The Moments of the Distribution for Normal Samples of Measures of Departure from Normality

If x 1... x n are the values of a variate observed in a sample of n , from any population, we may evaluate a series of statistics ( K ) such that the mean value of k p will be the p th cumulative moment function of the sampled population; the first three of these are defined by the equations; k 1 = 1/ n S ( x ), k 2 = 1/ n -1 S ( x - k 1)2, k 3 = n /( n -1) ( n -2) S ( x - k 1)3; then it has been shown (fisher, 1929) that the cumulative moment functions of the simultaneous distribution, in samples, of k 1, k 2, k 3,..., may be obtained by the direct application of a very simple combination procedure. The simplest measure of departure from normality will the be γ = k 3 k 2-3/2, a quantity which is evidently independent of the units of measurements, and in samples from a symmetrical distribution will have a distribution symmetrical about the value zero. In testing the evidence provided by a sample, of departure from normality, the distribution of this quantity in normal samples is required.