Summary: The Cramér-von Mises test statistic for the general \(k\) sample problem can be rewritten in terms of Gini’s mean difference with repetition, and this in turn leads to a derivation as a score test from a self-exciting point-process model. It also leads to a computationally useful form of the test statistic as a correlation between a function of an observation’s rank within its own group, and its rank in the ordered combined sample. The practical benefits of this viewpoint are: a better correction for ties; a recommended form for the corresponding stratified test; a new large-sample approximation for \(p\)-values, and a new test sensitive to localised differences between samples. The new correction for ties remedies a flaw in the existing test, in that when observations are tied, the \(p\)-value of the test is not invariant under a change in direction of measurement of the random variable \(X\to-X\).