In [1], the author proposed a "rank sum" criterion for testing whether or not a sample was drawn from a population having a completely specified continuous distribution. This criterion under the null hypothesis is distributed as the sum in a random sample from a discrete uniform population; it is otherwise distributed as the sum from a more general discrete population. In this paper we give a method of finding numerically the distribution of such random variables. Tables are given for the distribution of the sums from certain selected discrete uniform distributions. Normal approximations are investigated and applications are briefly discussed.