cxlviii Tables for Statisticians and Biometricians [XXVII XXX 



we integrate with regard to o~i we have the frequency curve for the distribution 

 of the correlation coefficient r in samples of any size n, namely 



jy(l-p)*(*-i) _ 4(B . 4) IT(l(n-l)) | /n-jN 2 (Zrp? 



./ \ 1 V * 1\1\9J 9! 



VTT l v]j " 1 /l \ z / ZJ 



-)' 



+ ... 



_l_ \ \ t // I i _|_ 



41 

 n\ z (2rp) 3 



+ -.. (xiv). 



22/5! 



This again is not a very profitable form for the surface as the series does not con- 

 verge rapidly. But it is a convenient form from which to ascertain the moments for 

 the frequency distribution of r. If we integrate zr l for an odd integer t the second 

 series vanishes, if for an even integer t, the first series. In either case we can 

 deduce fairly rapidly converging series for the moment coefficients and so find the 

 standard deviation and the /3i, /3 2 for the distribution of r for a given p and n. 



For example: multiply (xiv) by r and integrate from r = 1 to r = + 1, then 



r+i 

 only the second series remains, and since I zrdr= Nr we have 



J 1 



r = 



p* w 2 (n+2) 2 (n+4) 2 



_ 

 1.2(n + l)(n + 3) 4 1 .'2. 3(w + l)(rc + 3)(w+ 5) 8 



Multiply the series by the expansion of (1 p 2 )^"" 1 ) and we find 



I 2 2 1 2 .3 2 



1 2 .3 2 .5 2 



+...} < 



This is a sufficiently converging series for finding the values of r. It is by re- 

 duction to similar series that the moment coefficients of r about zero have been 

 computed*. Hence Tables XXXI XXXIV have been calculated; these are further 

 discussed on pp. cliv clxxx of this Introduction. 



7. Correlation and Regression of r xy and a~ l . 



We will now turn back to the frequency surface for r and <TI, namely equation 



(xiii). As we know the mean values of r, C7\ and also their standard deviations, it 



r+i r-o 

 is possible from the integral of zra-^drda to find the product-moment and 



J -i Jo 



so deduce the value of r r(fl , the correlation coefficient of r and ov This value is 

 * See Biometrika, Vol. xr. pp. 333336. 



