The failure of the F – test statistic and the Kruskal – Wallis (KW) H – test statistic to immediately identify the treatments that are responsible for the rejection of the null hypothesis of no difference can be seen as a shortcoming. However, very many post – hoc (i.e., multiple comparison) methods have been introduced to tackle this deficiency. Some of these also have their limitations especially when the power of the tests is considered since the error rates may differ. Statisticians have been living with these shortcomings and limitations. The work in this paper is an attempt at dealing with the shortcomings of the F – test, the H – test as well as the Friedman's test for the one – way and two – way ( or the single factor and two factors) analysis of variance.