  We study a rank based univariate two-sample distribution-free test. The test
statistic is the difference between the average of between-group rank distances
and the average of within-group rank distances. This test statistic is closely
related to the two-sample Cram\'er-von Mises criterion. They are different
empirical versions of a same quantity for testing the equality of two
population distributions. Although they may be different for finite samples,
they share the same expected value, variance and asymptotic properties. The
advantage of the new rank based test over the classical one is its ease to
generalize to the multivariate case. Rather than using the empirical process
approach, we provide a different easier proof, bringing in a different
perspective and insight. In particular, we apply the H\'ajek projection and
orthogonal decomposition technique in deriving the asymptotics of the proposed
rank based statistic. A numerical study compares power performance of the rank
formulation test with other commonly-used nonparametric tests and
recommendations on those tests are provided. Lastly, we propose a multivariate
extension of the test based on the spatial rank.
