Alternate Forms of the One-Way ANOVA F and Kruskal–Wallis Test Statistics - Johnson, Roger W. (Journal of Statistics and Data Science Education, 2022)
For ease of instruction in the classroom, the one-way analysis of variance F statistic is rewritten in terms of pairwise differences in individual sample means instead of differences of individual sample means from the overall sample mean. Likewise, the Kruskal-Wallis statistic may be rewritten in terms of pairwise differences in individual average ranks rather than differences of individual average ranks from the overall average rank. In unbalanced designs, it is seen that the contribution to either test statistic from a pair of samples is related to the product of the sample sizes multiplied by the square of the relevant pairwise difference. Supplementary materials for this article are available online.