XIV.—REMOVING LIMITATIONS OF SIMPLE SAMPLING. 281
7. Such an examination may be of service, however, as
indicating one possible source of bias, viz. great heterogeneity in
the original material. If, for example, in the first illustration,
the hair-colours of the children differed largely in the different
schools—much more largely than would be accounted for by
fluctuations of simple sampling—it would be obvious that one
school would tend to give an unrepresentative sample, and
questionable therefore whether the five, ten or fifteen schools
observed might not also have given an unrepresentative sample.
Similarly, if the herrings in different catches varied largely, it
would, again, be difficult to get a representative sample for a
large area. But while the dissimilarity of subsamples would
then be evidence as to the difficulty of obtaining a representative
sample, the similarity of subsamples would, of course, be no
evidence that the sample was representative, for some very
different material which should have been represented might
have been missed or overlooked.
8. The student must therefore be very careful to remember
that even if some observed difference exceed the limits of fluctua-
tion in simple sampling, it does not follow that it exceeds the
limits of fluctuation due to what the practical man would regard —
and quite rightly regard—as the chances of sampling. Further,
he must remember that if the standard error be small, it by no
means follows that the result is necessarily trustworthy: the
smallness of the standard error only indicates that it is not
untrustworthy owing to the magnitude of fluctuations of simple
sampling. It may be quite untrustworthy for other reasons:
owing to bias in taking the sample, for instance, or owing to definite
errors in classifying the 4’s and o’s. On the other hand, of course,
it should also be borne in mind that an observed proportion is not
necessarily incorrect, but merely to a greater or less extent
untrustworthy if the standard error be large. Similarly, if an
observed proportion =, in a sample drawn from one universe be
greater than an observed proportion =, in a sample drawn from
another universe, but m, — , is considerably less than three times
the standard error of the difference, it does not, of course, follow
that the true proportion for the given universes, p, and p,, are
most probably equal. On the contrary, py most likely exceeds p, ;
the standard error only warns us that this conclusion is more or
less uncertain, and that possibly p, may even exceed p,.
9. Let us now consider the effect, on the standard-deviation of
sampling, of divergences from the conditions of simple sampling
which were laid down in § 8 of Chap. XIII.
First suppose the condition (a) to break down, so that there is
some essential difference between the localities from which, or the