\\VII-XXX] Introditrilu,, cxlvii 



Write K ny, ;in<l the above expression 



> * 



n-1 1! (n- !)( + 2) 21 (n - 1 )(/< + !)( 

 and as 7* increases this approaches 



f H ~ 2l + X ! 4 ' 



Hence we have 



or 



approximately, when n is large. 



Neither (xi) nor (xii) denotes linear regression of o- 2 n <n, but (xii) indicates 

 that when w is large we may take 



2 = ^#2,0, 

 the latter being given by a linear regression equation. 



5. Correlation of Correlation Coefficient and Standard Deviation. 



We next turn to the correlation of the correlation coefficient r^ with the 

 standard deviation of one of the variates, say <TI. The correlation surface, which 

 has not hitherto been expressed in any simple form, is 



l n ) (* + I n 3l + I n(n + 2) 5l 



where y = \/2 rp . 



*i 



There is little approach in (xiii) to a normal distribution of the variates r and a\. 

 We can however find various properties of the surface. 



6. Frequency Distribution of the Correlation Coefficient. 



If we integrate with regard to r from 1 to + 1 the second series disappears 

 and the first reproduces the result (ii) above for the distribution of o-j*. Again, if 



* Put r = V^and double so that the limits of z are to 1 ; then, expressing the complete B- functions 

 in terms of complete T-fnnctions, it will be found that the series reduces to a factor x e~^ ff " t '*' . 



f S 



