Suppose we wish to test. H0: the X and Y distributions are identical versus Ha: the X distribution is less spread out than the Y distribution The accompanying figure pictures X and Y distributions for which Ha is true. The Wilcoxon rank-sum test is not appropriate in this situation because when Ha is true as pictured, the Y's will tend to be at the extreme ends of the combined sample (resulting in small and large Y ranks), so the sum of X ranks will result in a W value that is neither large nor small.
Consider modifying the procedure for assigning ranks as follows: After the combined sample of m + n observations is ordered, the smallest observation is given rank 1, the largest observation is given rank 2, the second smallest is given rank 3, the second largest is given rank 4, and so on. Then if Ha is true as pictured, the X values will tend to be in the middle of the sample and thus receive large ranks. Let W' denote the sum of the X ranks and consider an uppertailed test based on this test statistic. When H0 is true, every possible set of X ranks has the same probability, so 'the same distribution as does W when H0 is true. The accompanying data refers to medial muscle thickness for arterioles from the lungs of children who died from sudden infant death syndrome (x's) and a control group of children (y's). Carry out the test of H0 versus Ha at level .05.
Consult the Lehmann book (in the chapter bibliography) for more information on this test, called the Siegel-Tukey test.