the calculation of numerical and functional characteristics of empirical distributions. If in any group of objects the index of the characteristic being studied shows a change (a variation) from one object to the next, then each value of that index from x1,…, to xn (with n being the total number of objects) is given a corresponding index of probability equal to 1/n . This formally introduced “distribution of probabilities” is called an empirical distribution and may be regarded as the distribution of probabilities of an artificially introduced auxiliary random value xi with probability pi = 1/n (i = 1, …, n). This makes it possible, for the purposes of variational statistics, to use all the concepts and results of the general theory of discrete distributions, of which empirical distributions represent a particular case. For example, the relationships between moments of empirical distribution that are used in variational statistics are particular cases of analogous correlations for moments of random values. The more comprehensive and mathematically more rigorous mathematical interpretation of variational statistics can be made only in cases where the results of observations x1, …, xn are random values. Given a sufficiently large number of observations n, empirical distribution, owing to the law of large numbers, does serve as a good statistical estimate of an unknown theoretical distribution of random values xi and in this situation, variational statistics becomes a useful auxiliary to mathematical statistics. Attempts to justify variational statistics outside the boundaries of probability theory and mathematical statistics have not led to serious theoretical results.
L. N. BOL’SHEV