XVIL—SIMPLER CASES OF SAMPLING FOR VARIABLES. 347 
of n, we may take the following case. If the student will turn to 
the calculated binomials, given as illustrations of the forms of 
binomial distributions in Chap. XV. § 3, he will find there the 
distribution of the number of successes for twenty events when 
¢=09, p=0-1: the distribution is extremely skew, starting at 
zero, rising to high frequencies for 1 and 2 successes, and thence 
tailing off to 20 cases of 7 successes in 10,000 throws, 4 cases of 8 
successes and 1 case of 9 successes. But now find the distribu- 
tion for the mean number of successes in groups of five throws, 
under the same conditions. This will be equivalent to finding 
the distribution of the number of successes for 100 such events, 
and then dividing the observed number of successes by five—the 
last process making no difference to the form of the distribution, 
but only to its scale. But the distribution of the number of 
successes for 100 events when ¢=09, p=0-1, is also given in 
Chap. XV. § 3, and it will be seen that, while it is appreciably 
asymmetrical, the divergence from symmetry is comparatively 
small : the distribution has gained very greatly in symmetry 
though only five observations have been taken to the sample. 
We may therefore reasonably assume, if our sample is large, 
that the distribution of means is approximately a normal dis- 
tribution, and we may calculate, on that assumption, the fre- 
quency with which any given deviation from a theoretical value 
or a value observed in some other series, in an observed mean, will 
arise from fluctuations of simple sampling alone. 
The warning is necessary, however, that the approach to 
normality is only rapid if the condition that the several drawings 
for each sample shall be independent is strictly fulfilled. 1f the 
observations are not independent, but are to some extent positively 
correlated with each other, even a fairly large sample may con- 
tinue to reflect any asymmetry existing in the original distribution 
{¢f. ref. 32 and the record of sampling there cited). 
If the original distribution be normal, the distribution of 
means, even of smali samples, is strictly normal. This follows at 
once from the fact that any linear function of normally distributed 
variables is itself normally distributed (Chap. XVI. § 6). The 
distribution will not in general, however, be normal if the 
deviation of the mean of each sample is expressed in terms of the 
standard-deviation of that sample (cf. ref. 30). 
14. Let us consider briefly the effect on the standard error of 
the mean if the conditions of simple sampling as laid down in 
§ 2 cease to apply. 
(a) If we do not draw from the same record all the time, but 
first draw a series of samples from one record, then another 
series from another record with a somewhat different mean and