John Blakbman 
339 
Problem VI. To find the standard deviation of the values of ^ defined by the 
equation 
We have i^?= V^V ~ ''"S^- 
Hence iSf' = v''^n' + ''"-'-^ - 27;r5^,S,.72,,.. 
But . 
Hence il^' = (t - (V - + (xxvi). 
As in equation (xxiv) this is a formula always applicable. The formula for 2^ 
which corresponds to equation (xxiii) for 2^- is therefore 
i2f = = {(1 - rT - (1 - rj +1} (xxvii). 
Prohleni VII. To find the standard deviation of the values of ^ defined by 
the equation 
^ = 'T] — r. 
Proceeding exactly as before, we get 
2 ''=z2= + ^"-22 ^ R 
giving »?r2^" = {v- {''2^= - 'n%--] + ^"'"'S^' (xxviii), 
applicable in all cases. 
The corresponding simple formula is 
vrt^^=^''^^{r{l-nJ-v{'^-rJ + y] + r\ (xxix). 
Statistical Illustrations. 
I proceed to discuss, by appeals to actual statistical experience, the degree of 
closeness to which the simple formulae (xxiii), (xxvii) and (xxix), may be used 
instead of the complete formulae (xxii), (xxvi), and (xxviii). 
If, as regards any frequency distribution, we have tabulated for each a,-array of 
ys the rix^, yx^ and <t„„^ , we may proceed to calculate 2^- from the formula (xxii) 
without further reference to the sub-groups of the correlation table. This follows 
at once since we have, in general, 
Np,„, = s {ux^y^ - xT {ys - y)] 
= S {«.v,, (^P - {ys - yx^)} + S [nx^y^ (x, - xY" iyx^ - y)] 
= N\„„ 
[since S [iix^y^ {ys - yx)] = 0], hence 
i?m,i = Xni (XXX). 
43—2 
