XXVII XXX] Introduction cxlix 



known t.i i bo p/V2 in the case of large samplea. Tin- f'un.lam. -ntul formula, if an 

 lM-furo\ w -r(iw)/r(i(n-l)), is 



-f 



whero r is given by (xv) and o> by 



2 a 2.4 



2*. 4*. 6* p 



" H '" ..... 



a aeries which converges fairly rapidly. 



Table XXVIII has been calculated from equation (xvi); it will indicate how 

 far in any particular-sized sample the student is justified in assuming r rfl =p/*/2. 



When, however, we consider that the correlation coefficient is sufficiently close 

 to p/V2 to use this value, it does not follow that the mean value of ? for a given 

 <TI, i.e. r 9l , will be linearly related to <TI, although it is so when n is very large. The 

 regression equation of r ai on oi is given by 



r(jn) p gl ( _L__rf, 1.8 p*crs 



( + l)) V2 M (n + l)l! 2 j? ( + l}( + *)SI * *i 4 



"1 



1 V2 V" ~r *)) V Z *l I V" T A; A 



1.3.5 



which is fairly easy to compute. Remembering that i contains a factor - the 

 series in curled brackets is represented as n-*-oo by 

 , *1.3 1.3.5 



F! + TT 3~T +--- = ( 1 + 2 *)~ l > 



where K = i/>W/((l -p 2 )2x 2 ). 



Hence when w is large *, approximately, 



But when n is large, r = p and <?! = 2i, and accordingly l ^ -1 will be a small 



quantity ; hence expanding, and neglecting all but the first power of (<TI 2i)/2i , 

 we find 



f -c- P ^ n (a- - ff ) (xx) 



V2 S!/V2^ 

 the usual linear formf. 



* r fiJT+3) for " * " becomos nnity ' 



f It is probable that (xix) would give better results than (xx) for n fairly large. In Itiomctrika, 

 Vol. xvn. p. 185, I overlooked the point that (xix) could pass into (xx). 



