296 
Oil the Probable Error of Biserial rj 
arrays with linear regression and homoscedasticity, or even with a complete 
Gaussian distribution of frequency. 
L M N L 
In the first place let the division between the two series be the line LL. Let 
C be the centroid of the frequency surface and RR the regression line. Let x = CN 
and Xy = PM. Then by the equation to the regression line (correlation /• = 7;) 
(x - Xy)la^ = ry/ay, 
_ 1 _ ^!L\ 
1 
(r - rt/) (xv). 
Hence 
Thus, since we may suppose S (n^y') = S {Hsy'^) = 0 approximately: 
where jS^' is the second statistical ^-constant for the y'-avray, and since to our 
approximation r] = r: 
N 
N 
= vHy' + WiP2-m (xvi) 
Again S ("f = ^^-4-^,-^ S (y^ - 2ryy' + r^y"^)} 
(1 -r^) { N 
But as in the theory of biserial r 
N 
N 
N 
* Biometrika, Vol. vii. p. 97. 
