The moments of the distribution for normal samples of measures of departure from normality

Abstract If x1...xn 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 kp will be the pth cumulative moment function of the sampled population; the first three of these are defined by the equations; k1 = 1/n S (x), k2 = 1/n-1 S (x-k1)2, k3 = n/(n-1) (n-2) S (x-k1)3; then it has been shown (fisher, 1929) that the cumulative moment functions of the simultaneous distribution, in samples, of k1, k2, k3,..., may be obtained by the direct application of a very simple combination procedure. The simplest measure of departure from normality will the be γ = k3k2-3/2, 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.