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13 UN or PUN= CARD TAnJLAL!IN EQUIRPMJN 3 MuQimn cwsnoTIu n0oxms a U S DEPOSITORY a77III i Col lected and prepared for the use Iatimtilcans of the Bureau. Bradford B. tith, In Charge M iahis hbulation and Computing Section Washington, D. C. October, 1923. It: S i;,.: V C The use of pinched card tabulating e equipment is not new: I t mae used a number of years ago in the Bureau of the Census for the preparation of Correlatioa tables. More recently it "has been used in at least two of the, Bureaus of the Department of Agriculture in a similar way. The writer in no sense claims to be the sole author of the mthods herein set forth except insofar as the special tech nique Is devised to suit the equipment and problems of the Bureau of Agricultural Economic s. Especial credit is due to Messrs. Tolley and Ezekie. of this BTeau for devising the least square method (page 15 et seq.) of approaching correlation problems. The writer also wishes to express his appreciation to them for going over the following text, their suggestions 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, when these esti mated values are determined mathematically from two or more differ ent (independent) series.* The following pages describe the forms and arithmetic used in computing the value of B and in deriving the "regression equation" or forecasting formula. A five variable (four independent) example is given. *The coefficient of Multiple Correlation (B) may be defined as the coefficient of correlation (Pearsontan) between the dependent vari able and corresponding estimates thereof, computed from the inde pendent variables, i. e. if X', the estimated value of X, equals blA plus b2B and b3C....... plus a constant, where A. B, and C, are * *em tbe3dSpendent *varlblt n bl, b2, b3, arethe net regression coefficients, B equals the mean product of the deviations of X and X' from their respective averages divided by the product of the standard deviations of X and X1. B is also equal to the standard deviation of X' divided by the standard deviation of X. Also, the standard deviation of XI squared is equal to the mean product of the deviations of X and X' from their respective averages. 1 is esao here between ten and thirty, depeting upon t iscretiom 9f the person conducting thi investigation. The seller the vars0 tion, the simpler is the process of performing the calculatUos, for the whole success of using Hollerith iachines* lies in the grouping of like values.*" On the other hand, tod geat a groupingor re duction of variatJonremults in reduced accuracy. TAo coding my be accomplished either by subtraction or by division or both: Ius if the series under consideration varies between 120 and 175, Snb tract 120 from all items, and divide the differences by 5 thus re ducing the variation to from 0 to 11. This coding of the series in no way invalidates the calculation of R. Coding by division slightly reduces accuracy, but subtracting in no way affects accuracy., On the following pages are given the coded series used tn the example. A, B, C, & D are the independent variables. X is the de pendent variable (the values of which it is desired to estimate, or "predict".) The series should be so arranged that the dependent variable is placed in a column to the right of the independents. After the coded series have been listed, crossadd the values for A, B, C, CHECK and D, & X for each observation and list the sum in the "Check Su" Sum cnlun on tne right, Col. S in the tabulation. This check sum column is rnot essential to the solution but is of considerable aid in check ing the arithmetic.  l inhre are several types of punched card sorting and tabulating Ia chines. Those used in this bureau are for the most part Hollerith Machines. SMrrer;var, if the extensions are to be made by hand, figures that can be readily multiplied mentally greatly facilitate the process. w3 ea~eea~.. .   S10 14 7 00 10 31 2 3 10 11 3 11 38 10 5 12 00 6 33 .10 3 11 2 10 36 5 10 6 10 3 5 34 6 6 9 1 7 10 33 7 3 13 10 4 0 20 6 0 7 5 1 5 1i 9 3 6 10 81 28 to10 3 1 1 5 4 25 11 6 6 a o 6 26 12 10 6 2 4 6 28 3 1 3 5 2 6 4 20 li4 12 3 5 0 2 22 15 12 6 0 12 2 32 16 12 4 5 4 6 31 17 6 3 11 3 3 26 18 10 12 8 11 11 52 19 6 8 8 3 25 I0 10 3 4 3 5 25 21 .10 3 11 12 12 48 22 10 3 10 3 7 33 23 3 5 00 10 4 22 24 6 3 8 00 7 24 25 0 1 11 7 5 24 26 10 5 8 0 5 25 27 10 4 0 9 4 27 28 10 6 12 6 4 35 29 12 3 10 12 1 38 3 10 5 12 2 0 29 31 12 9 7 1 4 33 32 10 5 11 3 2 31 33. 7 6 4 4 11 32 34 6 4 10 0 12 32 35 7 4 12 6 12 41 36 3 5 0 10 a 26 37 3 5 5 4 2 19 35 3 1 0 12 4 20 3 0 1 8 1 13 12 3 11 9 4 39 41 10 1 8 0 12 31 42 10 3 10 1 12 36 6 6 5o 5 0 16 :110 3 11 1 5 30 '45 12 5 54 i123 '67 5 5 13 21 417 12 7 10 0 0 29 48 10 2 7 3 4 26 19 10 3 7 0 6 26 *50 10 3 4 4 U 3 51 12 4 5 1 2 24 52 3 4 4 3 '4 1 53 10 4 5 3 0 22 54 3 9 10 0 1 23 62 10O 6 1 27 Wco'raU O. A Ci E i x U    56 6 3 7 1 5 2a 57 10 3 11 2 11 37 58 6 6 5 0 3 20 59 10 3 5 0 12 33 60 12 3 4 0 23 61 10 7 5 0 2 a4 62 10 3 20 63 6 6 8 2 22 64 12 3 10 0 12 37 65 10 1 4 7 26 66 12 6 5 1 10 3 67 3 2 5 3 0 13 65 12 S 11 11 12 54 69 6 2 10 7 2 27 70 12 6 11 1 3 33 71 10 7 8 0 5 30 72 12 4 7 3 2 2A 73 10 3 10 4 3 30 74 12 4 0 0 24 75 12 5 7 3 12 39 7O 10 1 7 1 7 26 77 3 6 8 b 2 25 7 3 11 5 0 27 79 12 6 08 2 36 so 3 3 11 2 8 27 a1 10 6 7 b 1 30 32 10 4 1 6 27 83 b 3 8 5 3 30 84 10 3 11 7 2 33 65 10 4 2 7 5 28. 66 10 4 11 1 .1 27 37 6 3 51 6 21 M8 12 6 0 0 26 39 12 2 0 12 12 38 90 o 2 8 11 12 39 91 0 0 12 8 28 9p 10 42 b 2 N 93 3 10 3 a 22 U 6 3 1 7 2 19 95 3 1 7 1 8 20 90 10 3 3 8 12 41 97 o 0 0 4 11 21 93 10 6 0 4 25 39 3 7 7 3 4 24 100 7 S 3 12 0 31 101 u 3 3 23 J 10T3 3 1 2 21 103 0 3 0 3 1l 104 0 b 2 10 1:rj 0 46 0 1 2 7 100 3 b 0 7 6 22 1C7 10 1 10 1 0 22 100 10 o 3 2 25 Sf4 7 0 b 27 5  5 s ees e e s sea  as a 5  Sbod Ibo. A B C D Z S asno se  asss"" Q  110 0 1 0 3 0 4 111 0 5 2 3 0 10 112 10 0 11 2 2 25 i 6 4 0 7 0 13 1 0 5 7 1 3 115 3 3 8 0 7 21 116 10 5 11 1 0 27 117 10 1 0 7 2 20 118 10 a 5 3 5 25 119 o 6 12 8 9 35 120 6 3 11 1 9 30 121 12 3 5 4 0 24 122 10 5 11 3 0 29 123 10 3 8 0 4 25 a 0 4o 5 3 4 26 125 10 6 11 10 1 38 1.26 3 6 11 4 4 28 127 10 6 10 12 a 40 12 6 4 0 7 2 19 129 10 3 11 3 7 34 130 3 S 8 0 1 20 131 7 6 4 0 4 17 132 3 6 4 4 3 20 133 12 4 4 4 6 30 134 10 9 8 0 5 32 135 6 3 5 4 22 136 6 5 2 0 1 14 137 10 6 0 9 6 31 138 10 3 7 1 4 25 19 3 2 11 4 0 20 lUo 0 4 7 0 4 15 141 12 0 12 4 12 40 142 6 3 5 12 0 26 143 0 10 11 1 28 144 10 3 11 6 7 37 145 3 b 11 1 11 32 146 10 10 4 4 31 147 6 3 8 6 5 28 148 10 6 8 1 6 31 149 10 4 12 4 S 38 150 3 3 0 10 3 19 151 10 3 7 8 2 30 152 6 1 2 3 12 24 153 10 4 11 1 4 30 154 10 3 10 7 0 30 155 10 2 4 3 11 30 156 12 a 10 1 2 33 157 12 7 7 3 1 30 155 10 3 4 4 2 23 159 3 6 5 1 2 17 160 10 3 7 3 4 27 161 10 3 7 1 3 24 162 0 2 6 2 0 12 163 o10 4 11 1 2 2a 16& 10 3 10 4 0 27  (  e cordt, No A ......B ... c D.5"." 165 12 4 0 6 4 a6 166 6 4 5 a4 4 a6 167 10 4 S 3 2 27 165 10 4 0 7 4 25 169 3 4 11 3 5 26 170 3 1 4 3 11 22 171 7 0 0 11 5 23      (1) Original series minus 35 and reminder divided by 3 (2) I 350 4 (3) t2 6 (5) 2" 935 53 (5) 'II 935 aiI 33 ~ 7 SAfter the data has been coded and listed as on the preceding HPtf the values are then punched on punch cards, the record number i also being punched for the purpose of identification. There are 9bi .rtyfive columns on a punch card. In our example the record number vas punched in colums 3123; A in columns 345; B in 367, C in 3890, D in 401; x in 423; and S in 445. One card is used for each line for each observation, that is. After the cards have been punched, they should be sunaed: The sumn of A, of B, of C, of D, of X and of S should be obtained; also the number of cards should be counted. The results should be recorded : in some such form as follows: !Form 1  Suams and Means of the Variables.  Item : A : B : C : D : X : 3 171 1302 705 1149 680o 774 4610 i: ~Means: ::::i.23: 99 Seas7.6140 4.1 : 6.7193 : 3.9766 4.5b3 6.9591 The use of the Check Sum first becomes apparent here: Evidently the s of the sums of A, B, C, D, & X. should equal the Sumn of S; which is the case. The same is true of the means (averages). This checks the first additions used in building up the check sum itself; 1lt also  checks the accuracy of the punching; 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 coding by punching the orilgal values upon the punch cards. Then sort the 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 with the assigned class valuesuch as 0, 8, &c.. The check sum for the individual record then can be prepared by showing each card separately in the tabulator, adding across, and subsequently Ching upon the card, after which the procedure is as given above. The next step is to sort the cards upon the first variable, A;., the form for recording the tabulation .is given below: U' j SVar iable No. Class of A B C D Valug Items Sum&xt SuM Ext. aum Ext. SBn Ext. Sum Ext. &i Sm r) 5 ( .Lk ilL 4L (o 20.12) 1i i23 (1_i W51 iL .17 112 I"i __________________ ___ I ___________ ________ ______ _ i_ _ * lb. card@ being sorted nn the first variable, A. group them i+! S "into packsall the cards ;f fthe lowest value of A in the first pack, of the next value of A itx the second pack Ac. ac.. Li st in column : (l)IbYr 2the value of A in the first pack. Tabulate this pack. 1 Okn the first line in column (2) write the number of cards in the : Iack; in column (5) write the sum of the values of A in this first I+ pack; in coluam (7) write the sun of the values of B in this first Ip: pack; in column (9) the sum of the values of C in this pack; in colna (11), D; in colum (1i), 1; in column (1i) 5. Take the ... second pack, list the value of A in this pack on the second line of :, ...the form; and list the corresponding sum values as for the first I. pack. Repeat until all packs have been so treated. When this is completed mkJ the extensions for columns 6, 8, 10, 12, 14, & 16 as follows: Multiply the values listed respectively in columns 5, 7, 9, 11, 13 & 15 on any line each timesthe value listed on the same line in colam 1. List the products so obtained in columns 6, 8, 10, 12, 141, & 16 respectively. Do this for all lines. Add coluans 2, 6, 8, 10, 12,. 14 & 16. Take the cards and sort them again; this time on the second variable, B. Take a second sheet (Form 2; Sh.2); divide the sorted cards into packs, according to the values of B and list these values of successive packs in column 1. Tabulate, list and extend in a + manner exactly similar to that when the cards were sorted on A, ex cept no figures need appear in the two A columns 5 & 6. H!l : :i iiitil l Sort n C and proceed as for B on a n &heet (Form 2 S M. 2 No figures need appear in the A, or B colms; C0olums 5, 6, l7, A 0!7 Sort on D and repeat (Form 2; Ah 4): No. figures in the A, , or C colums: Columns to 10 inclusive. Sort on X andt repeat (Form 2, Sh 5). No figure in the 3* C, or D columns; Coluns 5 to 12 inclusive. (Note: The reason that an increasing. number of columns can be omitted is that to make the extensions and mn them would give fig ures already computed: Thus if we sort on C and extend its valtS times D. adding the extensions, we arrive at the same figure as If wk had sorted on D and extended its values times C.) In case difficulty i a encountered in making the figures check to the check sm  e plained later in connection with Form 3it may be advisable to O the extensions here directed to be omitted, for the sake of comprl sons and to help locate the errors.) Following are the tabulations of the five sorting*s ad In per forming the above steps for our example. Note that in each case a check is afforded by adding up column 2. This should add to the toatl number of cards in the problem as shown by the data on form 1. A further check may be afforded by adding the sum columns for each vari ablecoluans 5, 7, 9, 11, 13 & 15. These should on every sheet add to the same corresponding figures given on form 1. lajg ,l*J (5) 0 90 16s 42 690go 312 0 2034 1155 20470 =a 37M55 Sinr ted nn U nitm.A n .1. 5 13 12 45 30 19 .31 6 4 4 1 (171) Form 2 Sheet 0. 0 2 135 120 95 1l6 42 32 36 10 12 0 13 ~4s 405 480 475 iin6 294 256 324 100 144 3655 24 70 68 353 180 121 209 42 37 26 11 (1149) 0 70 136 1059 720 605 1254 294 296 234 110 96487 4_74 29 50 74 169 o109 59 144 7 17 9 3 11 (680) 0 50 148 507 436 295 864 49 136 72 30 129 2719 31 73 57 228 119 66 123 17 18 20 11 11 (774) 0 7 684 476 330 738 119 144 180 110 132 3100 122 284 295 1273 777 468 588 155 137 121 38 15) 0 284 590. 3108 23W0w 5325 1085* 1096 1089 350 S19743 19743 Sorted on C _ Form 2 Sheet M. 21 3 7 13 25 19 26 21 25 (17%j 0 14 52 125 133 208 210 308 0 3 625 205 2100 33899 1102 10099 166 22 57 31 93 109 .35) (6so) 0 22 5g 176 285 273 LBE 930 1199 42051 4051 89 13 30 79 82 90 127 6s 114i (774) 0 13 6o 316 41o 630 1016 ioi6 680o 1551 53366 5336 454 65 148 333 Z^9 748 627 894 an (4610) 0 296 1332 2515 ;34 5984 6270 9$34 33437 k I ) 190 67s 659 165 2047 (61) 12 30 28 6 69 (171) 0 270 1008 294 69o00 i744 12216 46 134 107 25 270 (71235) (705) 0 402 642 175 2700 14765395 5395 67 194 162 29 522 (1149) 0 582 972 203 5220 2109 9077 143 136 119 35 236 (6o80) 0 408 714 266 2360 1a96 5044 la&.; 34 124 133 31 329 It2l 0 372 798 217 3290 6153 (3) (9) (10) (111) () g (1) 13 The next step is to transcribe figures from the shouta (1 to 5) of Tun 2, to another sheet of the forum shown on the following peae (orn 3). On line Ai (lbzz 3) colwma A list the figure taken from Forrn 2; Sh. 1 Ooi. 6. last (total) line. the other figures on line Al aro taken from the sane shoot (lForn 2; Sh. 1) last line, coluzzms 8, 10. 12, 14 & 16. the figures filled into form 3 apply to our cxar aplo, so they ray be traced through the various forms. The data for line B1 comes from Form 2, Sh. 2, last linu, columns 8. 10, 12, 14, & 16. (TBits was the sheet used when the cards were sorted on B.) 'The data for line C1 cones from Form 2, Sh 3, last line, coluMns 10, 12. 14, & 16. (This was the sheet used when the cards were sorted on C.) The data for line D1 comes from Form 2, Sh. 4, last line, colmus 12, 14, & 16. (This was the sheet used when the cards were sorted on D.) The data for line XI comes from Form 2, Sh. 5, last line, ''lu.2.s 14, & 16. (Thlis was the sheet used when the cards were sorted on X) W U have nr.. 'filled i. rll 't has to go on thu lines ending 13. "1" (i.e.: A., 3I &c &c) on Form 3. Tb obtain the figures to go on the lines ending in "'" on Forn 3. we cnke computations fror the data on Form 1. The suLi of the vari able A 1'3iG in our example) is put into a coiputif mnachinu and mul tiplied sacccssivoly by the mean of A. 9f B. of C &c. *c. Transcripton of Sm of Ixtencionm. Line # A B , Ai 12a6.o 5395.0 9077.0 5044.0 6153.0 37885.0 A2 9913.4 5367.9 5714a.5 5177.5 5893.2 35100.5 A3 2302.6 7.1 3258.5 133.5 259.5 2784.5 B1 3655.0 4874.0 271 9.0 3100.0 19743.0 B2 2906.6 4737.1 2803.5 3191.0 19006.1 B3 748.4 136.9 54.5 91.0 736.9 Ci 10099.0 4051.0 5336.o 3J1437.0 c2 7720.5 45bo0.l 5200.7 30975.9 c3 2378.5 518.1 135.3 2461.1 D1 4786.0 3094.0 19694.0 D2 2704.1 3077.9 i5332:1 D3 2051.9 16.i 1361.9 x1 5590.0 23573.0 x2 3503.4 2866.2 x3 2386.6 27o6.8 V 15 the products so obtained are listed respectively in Columns A, B, C &c., on Form 3. line A2. Next the sum of the second Variable, (B in our example; or 705) is put into the computing machine and multiplied successively by the Man of B. of C. of D &c. &e.. The products so obtained are listed respectively in Columns B, C. D, &c. of Form 3, line 32. Next the aum of the third Variable. (C in our example; or 1149) is put into the computing machine and multiplied successively by the mean of C, of D &c. &c.. The products so obtained are listed respec tlvtely in Columns C. D, &c. of Form 3, line C2. In a similar manner the computations are made for the other lines ending in "2, Form 3. Note: In practice it is most convenient to prepare Form 3 on a asheet of paper which also carries Form 1 at the top. The figures for making the extension for lines of designation ending in "2" are then before the operator. Every item an a line ending in n2". is now subtracted from the figure directly above it on a line ending in "I1." (Note: naturally should the minuend be greater than the subtrahend the difference will be a negative value.) These differences are listed on the lines of designation ending in "3" These lines are then transcribed to Form 4. The differences thich have Just been secured are the product moments and squared standard deviations (times I1); and are the neces sary data for any solution of multiple, and partial correlation coef ficients, or gross and net regression coefficients. The usual solution may be found in Yule: "Introduction to Statistical Method." The solu tion given in the following., however. Is a nLeast Square" method, first conceived of and developed by Wssrs. Trolley and Ezekiel of the Bureau of Agricultural SEconomics. An aruticl ascribing the theory of 1W S method is published by them in the Journal of the Am. Stat. Asoo., ::.. for December, 1923. Note: In cas it is decided to mako the extensions by head rather than to use punch cards and tabulating machine sfrequsntll : the case when shabort series. such as time series, are being analy., a multicolumnar form should be used. In the six lefthand colua) list the data; column beadingo s would b1 B, C D, X.& 8. The remaining co Lims should be headed: A, XAJV AC, AD# AX. AS; Bic, SGI ED, BX, B5; Cc, CD. CX, CS; D', DX, DS; X, X5. Inthe A column write the squares of the values in the A Colum. In the AB column write the products of the A items times their corresponding B items, Ac Sc. When all columns have been extended, add them, listing t1e totals below, in their respective colbms. SFind the means (averages) of the.A, B, C. D, X, & &aolunls. Multiply the sum of the A column times each of the meane of the A, B C D, X, & S columns and write the products below the 8us of the A. AB, AC, AD, Al, & AS columns respectively. Multiply the sum of the B column times the means of the B, C, D, X, & S columns and write the products below the sums of the 32 BC, BD, BX .'& BS33, colums. In a simi lar manner extend the sum of the C colum times the means of C, D, X, & S; and inscribe the products in the C2, CD, CX, & CS columns. Also the sums of the D, & X colums. It is not nt&V1  To multiply the sun of the S column times anything. Now subtract the last values listed in the A2 column and in columns to the right thereof from the figures just above them. If the minuend should be greater than the subtrahend the. difference would naturally be a negative value. These differences are now to be trnse ferred to a new sheet of the arrangement shown in form 1. The dif ferences in the columns comnencing with an "A" in their designation are transferred to the first lineof form 4, designated as line A). The differences in the columns commencing with a "B" in their desig nation (this of course includes the B2 col.) are transferred to line B3 of form 4. The remaining differences are transferred in a sim ilar manner. The so arranged differences constitute the Normal EVLua t ,o"Bor a least square solution for the value of the net rg4s.tO . of A, B, C, & D, on X. F *  At this point the ue or tMe ,imAcE 1am IeL. Q a I EO 6 a. the work to this point may be shown: On Line Ai (Forn 3) the Sm of the items in columme A. B, C. D & X should equal the figure in Column 8 (Line Al), thus chcking the extension and addition of all the figures used in connection with d4oviag these valuese. Line Colunm The Sa of the following' A1 B B1 B .. C D H 7 D X Should check to: B1 S The Sum of the following: A1 C B1 C C1 C D n X Should check to: Ci S The SaB ofthe following: Ai D B1 D Ci D D1 D P X Should check to: D1 S The sum of the following: A1 X Bi X CI X D1 X X1 X Should check to: X1 S. By substituting "2"' for "I"; and also "3" for "1" In the above schedule, a further check my be secured. It is essential that the values should all check, before the work is carried to a further stage. J, 2302.6 I 27.1 748.4 328.5 136.9 2375.5 A D 133.5 &r.5 518.1 2081.9. I 259.5 91.0 1)5.3 1.6.1 21afi6 9'86.6 2IRS..6 flt~F .8 mttiYInV . 14 7361.9 1361.9 27069g 2302.6 1.0000 27.1 .0115 .3 7148.1 1.0000 1 2 3 4 5 6 7 5 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 33.5 1369.9 e39 3.9' 133.0 .1778 237s.5 46.9. 23.7 2307.9 1.0000 6 l J 1. .0495 .0097 .0078 248.45 3.60 17.9 Square root of 2386.6 133.5 259.9 ,0q .112 8W.5 91.0 1.6 3.1 82.9 94.1, .1108 .lc1 58..1 135.3 19.0 .37.1 14.8 16.7 4.5 3 114.9 .20O8 .049 )58.9 16.1 7.7 15.1 9.2 10.4 L0o.6 j24..1 163.4 *44:9 0000 .o229 D : .0o29 .o048 : .0546 ,0025 .1330 0013 : .1079 a 2784.5 1.20NO 736 .9 32.8 704.1 397.2 125.2 1935.7 .8400 1361.9 161.4 76.0 406.6 2006.1 1.0228 z 16.1; 2 135.3 : x 91.0 : x 259.8 : P.M. Sq. Root : 3.05 : 250.79 (See Line X3; Col.r equals 6.94  45.8 : o.8 or .142 14=t A3 33 C3 D3 X3 SC: B: S .1258 A .1128 .0016 a 1,L 6i.: 6$9 m mm maim 7 w~ Ai MCM tion is given on lines 1 to 25, as described below.* On Line 1 write the first normal equation, i.e., copy Line A3. now, divide every item on Line I by the first Item of Line 1, reverse the algebraic sips and list the quotients on Line 2. In our emanple, we divide by 2302.6. The algebraic sum of the items on Line 2 Colunns Al B, C D, and I. should equal the quotient appearing in Column S on the ama line. This will not always check to the last digit, owing to the dropping of places in the division. Now, draw a line under the figures Just written in. On Line 3 copy in the second normal equation, that is, copy Line B3. Now put the figure on Line 2, Column B (i.e. .0118) into the multiplying machine and multiply it consecutively by the items on Line 1 in Columns B, C, D, X, and S, listing the products in the respective columns on Line 4. In our example, we multiply .0118 by 27.1 by 328.5 by :133.5 by 259.9 and by 2734.5, giving as quotients the items appearing on Line 4, i.e., giving .3, 3.9; l.6; 3.1; 32.5. How add the items on Line 4 to the items immediately above on Line 3, giving Line 5. Careful attention must be given to the algebraic signs. sow, divide the figures on Line 5 by the first figure on Line 5, and reverse the algebraic signs, listing the quotients on Line 6. In our exampl, e divide by 748.1. The algebraic m of the first four itdes on Line 5 should check to the last Item of the line, 704.1, in the S Column. In lik mnmwr, the algebraic sum of the first four items on aac aeeee 'This is the tDoolitt$1 Method" See Oscar 5. Adams "Geodesy Appli cation of the Tbooty of Least Sqmure .to the Adjustment of Triangula tion.a 1912. special publication 52, Geodetic Survey. machine and multiply it consecutively by the items in the C, D), X 6 col=as of Line 1, listing the product s in the corresponling cop on LIve 8, giving careful attention to the algebaic algps. INt,.I".^ the item in the C colm= of Line 6, .1778, into ,the multiplying af.i t .."ii ........ .. and multiply it consecutively by the C, D. 1, and 8 colv= figures e9 ii Line 5, listing the products in their respective colnsm on Limn 9,s .:: *iliipi giving careful attention to algebraic signs. Now, add together fo each column the values in Lines 7, 5, aad 9, giving Limw 10. the ftr"s :i items of this line should check to the last, simlar to the cae for Lines 5 andM 1. Divide each of the items of this line by the first its in the line; that Is, divide by 2307.9, reverse the sigmas and list tM quotient. on Line 11. In a msanmer mimllar to Line .6, the first items on this line should check to the last when added together. mDrw mothW line. On Line 121 write the fourth normal equation, that Is coy Line D3. Put into the multiplying machine the value on Lined 2, CoLa D, and multiply it consecutively by the D, 1, and S column values of Lim 1, listing the products on Line 13 in their respective columns, givibT careful attention to the algebraic signs. Next, place the value on Line 6 colunn D into the multiplying machine. .1108, and multiple' con secutively by the values on Line 5, collins D, azd S, listing the products in their respective column on Line 14, giving careful atten tion to the algebraic signs. Next, put the figure on Line 11, column D. .2.09, into the multiplying machine q4d multiply consecutively by the values on Line 10 colms D, X, and.S,8 listing the products in their I Ii reaCstive *ohnm on AA 16. giving careful attention to algebraic siM. Aitt, ndtld or colwzis 3) nDA S the vrlucs of Lines 12, 13, 14, rnd 15, S 11sting the algebraic sis on Lino 16. CGruful uttontion should to Given **ii" iiE:: ** "i to al brnic sinjs. The flfobrmic am: of the first itc:.a on LUnc 16. should check to the last itor, as was the caso for Lines 10, 5, rnL 1. Divi'.o the ities on Line 16 by the first ite:. on LLUe 11, 19r3.,. rovcrsc tho aljbraic sits. a M list the qrotntions on Linu 17. *hu first itcom oi Lino 17 shoul. choc: to the Thst ito. of Line '17, ms fls thL. creo for ses 11., 6, and 2. Te have now finished the 'forward!' solution for the 4 worral omations, and. by this tinc the rcdor prosuarbly undcrstPnda tho mtbod so tkr.t the eZtcnsion of tho solution to r. rrt;.'tor nttbor of v'ri abless ill to n co:.:preratively si..plc zatter. 1e are now ready for the 'beck solution, which in iven on Lines 1, ct sc. On Lino 1., coluL X writo the vplu or.n Liae 17 colmI2n X; rcvorse the sign. This value is tLe net repression coefficient of the variable D on X. Bext in Column % ULaas 18 tV 21, inclusive, list in inverse order the values in colun X,. Lines A3. 3 3 3, r3. Ncxt on Line 19 Colua C. write the vqluo on Linc 11 Colu.n X, rovarsinr the sign. :n Line 20, clzim 7, writc the vrluc. on Line G, column X, revereir; the sim. On Lim; 2J1. colum A. t.Tit the vnluc on Linm. ;l cnlu.n X. rt.vcrsin,' the suin. Mt thc. vrluc o, Lin.; 18. cluL X., into the :.xltiplylat rw chine nd iultip.ly it tL..s th vlIuc on tc s. ii. in colI.;.n 5, r list the 'rojuct in Coluim 7.. (Protuct Rnr' t). Th en Lultiply it tir.,S the' vrl'2.3 ia colu.i D), LUes 11, 6 "n. C. l.stin" the proAucts rswTc tively on tLan 19. 20 n's 21 in c'lw.n '. l.ultiply it "Iso ti:zsa the value in colnum D on Line A3, listing the product in Colurm v) on Li.n, 2/. nm in 0colvM A, Line 19. Ths least sum is the net regression co.fiii of the variable C on X. f lL:IX Put the net regression coefficient of the variable C on.X into tO:.h:u multiplying machine, and, having a care for algebraic sigas, multiple itl times the value listedL beside It in column', writing tie product on to I same line in column PHM. Then, multiply it IF the values In column Co .an Lines 6,2 and A3, writing the products on Lines 20, 21 aMd 24, reqxpe tively of the same column. Now, add the values on Line 20, colume J, C and D together writing the sum in column X on the sai line, having a p ticular regard to algebraic signs. This last am 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 times the value listed beside it in column S, writing the prodnt on the same line in column PM. Then, multiply it by the values in colan B on lines 2 and A3, listing the products respectively on Lines 21, as 24 of the sme column. Now, add the 'Values on Line 21, and in colums A, B, C, and D, writing the sum in Column X on the same line. This aml is the net regression coefficient of A on X. Place it in the mualtiplying machine and multiply it times the value listed beside it in column S, hrv ing a care for algebraic signs, and list the product in column S on the same line. Also, multiply it times the value on Line A3' column A, liAst ing the product in the ama column on Line 24. Now, the values on Lime a In columns A, B. C, D added together algebraically, should equal the value' listed on Line A3, column X., which serves as a check upon the derivationU nf other net regression coefficients than D on X. There is a difference  23  tof 9 beblsa the We values in our seample. A greater accuracy may be a  cred Marryin the arithmetic to a greater number of places throughout the entire solution. It as deemed e*pedIent to make the example as simple as We have now secured the net regression coefficients, which are es eatial& to the forming of the regression equation for predicting or esti mtftgte values of S. To ascertain to how great a degree these predic tions conform to the actual values, it is necessary to obtain some measure of agreement between them, the predictedand the actual. This measure of aunenat is the coefficient of multiple correlation, 5, defined on Page 1. o secure this coefficient of multiple.correlation, add the values in the FN cola, listing the sum on Line 22. Next, secure the square root of Sthis aM, given on Line 23. Finally, secure the square root of the value listed on Line 13, colsum X, listing this in the PM column on Line 25. This value divided into the value Jmlidiately above it on Line 23, gives the coefficient of multiple correlation. There are certain aids and other checks in the solution which can be applied to help in locating errors. The diagonal terms of the normal equations (2j02.6, 714S.4, 2378.5 *\are always positive in sign. In mak Ing the solution the figures listed Immediately below these figures (to be added to them in the course of the solution) are always negative in sign. Me sms appearing above the 1.00000 terms are always positive, (i.e. 74.1, 2307.9, 196314). accuracy is increased if comparatively small diapnonal terms are avoided. (This can be controlled by controlling the original coding.) %e Product Moent, Line 22, Is always positive in sign. uNIVERSiTY OF FLORIDA q C Cd UWd U UI l 191 S.... ..... vs nintszou wasass. : sbe fial step in the arithmetic i1s to write t nr.grq"e. or "preticitire or Uestimting.R equation as it is various ly ealls Av first write down the retgrossion coefficient of A on X1 lar in o this is .1079. Beside this value write tbat wae donL*..w  '. .,a 4.... simpltfying the A series: thi XUl b" .ot,4f fom. A 3 5.( = 6, note (1)). Then further write the subtraction of the &nflg a("4i the A series as taken from farm 1. enclosing all in pare#tbhes s.1 algebr c for. so far will look like this: .1079 (A I ... 3 ,.?:.:. ]mtt 3 In an exactly similar manner treat the B, C, D. & X series. (Dlikq. the S series) There trill be no regression coefficient for the X .... The algebraic sum of. the A. B. C, & D expressions hsuld be equated . the X expression as follows: ; . C ) ( : ) (y .1079(L25 7.614) .1330(%M2 4.1226) + .o00A 6 (3) (41 ) (6^:: + 0229(D 5397f6 6 I z..2 4 *E '" l ) ::.: This is the "rai" regression equation. It is only necessary now to.. evaluate for X, involving only elementary algebra, to put the eqatis in its most useful form: X .1079 A .0998 B' .0273 c 0 .0137 D 99055 This is the predictingg" equation mentioned in the definition,.. of R in the note on page 1. 
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