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 population.
 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 deathrates
 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 deathrates
 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 deathrates
 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)