The Distribution of the Extreme Deviate from the Sample Mean and Its Studentized Form

Abstract

The distrs. of McKay''s statistic [nu] = (xn - [image])/[sigma]rand of its studentized form (xn - [image])/s[nu] , where xn is the largest observation and [image] the mean in a sample of n from a normal distr. and sp is an independent estimate of cr based on y degrees of freedom, are obtained. An extensive table of the probability integral of [mu] and a less extensive table of the first 3 coefficients in Hartley''s (Biometrika, 1944) expansion for the studentized form including terms in [nu]-2 are appended for n = 3-9, inclusive. Approximations are given which may make it possible to extend the tables to larger n. The application of the studentized statistic to analysis of variance is illustrated by a test of the significance of deviation of the most extreme of a set of group means. Godwin''s G-functions are used to obtain the distr. of the difference between the means of the k smallest and the 1 largest values in a sample of n from a known normal population for the cases k + 1 = n, (n-1) and (n-2).