: THEORY OF STATISTICS. 
variation on the standard-deviation of simple sampling is quite 
small, for, as calculated from equation (4), 
#=—(18 x 982 — 900) 
s=130/n/n 
as compared with 133/s/n. 
13. We have finally to pass to the third condition (c) of § 8, Chap. 
XIIL, and to discuss the effect of a certain amount of dependence 
between the several “events” in each sample. We shall suppose, 
however, that the two other conditions (a) and (0) are fulfilled, 
the chances p and ¢ being the same for every event at every trial, 
and constant throughout the experiment. The problem is again 
most simply treated on the lines of § 5 of the last chapter. The 
standard-deviation for each event is (pg)! as before, but the events 
are no longer independent: instead, therefore, of the simple 
expression 
0? =n.pg, 
we must have (cf. Chap. XL. § 2) 
o2=npq+2pq(rg +r t «oo Togt ooo) 
where, 7,4, 7,4, etc. are the correlations between the results of the 
first and second, first and third events, and so on—correlations 
for variables (number of successes) which can only take the 
values 0 and 1, but may nevertheless, of course, be treated as 
ordinary variables (¢f. Chap. XI. § 10). There are n(n —1)/2 
correlation-coefficients, and if, therefore, 7 is the arithmetic mean 
of the correlations we may write 
a? =mnpg[l +7(n—-1)]. ; . (Bb) 
The standard-deviation of simple sampling will therefore be 
increased or diminished according as the average correlation 
between the results of the single events is positive or negative, 
and the effect may be considerable, as o may be reduced to zero 
or increased to m(pg)t. For the standard deviation of the propor- 
tion of successes in each sample we have the equation 
s2 = +r(n-1)] . (8) 
It should be noted that, as the means and standard-deviations 
for our variables are all identical, » is the correlation-coefficient 
for a table formed by taking all possible pairs of results in the 
n events of each sample. 
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