This article derives analytic finite sample approximations to the bias and standard error of a class of statistics which test the hypothesis of no serial correlation in market returns. They offer an alternative to both the widely used Monte Carlo approach for calculating the bias, as well as asymptotic standard error calculations. These approximations are calculated under the assumption that returns are spherically symmetrically distributed (such as Gaussian) and also under the weaker assumption that returns follow any arbitrary continuous distribution. The class of statistics examined here includes many of those employed in the finance and macroeconomics literature to test for the existence of random walk, including the variance ratio and the multi-period return regression on past returns. The accuracy of the approximations is benchmarked using simulated data, where arbitrarily tight estimates of the bias and standard error can be calculated. The approximations are then applied to adjust the statistics calculated using returns on the NYSE from 1926-1991.