The data for the tests considered here may be presented in two-way contingency tables with all marginal totals fixed. We show that Pearson's test statistic X P 2 ( P for Pearson) may be partitioned into useful and informative components. The first detects location differences be tween the treatments, and the subsequent components detect dispersion and higher order moment differences. For Kruskal-Wallis-type data when there are no ties, the location component is the Kruskal-Wallis test. The subsequent components are the extensions. Our approach enables us to generalise to when there are ties, and to when there is a fixed number of categories and a large number of observations. We also propose a generalisation of the well-known median test. In this situation the location-detecting first component of X P 2 reduces to the usual median test statistic when there are only two categories. Subsequent components detect higher moment departures from the null hypothesis of equal treatment effects