For testing the hypothesis that successive members of a series of observations are independent J. von Neumann [5] (see also B. I. Hart [4]) and R. L. Anderson [1] have proposed test statistics and tabulated their significance points. von Neumann's criterion seems well designed to detect deviations from the null hypothesis which might be encountered in practice but its exact distribution is unknown. On the other hand Anderson's statistic, while it has a known distribution, is based on a circular conception of the population which is rarely plausible in practice. In the present note certain noncircular statistics are proposed for which exact distributions can be obtained from Anderson's results. The statistics are derived from the usual noncircular statistics by sacrificing a small amount of relevant information. Their application is noted to certain regression problems for which no satisfactory tests are at present available. Finally, some general remarks are made about the choice of best statistics for the problems discussed.