This article explores the identical distribution hypothesis for stock-price changes through a set of optimum non-parametric tests for randomness against location- and scale-shift alternatives. An application of these tests to the daily, weekly, monthly, and quarterly rates of return from 1966 through 1976 for each of the first 15 stocks in the Dow Jones Industrial Average reveals that these series may have constant location parameters throughout the sample period, but the stability of their scale parameters is questionable.