284 THEORY OF STATISTICS.
certain registration districts of England, in § 10 of Chap. XIII.
p- 263. It will be seen that in the first group of small districts
there appears to be a significant standard-deviation of some 6
units in the proportion of male births per thousand, but in the
more urban districts this falls to 1 or 2 units; in one case only
does s fall short of s,. In the table on p. 283 are given some
different data relating to the deaths of women in childbirth in the
same groups of districts, and in this case the effect of definite
causes is relatively larger, as one might expect. The values of
Js? — st suggest an almost uniform significant standard-deviation
o,=0'8 in the deaths of women per thousand births, five out of
the eight values being very close to this average. The figures of
this case also bring out clearly one important consequence of (2),
viz. that if we make » large s becomes sensibly equal to o,, while
if we make » small s becomes more nearly equal to p,g,/n. Hence
if we want to know the significant standard-deviation of the pro-
portion p—the measure of its fluctuation owing to definite causes
—n should be made as large as possible ; if, on the other hand, we
want to obtain good illustrations of the theory of simple sampling
n should be made small. If » be very large the actual standard-
deviation may evidently become almost indefinitely large com-
pared with the standard-deviation of sampling. Thus during the
20 years 1855-74 the death-rate in England and Wales fluctuated
round a mean value of 222 per thousand with a standard-devia-
tion of 0:86. Taking the mean population as roughly 21 millions,
the standard-deviation of sampling is approximately
22 x 978
vo 3 106 =0052
This is only about one twenty-seventh of the actual value.
11. Now consider the effect of altering the second condition
of simple sampling, given in § 8 (8) of Chapter XIII., viz. the
condition that the chances p and ¢ shall be the same for every
die or coin in the set, or the circumstances that regulate the
appearance of the character observed the same for every individual
or every sub-class in each of the universes from which samples
are drawn. Suppose that in the group of n dice thrown the
chances for m, dice are p; ¢,; for m, dice, p, ¢,, and so on,
the chances varying for different dice, but being constant
throughout the experiment. The case differs from the last, as
in that the chances were the same for every die, at any one
throw, but varied from one throw to another: now they are con-
stant from throw to throw, but differ from one die to another as
they would in any ordinary set of badly made dice. Required to
find the effect of these differing chances.
