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