On Hypothesis Testing Under Unequal Group Variances: The Use of the Harmonic Variance

Abstract

The assumptions of equality of group variances required for ANOVA often fail for real life data. However, because the major aim is to test equality of group means, a single summary value for these group variances is a necessity. Previous works in literature have zeroed in on the choice of either the Harmonic or Geometric mean as a proper mean especially when extreme observation(s) are present which renders a simple mean inappropriate.

In this work, the asymptotic sample distribution of harmonic mean of group variances is established to be a chi- square distribution though the degrees of freedom need not be an integer.

Keywords:  Harmonic mean of group variances, three - parameter Beta distribution, chi – square distribution.