XIV.—REMOVING LIMITATIONS OF SIMPLE SAMPLING. 285
For the mean number of successes we evidently have
M=mp, +m,py + maps +... .
=n a,
Pp, being the mean chance Z(mp)/n. To find the standard-deviation
of the number of successes at each throw, it should be noted that
this may be regarded as made up of the number of successes in
the m, dice for which the chances are 21 9; together with the
number of successes amongst the m, dice for which the chances
are p, q,, and so on: and these numbers of successes are all
independent. Hence
0 =m P1qy + MoPoy + MgPigs + +o
= Z(mpq),
Substituting 1-p for ¢, as before, and using o, to denote the
standard-deviation of p,
ol =npyg, — no; v2.3)
or if s be, as before, the standard-deviation of the proportion of
successes,
_Puls_0y
$= Si . 4)
12. The effect of the chances varying for the individual dice or
other “events” is therefore to lower the standard-deviation, as
calculated from the mean proportion Ppp and the effect may
conceivably be considerable. To take a limiting case, if p be zero
for half the events and unity for the remainder, Po=9,=3, and
o,=3, so that s is zero. To take another illustration, still somewhat
 extreme, if the values of p are uniformly distributed over
the whole range between 0 and 1, p,=9,=4 as before but =
1/12=0-0833 (Chap. VIIL § 12, p. 143). “Hence §2=01667/n,
s=0408//n, instead of 0-5/n/n, the value of s if the chances are
$ in every case. In most practical cases, however, the effect will be
much less. Thus the standard-deviation of sampling for a deathrate
 of, say, 18 ver thousand in a population of uniform age and
one sex is (18 x 982)}/s/n=133//n. Ina population of the age
composition of that of England and Wales, however, the deathrate
 is not, of course, uniform, but varies from a high value in
infancy (say 150 per thousand), through very low values (2 to 4
per thousand) in childhood to continuously increasing values in
old age ; the standard-deviation of the rate within such a population
 is roughly about 30 per thousand. But the effect of this