XI.—CORRELATION : MISCELLANEOUS THEOREMS. 223 
For the birth-rate, on the other hand, assuming that o,/@ 
is approximately the same for the decade 1881-90 as in 1891, 
we have, o, being 4:08, 
32-34 — 30:34 459 
ol 50d 
= 4 40, 
The closeness of the numerical values of # in the two cases is, 
of course, accidental. 
18. The principle of weighting finds one very important 
application in the treatment of such rates as death-rates, which 
are largely affected by the age and sex-composition of the popula- 
tion. Neglecting, for simplicity, the question of sex, suppose the 
numbers of deaths are noted in a certain district for, say, the 
age-groups 0-, 10-, 20, ete., in which the fractions of the whole 
population are pg, p,, p,, etc, where 2(p)=1. Let the death- 
rates for the corresponding age-groups be dy, d,, dy, ete. Then 
the ordinary or crude death-rate for the district is 
D=3(d.p) i (16) 
For some other district taken as a basis of comparison, perhaps 
the country as a whole, the death-rates and fractions of the 
population in the several age-groups may be 6, 8,8; . . . , m 7, 
ms « « + , and the crude death-rate 
A=3(5.7) x (17) 
Now D and A may differ either because the @’s and &'s differ 
or because the p’s and =’s differ, or both. It may happen that 
really both districts are about equally healthy, and the death- 
rates approximately the same for all age-classes, but, owing to a 
difference of weighting, the first average may be markedly higher 
than the second, or wvice vers. If the first district be a rural 
district and the second urban, for instance, there will be a larger 
proportion of the old in the former, and it may possibly have a 
higher crude death-rate that the second, in spite of lower death- 
rates in every class. The comparison of crude death-rates is 
therefore liable to lead to erroneous conclusions. The difficulty 
may be got over by averaging the age-class death-rates in the 
district not with the weights Py Po pp - - . . given by its own 
population, but with the weights, mT, Ty Ty . . . . given by the 
population of, the standard district. The standardised death rate 
for the district will then be 
D' =3(d.w) 
18)