220 Variation of the Multiple Correlation Coefficient. 

 So that correct to 1 /n, 



x 2 -x 2 fSx= 2(1 ~ X2)2 - • . • m 



But to the same degree of approximation, we have already 

 shown that 



An A. 



— (1 — X 2 ) 2 

 Hence X 2 = X 2 + } — correct to 1/n. 



Hence Z^^l (25, 



correct, to 1/n. 



We have thus established that for samples of fair size 

 extracted from a population in which R^ is not very small, 



(1 — R 2 ) 2 

 the mean value of R exceeds the true by _ -. ■ , and the 



J 2nR 



standard deviation of the distribution of R about this mean 



value is 



\/n 



It is important to observe that where the true value is not 

 known, and R necessarily has' to be determined from samples, 

 we may replace R by R in these results within the same 

 limits of accuracy — thus : 



Kr=R --mr' (2b) 



an< ' V (1-R 2 ) ,o-, 



V n 



