This study sought to determine the effect upon Tukey's Honestly Significant Difference (HSD) statistic of concurrently violating the assumptions of homogeneity of variance and equal sample sizes. Two forms for the unequal sample size problem were investigated. Kramer's form and the harmonic mean approach were the two unequal sample size procedures studied. The study employed a Monte Carlo simulation procedure which varied sample sizes with a heterogeneity of variance condition. Four thousand experiments were generated. Findings of this study were based upon the empirically obtained significance levels. Five conclusions were reached in this study. The first conclusion was that for the conditions of this study the Kramer form of the HSD statistic is not robust at the .05 or .01 nominal level of significance. A second conclusion was that the harmonic mean form of the HSD statistic is not robust at the .05 and .01 nominal level of significance. A general conclusion reached from all the findings formed the third conclusion. It was that the Kramer form of the HSD test is the preferred procedure under combined assumption violations of variance heterogeneity and unequal sample sizes. Two additional conclusions are based on related findings. The fourth conclusion was that for the combined assumption violations in this study, the actual significance levels (probability levels) were less-than the nominal significance levels when the magnitude of the unequal variances were positively related to the magnitude of the unequal sample sizes. The fifth and last conclusion was that for the concurrent assumption violation of variance heterogeneity and unequal sample sizes, the actual significance levels significantly exceed the nominal significance levels when the magnitude of the unequal variances are negatively related to the magnitude of the unequal sample sizes.