FOUNDATIONS OF THEORETICAL STATISTICS. 
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mental Problem of Practical Statistics,” in which one of the most eminent of modern 
statisticians presents what purports to be a general proof of Bayes’ postulate, a proof 
which, in the opinion of a second statistician of equal eminence, “ seems to rest upon a 
very peculiar—not to say hardly supposable—relation.” (2.) 
Leaving aside the specific question here cited, to which we shall recur, the obscurity 
which envelops the theoretical bases of statistical methods may perhaps be ascribed 
to two considerations. In the first place, it appears to be widely thought, or rather 
felt, that in a subject in which all results are liable to greater or smaller errors, precise 
definition of ideas or concejhs is, if not impossible, at least not a practical necessity. 
In the second place, it has happened that in statistics a purely verbal confusion has 
hindered the distinct formulation of statistical problems ; for it is customary to apply 
the same name, mean , standard deviation, correlation coefficient, etc., both to the true 
value which we should like to know, but can only estimate, and to the particular value 
at which we happen to arrive by our methods of estimation ; so also in applying the 
term probable error, writers sometimes would appear to suggest that the former quantity, 
and not merely the latter, is subject to error. 
It is this last confusion, in the writer’s opinion, more than any other, which has led 
to the survival to the present day of the fundamental paradox of inverse probability, 
which like an impenetrable jungle arrests progress towards precision of statistical 
concepts. The criticisms of Boole, Venn, and Chrystal have done something towards 
banishing the method, at least from the elementary text-books of Algebra ; but though 
we may agree wholly with Chrystal that inverse probability is a mistake (perhaps the 
only mistake to which the mathematical world has so deeply committed itself), there 
yet remains the feeling that such a mistake would not have captivated the minds of 
Laplace and Poisson if there had been nothing in it but error. 
2. The Purpose of Statistical Methods. 
In order to arrive at a distinct formulation of statistical problems, it is necessary to 
define the task which the statistician sets himself: briefly, and in its most concrete 
form, the object of statistical methods is the reduction of data. A quantity of data, 
which usually by its mere bulk is incapable of entering the mind, is to be replaced by 
relatively few quantities which shall adequately represent the whole, or which, in other 
words, shall contain as much as possible, ideally the whole, of the relevant information 
contained in the original data. 
This object is accomplished by constructing a hypothetical infinite population, of 
which the actual data are regarded as constituting a random sample. The law of distri¬ 
bution of this hypothetical population is specified by relatively few parameters, which 
are sufficient to describe it exhaustively in respect of all qualities under discussion. 
Any information given by the sample, which is of use in estimating the values of these 
parameters, is relevant information. Since the number of independent facts supplied in 
2x2 
