XIV.—REMOVING LIMITATIONS OF SIMPLE SAMPLING. 279 
between two observed proportions by equation (6) of that chapter, 
this may be taken, provided » be large, as approximately the 
standard-deviation of true differences for the given observed 
difference. 
4. The use of standard errors must be exercised with care. It 
is very necessary to remember the limited assumptions on which 
the theory of simple sampling is based, and to bear in mind that 
it covers those fluctuations alone which exist when all the assumed 
conditions are fulfilled. The formule obtained for the standard 
errors of proportions and of their differences have no bearing 
except on the one question, whether an observed divergence of a 
certain proportion from a certain other proportion that might be 
observed in a more extended series of observations, or that has 
actually been observed in some other series, might or might not 
be due to fluctuations of simple sampling alone. Their use is 
thus quite restricted, for in many cases of practical sampling this 
is not the principal question at issue. The principal question in 
many such cases concerns quite a different point, viz. whether the 
observed proportion = in th: sample may not diverge from the 
proportion p existing in the universe from which it was drawn, 
owing to the nature of the conditions under which the sample was 
taken, = tending to be definitely greater or definitely less than 
p. Such divergence between 7 and p might arise in two distinct 
ways, (1) owing to variations of classification in sorting the 
4’s and os, the characters not being well defined—a source of 
error which we need not further discuss, but one which may lead 
to serious results [cf. ref. 5 of Chap. V.]. (2) Owing to either 4’s 
or as tending to escape the attentions of the sampler. To give 
an illustration from artificial chance, if on drawing samples from 
a bag containing a very large number of black and white balls 
the observed proportion of black balls was =, we could not 
necessarily infer that the proportion of black balls in the bag was 
approximately =, even though the standard error were small, and 
we knew that the proportions in successive samples were subject 
to the law of simple sampling. For the black balls might be, 
say, much more highly polished than the white ones, so as to 
tend to escape the fingers of the sampler, or they might be re- 
presented by a number of lively black insects sheltering amongst 
white stones: in neither case would the ratio of black balls to 
white, or of insects to stones, be represented in their proper pro- 
portions. Clearly, in any parallel case, inferences as to the 
material from which the sample is drawn are of a very doubtful 
and uncertain kind, and it is this uncertainty whether the chance 
of inclusion in the sample is the same for 4’s and o’s, far more 
than the mere divergences between different samples drawn in