Karl Pearson 
157 
We find 
pp |„ PA, fd, d:\^ 2P /d, 
p + p 
pp' 
p + p 
7 'Si" 
DsL, \p 
PA, 
P 
p p' / \p 
a, a, \ _^ PD, _ 
AJ.S \p p'J } 
P 
dj Dg ''a, «/ 
DsLs Ip / A,\i 
P 
Let = Ps>~T ^ 1s> then we may write 
P fds - Ps(h d,' - p,aj\^ 
p + p 
{A.Psqs 
P 
P 
Now if we know the population sampled, we have only to insert the values 
of P, Ag, and p^, to obtain the value of Xo^- But if we do not, and this is usually 
the case^ then our problem is : Are the two districts samples of the same unknown 
general population? For example, we might enquire whether the death distribu- 
tions in Bradford and Leeds were, correcting for age, significantly different. It 
might be supposed that it would be correct to give A^/P, ^-i^d 1s> the values 
found for all England and Wales. But it may be doubted whether this would 
be satisfactory. The populations of Bradford and Leeds might fairly be considered 
as samples of a general population which is very far indeed from being that of all 
England. Accordingly it seems much more reasonable to suppose that they are 
samples of a population whose mortality characters will be best represented by 
the combination of those of the two districts themselves, or we take 
Ps 
d, + d' 
As 
P 
a, + a. 
Substituting the first of these relations we find 
P 
a, a, — 
a. 
(xxxix). 
,p+p' P/ Y"" ' "'V as + (^s' 
If we substitute the second relation, (xxxix) becomes 
p p' ' ' \a, as 
Xo' = >Si« 
y\p + p 
rj ids 
+ ds'){l 
ds + ds 
a, + a-s 
.(xl). 
and this, I take it, is the best measure of significance in the difference of the 
death distributions allowing for age groups in the two districts. 
It is clear that the first factor will in many cases differ but little from unity. 
For we should anticipate that approximately 
