ON THE MATHEMATICAL EXPECTATION OF THE 
MOMENTS OF FREQUENCY DISTRIBUTIONS. 
By professor AL. A. TCHOUPROFF of Petrogmd. 
INTRODUCTION 
I 
(1) One of my pupils, O. Anderson, in a brief exposition * of his researches on. 
the Variate Difference Correlation Method in Biometrika (1 914<), draws attention 
to the supeiiority of the method of mathematical expectation over the methods 
usually employed by English statisticians. The small popidarity enjoyed by the 
method of mathematical expectation in England is not of course accidental. 
English scientific tradition rejects the concept of " mathematical probabilitj'." 
From the time of R. L. Ellis and of the first edition of John Stuart Mill's 
Si/stem of Logic, the logician's basis of probability has, in England, been the notion 
of empirical frequency. English mathematicians have followed the lead of the 
writers on logic in their preference for the idea of statistical frequency, and the 
method of mathematical expectation has naturally shared the fate of the concept 
of mathematical probability on which it rests. 
Notwithstanding its deep-rooted historical basis, English statisticians should 
break with this tradition. The substitution of statistical frequency for mathe- 
matical probability does not obviate the logical difficulties in laying the foundations 
for a statistical study of Causation, but merely shifts them elsewhere. The gain 
from the point of view of philosophical representation is sufficiently doubtful, 
while from the purely mathematical point of view the rejection of the ideas of 
mathematical probability and mathematical expectation is accompanied by very 
substantial disadvantages. Verbal formulation becomes very complicated, leading 
to loss of economy of attention : it is continually necessary to speak of " the statis- 
tical frequencies which would become established if the number of occurrences 
were infinitely great." The absence of a sharp distinction in terminology between 
statistical fi-equency in the exact meaning of the term and those quasi-empirical 
* Anderson's research was carried out under my supervision in the statistical seminary attached to 
the Economics Departnjent of the Petrograd Polytechuic Institute; tlie results he obtained were to 
have been published in extenxo in the Proceedingn (Stuilejits' Section) of the Economics Department, 
but the War dr. \v Mr Anderson away from his scientific pursuits to otlier work of a more practical 
character and the complete publication of his researches had to be postponed. 
