cxlvi Tables for Statisticians and Biometricians [XXVII XXX 



then if \ n = F (^7i)/F (^(n 1 )), 



we have r \<*i = ~ l 2 ( v ii)> 



which tends as n-*-<x> to become p 2 . Table XXVII gives the values of r<, 1<r2 for 

 samples of various sizes when the sampled population has a correlation p. 



3. Correlation Surface of Variances. 



If instead of dealing with the standard deviations, a-^ and o- 2 , we use yu. 2( o, y"o,2, 

 the variances, the correlation surface is 



\-3 

 JM 2 0^0 2 \ ~~7T~ 



z = - - - - _-' - - 6 - v *i a * 9 z / 1 A -= L ^~^ i 2 



* f + 1 ! Sl 2 s 2 2 (2n - 2) + 2 ! s^.s'/ (2/i - 2) (2n + 2) 



+ ...+- ^2.0^0.2 _ + 



Here the correlation of /z 2i0 and /* 0)2 is simply 



^2,0^0,2 = P 2 ( ix )' 



and the regression of t/ 2(0 on /i ,2 is given by the linear relation 



1 ^ 2 



/(. J. ^A Q /I Ov ^^2 9 / \ 



/"0,2 = 2 2 2 (1 - p 2 ) + g- a p 2 /"2,0 (X). 



4. Regression of Standard Deviation of one Variate on that of a second. 



On the other hand the regression curve of o- 2 on ai is given by the following 

 equation, where <r 2 = mean value of <r 2 for constant a\\ 



l M _. 71 (n + 2) 2 71 (/i + 2) (n + 4) K 3 | 

 " T7 1 ri + (w - 1) (n + 1) 2 ! (u - 1) (K + 1) (n + 3) 3 ! 



where 



-^ 22 



For 71 large the expression r vf7- *yvi rapidly approaches unity. 



Multiply out the expression in the curled brackets by the expansion of e~ K and 

 we find 



7i (n + 2) (M + 4) 



j 4. 



71-11! + w-ln + l2! 



3 \ 

 3! "V 



1.3 



11 (7i-l) (Tt + 1) 2! 

 1.3.5 



(n - 1) (n + 1) (n + 3) (n + 5) 4 ! 





