Karl Pearson and J. F. Tocher 
161 
and there will be no reason for supposing any correlation between d' ^ and d^. Thus 
it will follow that the standard deviation of M' — M is Vaj/^ + aj/'^. Hence: 
A 
a^3r-M = ^ I r- + I ^ ^ (iv) 
Now assuming p, and qs for the moment to be known, can we learn more from 
the relative values of M' — M and o-j^-ij/ than we thought possible from the 
distribution of the deaths in a single age group owing to the latter's non-Gaussian 
form? 
It seems probable that we can for the following reasons. Let z be the sum of 
u variates x-i_ + ... + x^^, these variates following arbitrary laws of frequency 
and being in no way correlated together, i.e. z is to be found by taking a random 
selection of each of our -w-variates and adding them together, then from 
Z = X-i -\- X2 ... -|" 
we can find the moments of z. Obviously we can measure all variates from their 
means, and accordingly we find : 
zf^i = S ixfX^) + QS (^/X2 . 
„ W (m — 1) , _ ,„ 
= M X + 6 2~ - i^/X^Y, 
where ^.ji^, ^J^s and Jl^ denote the mean values of the moment coefficients for the 
various x-distributions. Accordingly if and B2 be the ^-coefficients for 2 : 
5, = ^^J=i^„ 
V 
^2-3=^^(^2-3) (vi). 
where and denote ^-coefficients found from the mean moments. Now in the 
case of the Poisson's Exponential Limit to the Binomial we have* 
^^ = ^2-3 = — =- (vii), 
where m is the mean number of deaths in the age groups. Hence, if we dealt with 
a fairly large population, where the number of deaths in any group were say 5 to 
10 and we made 10 age groups, we could reckon on and i?2 — 3 being of the 
order -02 to -01, or the distribution would be closely Gaussian. Thus there need 
be small hesitation in applying the tables of the probability integral to the 
investigation of the relationship of ikf' — M to om'-m- 
* Biometrika, Vol. v. p. 353 and Vol. x. p. 39. 
Biometiika xi 1] 
