XIIL.—SIMPLE SAMPLING OF ATTRIBUTES. RN 
We have therefore M =p, and 
oi=p-p*=pq. 
But the number of successes in a group of # such events is the 
sum of successes for the single events of which it is composed, 
and, all the events being independent, we have therefore, by the 
usual rule for the standard-deviation of the sam of independent 
variables (Chap. XI. § 2, equation (?)), o, being the standard- 
deviation of the number of successes in z events, 
oa=npq . : : L(Y) 
This is an equation of fundamental importance in the theory of 
sampling. The student should particularly bear in mind that 
the standard-deviation of the number of successes, due to 
fluctuations of simple sampling alone, in a group of mn events 
varies, not directly as n, but as the square root of n. 
6. In lieu of recording the absolute number of successes in each 
sample of n events, we might have recorded the proportion of 
such successes, 7.e. 1/nth of the number in each sample. As this 
would amount to merely dividing all the figures of the original 
record by 7, the mean proportion of successes—or rather the value 
towards which the mean tends to approach—must be p, and the 
standard-deviation of the proportion of successes s, be given by 
S=cifnt=pgin . . . . (2 
The standard-deviation of the proportion of successes in samples 
of such independent events varies therefore inversely as the square 
root of the number on which the proportion is calculated. Now 
if we regard the observed proportion in any one sample as a 
more or less unreliable determination of the true proportion in 
a very large sample from the same material, the standard-devia- 
tion of sampling may fairly be taken as a measure of the 
unreliability of the determination—the greater the standard- 
deviation, the greater the fluctuations of the observed proportion, 
although the true proportion is the same throughout. The 
reciprocal of the standard-deviation (1/s), on the other hand, may 
be regarded as a measure of reliability, or, as it is sometimes 
termed, precision, and consequently the reliability or precision of 
an observed proportion varies as the square root of the number of 
observations on which it is based. This is again a very important 
rule with many practical applications, but the limitations of the 
case to which it applies, and the exact conditions from which it 
has been deduced, should be borne in mind. We return to this 
point again below (§ 8 and Chap. XIV.). 
7. Experiments in coin tossing, dice throwing, and so forth 
have been carried out by various persons in order to obtain ex- 
ne 
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