122 MCEWEN AND MICHAEL. 



which agrees well with the observed value of 4.0 given in table 5. For 

 comparison, it is well to add that the linear regression equation (49), 

 derived by the method of multiple correlation, is in our notation 



w = 11.2 - 0.48 {x - 65.9) + 0.31 {y - 6.8) (85) 



Putting X = 73.4 and y = 3.6, as above, gives w = 6.6, a value 

 departing more widely from that observed. 



In illustrating the procedure when variability within the group is 

 taken into account, the wheat data are arranged and grouped as shown 

 in table 5. In addition, for the purpose of computing the required 

 regression coefficients, the deviations yl* — A*, and x — Xi in group 1, 

 B^ — B' and .r — X2 in group 2, and so on to F^ — F* and 7/ — ye in 

 group 6 are entered. However, in order to save labor in computing 

 S(^' — A') (x — Xi), etc., the averages A*, Xi, B^, X2, etc., are replaced 

 by the approximate values Af, x^, B{, Xg, etc., carried only to as many 

 places as the individual entries, and deviations from these values are 

 the ones that are entered. To save space table 5, including these 

 deviations, is not reproduced. 



Owing to substitution of the above approximate values for the 

 true averages in equations (29) to (40), certain initial corrections must 

 be applied before they are ready for solution. Let Xi = x^ + Ax^ 

 and A* = Aj + AAj where Ax^ and AAj are the respective correc- 

 tions that must be added. Substituting these equivalents into the 

 expression 



2(.4^ - AO (.T - xO p, 1 . , .^e . 



- = Kl, as denned on page 108, gives 



Ei = 



S(x - Xi)2 



2 W - (Af + AAOl \x - (x; + Ax;)] 



S[.r- (x; + Ax^P 



^ ^{A^ - Af) (.r - x;) - Ax; 2(yl^ - AQ - AA' S(j- - xj - Ax;) 

 S(.r - x;)2 - 2Ax;s(a- - x;) + 2(Ax;)2 



^ nA' - AQ (.r - X,) - A^(AAl) (AxJ 



s(x - x;)^ - ^^(Ax;)^ ^^^^ 



smce 



S(zl^ - Aj) = 2AAi = A'lAAj 



s(x - x;) = 2 Ax; = iViAx; 



2(0; - x; - Ax;) = 2(0; - xi) = . 



(87) 



