Ea THEORY OF STATISTICS. 
conditious under which, samples are drawn, or that some essential 
change has taken place during the period of sampling. We may 
represent such circumstances in a case of artificial chance by 
supposing that for the first f; throws of = dice the chance of 
success for each die is p;, for the next f, throws p,, for the next f, 
throws pg, and so on, the chance of success varying from time to 
time, just as the chance of death, even for individuals of the same 
age and sex, varies from district to district. Suppose, now, that 
the records of all these throws are pooled together. The mean 
number of successes per throw of the n dice is given by 
n 
MU = (apy +1oPy + 13Ps + ltl ge ) = 1.0, 
where V=23(f) is the whole number of throws and p, is the mean 
value 2(fp)/N of the varying chance p. To find the standard- 
deviation of the number of successes at each throw consider that 
the first set of throws contributes to the sum of the squares of 
deviations an amount 
Alnengy + ney =o’) 
n.p,q, being the square of the standard-deviation for these throws, 
and n(p, -p,) the difference between the mean number of 
successes for the first set and the mean for all the sets together. 
Hence the standard-deviation o of the whole distribution is given 
by the sum of all quantities like the above, or 
No? =n2(fpqg) + n* 2f(p — Py) 
Let o, be the standard-deviation of p, then the last sum is 
N.n2, and substituting 1 — p for ¢, we have 
ol = np, — np; — no> + Noy 
= npg + n(n —1)as . . + AD) 
This is the formula corresponding to equation (1) of Chap. 
XIII : if we deal with the standard-deviation of the proportion 
of successes, instead of that of the absolute number, we have, 
dividing through by 7? the formula corresponding to equation 
(2) of Chap. XIIL., viz.— 
e Loto ol 
Bn + Sa : (2) 
10. If » be large and s, be the standard-deviation calculated 
from the mean proportion of successes p), equation (2) is sensibly 
of the form 
S2= 5 = oo 
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