Wilcoxon two-sample test. Let Ui j = 1 or 0 as Xi < Yj or Xi > Yj , and let U = Ui j be the number of pairs Xi , Yj with Xi < Yj .

(i) Then U = Si − 1 2 n(n + 1), where S1 < ··· < Sn are the ranks of the Y ’s so that the test with rejection region U > C is equivalent to the Wilcoxon test.
(ii) Any given arrangement of x’s and y’s can be transformed into the arrangement x ... x y ... y through a number of interchanges of neighboring elements. The smallest number of steps in which this can be done for the observed arrangement is mn − U.