FOUNDATIONS OF THEORETICAL STATISTICS. 
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It should he noted that there is no falsehood in interpreting any set of independent 
measurements as a random sample from an infinite population ; for any such set of 
numbers are a random sample from the totality of numbers produced by the same 
matrix of causal conditions : the hypothetical population which we are studying is an 
aspect of the totality of the effects of these conditions, of whatever nature they may be. 
The postulate of randomness thus resolves itself into the question, “ Of what population 
is this a random sample ? ” which must frequently be asked by every practical statistician. 
It will be seen from the above examples that the process of the reduction of data is, 
even in the simplest cases, performed by interpreting the available observations as a 
sample from a hypothetical infinite population ; this is a fortiori the case when we have 
more than one variate, as when we are seeking the values of coefficients of correlation. 
There is one point, however, which may be briefly mentioned here in advance, as it 
has been the cause of some confusion. In the example of the frequency curve mentioned 
above, we took it for granted that the values of both the mean and the standard deviation 
of the population were relevant to the inquiry. This is often the case, but it sometimes 
happens that only one of these quantities, for example the standard deviation, is required 
for discussion. In the same way an infinite normal population of two correlated variates 
will usually require five parameters for its specification, the two means, the two standard 
deviations, and the correlation ; of these often only the correlation is required, or if not 
alone of interest, it is discussed without reference to the other fonr quantities. In such 
cases an alteration has been made in what is, and what is not, relevant, and it is not 
surprising that certain small corrections should appear, or not, according as the other 
parameters of the hypothetical surface are or are not deemed relevant. Even more 
clearly is this discrepancy shown when, as in the treatment of such fourfold tables as 
exhibit the recovery from smallpox of vaccinated and unvaccinated patients, the method 
of one school of statisticians treats the proportion of vaccinated as relevant, while 
others dismiss it as irrelevant to the inquiry. (3.) 
3. The Problems of Statistics. 
The problems which arise in reduction of data may be conveniently divided into three 
types :— 
(1) Problems of Specification. These arise in the choice of the mathematical form of 
the population. 
(2) Problems of Estimation. These involve the choice of methods of calculating from 
a sample statistical derivates, or as we shall call them statistics, which are designed 
to estimate the values of the parameters of the hypothetical population. 
(3) Problems of Distribution. These include discussions of the distribution of 
statistics derived from samples, or in general any functions of quantities whose 
distribution is known. 
It will be clear that when we know (1) wliat parameters are required to specify the 
