THE DISTRIBUTION OF THE RATIO, IN A SINGLE NORMAL SAMPLE, OF RANGE TO STANDARD DEVIATION

vations. While the first ratio can play a useful part in providing certain rapid tests in the analysis of variance, the second, whose value depends only on the configuration of a particular sample, may be useful (with suitable tables) in detecting heterogeneity of data or departure from normality. Our interest in this matter arose as a result of correspondence between one of us and Dr Joseph Berkson who has for some time carried out a routine scrutiny of data by making a comparison of the range and standard deviation estimators of oa; in this connexion he initiated an empirical investigation into the correlation between the estimators in order to determine the standard error of the difference between them. Some years earlier, however, G. A. Baker (1946) suggested the use of the ratio w/s as a means of detecting lack of homogeneity, and showed by an artificial sampling experiment that the distribution of this ratio might be expected to change considerably with the form of the parent population. The object of the present paper is to provide tables of certain upper and lower percentage points of u = w/s for samples of n observations from a single normal population. Two methods of attack will be used: (a) We shall show that the exact moments of the distribution of u can be derived quite simply from the known moments of s and w. Hence approximations to the percentage points may be obtained from representing the distribution by any suitable frequency curve having the same moments, e.g. by a curve of the Pearson system. (b) Using a method employed by Pearson & Chandra Sekar (1936), we shall show how in small samples values of the upper percentage points of u may be calculated exactly. In an overlapping region of the table, these two methods provide some confirmation of the accuracy of the results. Finally, we shall give some numerical illustrations of how the ratio might be used to provide a quick assessment of the homogeneity or normality of data.