Wilcoxon two-sample test. Let Uij = 1 or 0 as Xi < Yj or Xi >

Yj , and let U =

Uij be the number of pairs Xi, Yj with Xi < Yj .

(i) Then U =
Si − 1 2n(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 . . . xy . . . 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.