On the Distributions of Certain Statistics

Abstract

Introduction. It is the purpose of this paper to present certain general theories on the distributions of such statistics as the arithmetic mean, harmonic mean, geometric mean, median, quartile, decile and range of samples of n items selected at random from a rather arbitrary universe. We shall also develop, without appeal to n-dimensional geometry the distribution of the Pearson x2. Exact distributions are known only for a few statistics and these in case the sampled universes are of special types. For a normal parent population, the distribution of the means of samples of n was probably known to Gauss, while Czuber f seems first to have obtained the distribution of the sum of the squares of the deviations of the n items of the sample from the sample mean. Czuber's derivation was closely related to that of Helmert I who dealt with the distribution of sums of squares of deviations from a fixed point. These results of Helmert and Czuber were apparently unknown to Student ? who found the distributions of the standard deviation and of the ratio of the deviation of the mean of the sample from the population mean to the standard deviation of the sample in case the parent population is normal. Student's results, somewhat empirically obtained, were later verified by Karl Pearson.? By geometrical methods, R. A. Fisher also verified Student's distribution anld obtained the distributions of the correlation coefficient,11 the correlation ratio,** the regression coefficient,** the partial correlation coefficient tt and the multiple correlation coefficient 11