a coefficient C - on X into I 
 
 multiplying .ie, and, having 1 can I or al£- it 
 
 times I ie j t i . I iting the product r. the 
 
 same line in column rM. Then, multiply it by tne values In -- on 
 
 Lines nd A-3, .-.nting the products on Lines 20, 21 ana. 2k, respec- 
 
 tively of the same column. No. id the values on Line 20, columns B, C 
 
 1 C together ..riting tr.e sum in column X on tne same line par- 
 
 ticular regard to algebraic signs. This 1-i^t ran written on Line 20 in 
 column X is the net regression coefficient of the variable B on X. Place 
 it in the multiplying machine, and, having a care to algebraic signs, mul- 
 tiply it tirr.es th° value listed beside it in column S, writing the proauct 
 on the same Line in column FM. Then, multiply it by the values in cclu 
 B on Lines 2 and A-3, listing the products respectively on Lines <_' 
 
 - of the same column. Now, add n Line 21, and ir. columns 
 
 '., B, C, and D, writing the sum in Column X on the same line. .urn 
 
 net re^rrssion coefficient of A on X. Place it in the iying 
 
 machine a . tiply it times the vulu ide it in column S, hav- 
 
 ing a l r algebraic signs, and libt tne product in column FM on I 
 
 same line ill altiply it times -lue on Line A-3, column A, list- 
 
 roduct in the same column on Line 2^. No.' values on Lj 
 
 columns A, B, C, D I toge* ..ould e^ual the value 
 
 Line A-), column X, which serves as a check upon the a 1 ?n 
 
 r net regression C X. There is a d .ice 
 
of ? between the two va . wur pie, A greater ... y may be se- 
 
 the arithmetic to a greater number of pla^ ..jhout I 
 
 entire solution. It was deemed expedient to make the example as simple as 
 possi i 
 
 We have ao the net regression coefficients, ehicfa ire es- 
 
 sential to the forming of the regression equation for predicting or esti- 
 azi' oes of S. To ascertain to how great a degree these preaic- 
 
 ~ns cc: -o the actual values, it is necessary to obtain some measure 
 
 of agreement between them, the predicted and the actual. This measure of 
 agreemen* - .e coe: .t of multiple correlation, R, defined on Page 1. 
 
 To secure this coefficient of multiple correlation, add the values in the 
 PM coluaan, listing the sum on Line 2.2.. Next, secure the square root of 
 
 .5 sum, given on Line 2}. Finally, secure the square root of the value 
 listed on Line X-3 ( column X, listing this in the PM column on Line 25. 
 
 valur - .led into * lue immediately above it on Line 23, gives 
 
 the coefficient of multiple correlation. 
 
 There are certain aids and other checks in the solution which can 
 be app liea to h^lp in locating errors. The diagonal terms of the norma. 1 
 equations (23C2.0, 7 4 ^--, 2373-5 £»> are always positive in sign. In me 
 ing the s-lution the figures listed immediately below these figures (to be 
 added to them in the course of the solution) are always negative in sign, 
 .ppearing above the -1.00000 terms are always positive / i .e. 
 
 1, 2307- I -3>)- 
 
 icy is increased if comparatively small diagnonal terms are 
 3 can be controlled by controlling the origins 1 coding.) 
 The Product Moreen* 22, is al-vays positive in sign. 
 
UNIVERSITY OF FLORIDA 
 
 3 1262 08918 7172 
 
 - 2U - 
 
 THE REGRESSION E- 
 
 The final step in the arithmetic ia to tha "r ion," 
 
 or citing" or " equation as it is vario 
 
 .e down t: ;ion coefficient of A on X; ir. 
 
 \ 
 
 ~"3. P> alue write what V7as done algebraically in 
 
 thU \ ill be of the forr., ~_: 15. . (See page 
 
 6, note (1)). Then furtr rite the subtraction of the average of 
 
 en from form 1, enclosing all in pa: • . . 
 
 ) 
 
 r c form so far will look like this: .1079 ( A - 35 _ <,, ) 
 
 / ■» " -7 .dIhO ( 
 
 In an exact! nner treat the B, C, D, & X scriec. (Disregard 
 
 1 be no regression coefficient for the X B< 
 of the A, B, C, & D expressions should be equated to 
 xpression as follows: 
 
 ♦ .1079(^=21 - 7.61U) - .1330(3-30 - U.1226) ♦ .05U6( C ; 120 - 6.71 
 
 
 ) 
 
 ( 
 
 D-23S 
 
 .:'22?(— c^--3- 9766 ) 
 ° ) 
 
 ( ° ) 
 
 H -5263 
 
 This is ' 1 " :■ Ion equation. It is only nee on to 
 
 . uate for X, involvin. -, to put .uation 
 
 : ul foi 
 .1079 A - .0998 B • .0137 D ♦ 9^-0552 
 
 ia is • definition 
 
 of R in the note on par.o 1*