On the Distribution of the "Student" Ratio for Small Samples from Certain Non-Normal Populations

Abstract

deviation of a sample of N items, say x1, X2, * , XN, taken at random from a normally distributed parent population of mean, m. The investigations of certain non-normal parent distributions by Shewhart and Winters [1], Rider [2], E. S. Pearson [3], M. S. Bartlett [4], and R. C. Geary [5] indicate that applications of the Student theory give more satisfactory results than the classical theory for a considerable variety of non-normal parent distributions, but some of these investigators find that the theory fails in certain cases to describe the facts to an extent that suggests further experimental sampling investigations along this line whenever suitable data are available. Others infer that a completely satisfactory analysis of the position of the Student z-test will be possible only if the theoretical distribution of z in samples from the non-normal distribution in question becomes known. Several of the above named statisticians have attributed the failures of the distributionl (1) to describe their data, in large part, to the correlation between x = m and s. For this reason, there is considerable interest in the degree of correlation betw-een x = m and s, and especially in the nature of the regression of s or of 82 on x. The present paper gives an analysis of data obtained by experimental sampling from two non-normal distributions whose soturces we slhall liow describe. The parent distributions with which the paper is concerned are theoretical distributions resulting from certain urn schemata devised [6] by the writer some years ago. In 1925, Leone E. Chlesire, in an uiipublislhed thesis for the degree, Master of Science, at the University of Iowa, obtained data by experimental sampliing. that seem to be appropriate material for a study of the correlation of mean and standard deviation for small samples from certain non-normal distributions. One of the original bivariate pareiit populations, wlhose margiinal totals we are