The Exact Distribution of Bartlett's Test Statistic for Homogeneity of Variances with Unequal Sample Sizes

An expression is derived for the null density of the Bartlett test statistic for testing equality of variances of n normal populations, when random samples not necessarily of the same size are taken. The expression permits computation of exact Bartlett critical values. As a generalization, an expression is derived for the density of the normalized ratio of an arbitrarily weighted geometric mean to the unweighted arithmetic mean, based on a sample of n independently distributed gamma random variables having common scale parameter but different shape parameters.