4 On the Probable Errors of Frequency Constants 
Let us sum for s' and keep x s constant ; then 
2 {{y* - 2/)' 2 ? w} = {» s (o- 2 j, a + (y. - y)% 
where <r is the standard deviation and y s the mean of the array of y's for a given 
oc s . If the regression be linear and homoscedastic, then 
- _ _ fj// ~\ 
2/s 2/ — r xy \ x s x )> 
and <r 2 ys = ay* (1 - r\ y ). 
Hence p 22 = ~ # j?; s (# s - «) 2 ( 1 - r- xy ) + n s (a; s - xf 
= o-^-o-,;-' {1 - r\ v + r 2 xy ft 2 }, 
z^--l = rV(A-l). 
Similarly - 1 = r 2 ^ (#.' - 1), 
on the assumption that the other regression is also linear and homoscedastic. 
It is accordingly impossible for two variables to have a regression linear and 
homoscedastic in both senses unless /3 2 has the same value for both variables. 
Clearly for most practical purposes we may take 
P22 
^-1 = V(& -!)(&'- 1). 
^20^02 
Thus approximately r a ff = 7- 2 xy (xvii). 
This is identical with the result found by Pearson and Filon (Phil. Trans. 
Vol. 191, A, p. 242) on the assumption of normal correlation. It is now seen to 
be true, whenever we may assert linearity of regression and homoscedastic dis- 
tribution for both variables. 
(7) Probable error of a coefficient of correlation. 
Put r ^ = ^_ 
&r xy _ Spn _ 1 8pw _ 1 Spw 
1'xy pn 2 p. 20 2 pta * 
Square, add for all random samples and divide by their number : 
a \ u _ _1 {P^zPlL + 1 P*> ~ P*> 2 + I P Qi ~ P<* + I P*~P*>P<* 
r" xy N\ p u 2 4 p.J 4 p 02 2 2 p 20 p 02 
_ P ti-p uPio Pu-pnPu \ # > / xviii )_ 
This is the most general value of the standard deviation a r of a correlation 
coefficient. It was first given by Sheppard (Phil. Trans. Vol. 192, A, p. 128, with 
