The efficient market hypothesis has become a controversial subject due to empirical results against market efficiency in various stock markets. The measurement of the Nigerian stock market is considered crucial for several reasons, including investors and participants in the securities market decried non-reflection of the firm performance in the prices of their stock traded in the Nigerian stock exchange. The focus of this study is to assess the weak form efficiency of the Nigerian stock exchange market. Nigerian stock exchange all share historical daily, weekly and monthly returns were employed for the analysis. The data was analyzed using unit root tests of stationarity and random walk, Jarque-Bera for normality and graph presentation. The results revealed that, the Nigerian stock exchange all share historical daily, weekly and monthly returns exhibited significant random walk. The study concludes that the Nigerian stock exchange market is efficient in the weak form. It is therefore recommended that efforts should be made by government to develop and implement stock exchange policies that will ensure active trading and vibrancy in the market.