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.