£ THEORY OF STATISTICS. 
identically similar throughout the experiment, so that the chance 
of throwing “heads” with the coins or, say, “six” with the dice 
was the same throughout: we did not commence an experiment 
with dice loaded in one way and later on take a fresh set of dice 
loaded in another way. Consequently if formula (2) is to hold 
good in our practical case of sampling there must not be a 
difference in any essential respect—1.e. in any character that can 
affect the proportion observed—between the localities from which 
the observations are drawn, nor, if the observations have been 
made at different epochs, must any essential change have taken 
place during the period over which the observations are spread. 
Where the causation of the character observed is more or less 
unknown, it may, of course, be difficult or impossible to say what 
differences or changes are to be regarded as essential, but, where 
we have more knowledge, the condition laid down enables us to 
exclude certain cases at once from the possible applications of 
formula (1) or (2). Thus it is obvious that the theory of simple 
sampling cannot apply to the variations of the death-rate in 
localities with populations of different age and sex compositions, 
nor to death-rates in a mixture of healthy and unhealthy districts, 
nor to death-rates in successive years during a period of con- 
tinuously improving sanitation. In all such cases variations 
due to definite causes are superposed on the fluctuations of 
sampling. 
(6) In the second place, we have also tacitly assumed not 
only that we were using the same set of coins or dice throughout, 
so that the chances p and ¢ were the same at every trial, but 
also that all the coins and dice in the set used were identically 
similar, so that the chances p and ¢ were the same for every coin 
or die. Consequently, if our formule are to apply in the practical 
case of sampling, the conditions that regulate the appearance of 
the character observed must not only be the same for every 
sample, but also for every individual in every sample. This is 
again a very marked limitation. To revert to the case of death- 
rates, formule (1) and (2) would not apply to the numbers of 
persons dying in a series of samples of 1000 persons, even if these 
samples were all of the same age and sex composition, and living 
under the same sanitary conditions, unless, further, each sample 
only contained persons of one sex and one age. For if each 
sample included persons of both sexes and different ages, the 
condition would be broken, the chance of death during a given 
period not being the same for the two sexes, nor for the young 
and the old. The groups would not be homogeneous in the sense 
required by the conditions from which our formule have been 
deduced. Similarly, if we were observing hair-colours, our formule 
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