Karl Pearson and J. F. Tocher 
163 
Another solution of the-standard-population-problem, which seems to us of 
some importance, arises from the consideration that we ought to select our standard 
population so that the probability that the two districts or classes under considera- 
tion are samples of one and the same population should be a minimum. In other 
words we ought to select AJA (= X^) so that 
Q = S {X,X,)IVSKXJ), 
where \ = d'Ja\ - dja,, 
V, = V^qs (l/«'s + 
is a maximum, subject to the relation S (Z^) = 1. 
Proceeding by the usual rules for finding a max.-min. we have 
S (SX,) = 0. 
Therefore if P be an indeterminate multiplier : 
'Sixxl) ~ sjvjcj) + ^ = ^• 
Multiply by X, and sum all such equations and we find P = 0. Hence : 
v/S{KX,)- 
Sum all such equations and we have 
1 
Thus we deduce : X^ 
whence S (A,Z,) 
S (v,X,^) 
and accordingly Q 
A little consideration shows that this is a maximum value of Q. The argument 
we then use is that if on the standard population which provides a maximum for Q, 
there be no significance in the deathrate difi^erence, there cannot be any significance 
at all in the difference. While on the other hand if on any population whatever 
used as standard, we do find a significant difference, such a difference really exists. 
