the Multiple Correlation Coefficient. 213 



£w„ — *0"«)O») r . . . (7) 



E A*1. =('11 + ^2-!^) /2'' 231 R 23 (12) 



R lE23 ,r 31 =(iRl 3 + '-3i- 3 ? f 2 )/2 1 E 23 '- 31 (9) 



R ,e 2 3,^s=(2'1i+^ 2 -iR1 3 )/2( 1 R 23 )( 1 ^) .... (14) 



R l w a ,=(i E l,+^i-^/2(A>X 3 '- 81 ) (10) 



RE 2S , Al = H 2 + 3 ry/2( 1 E 2 3)( 2 R 31 ) (15) 



Si =2A(,f 2+ ,.| 3 + ^ 1 )/ J! (11) 



2v =2^'( 3 'f 2 -r,)| 3 + 2 r| 1 )/ re (13) 



2 R =(i-,Ry 2 /» (8) 



§ 3 (i.) The underlying assumption of the methods 

 employed to obtain the results in the preceding section, a 

 method valid over a large range of cases, is that the mean 

 value in many samples of the statistical constant whose 

 probable error we are determining is equal to or only differs 

 by small quantities of a high order from the true value, i. e. 

 the value in the sampled population. We have already seen 

 in § 1 that this is unlikely to be exactly true in the case of 

 a multiple correlation coefficient, and we shall in this section 

 find the mean value of Hi-23 in samples of size n extracted 

 out of an infinite population with normal distribution, and 

 also the corrected value of the probable error when the 

 position of the mean has been allowed for. We shall 



consider n to be so large that — 2 may be taken as negligible 



in our results, and will reserve for a separate paper what 

 turns out to be a much more difficult investigation, the 

 determination of these quantities correct to terms in 1/n 2 , 

 an investigation of great importance when we come to deal 

 with moderately small samples. The justification of the 

 separate treatment of the easier problem lies in the result, 

 obtained comparatively simply, that the formula 



i-23 n 





remains true (if 1/n 2 be neglected) after being corrected for 

 the deviation of the mean value in many samples from the 

 value in the sampled population. 



