A distribution-free test of the hypothesis that identically distributed paired observations are independent may be based on a test statistic,D, which measures the variation between the ranks in each pair of observations. Examples of applications of this proposed test are discussed. Further, since the statistic is related to the one used in the Kruskal-WallisH-test applied to samples of 2, some results are of interest for what they suggest about theH-test. Properties of the proposed test under the null hypothesis that are considered include exact sampling distributions, some exact probabilities, moment coefficients, the asymptotic distribution, and approximations to critical values. Large-sample theory for describing alternative hypotheses, for approximating the power of the test, and for determining sample size is also suggested.