In this paper, we study the power of moment‐based normality tests which include Jarque Bera (JB) test and D′Agostino and Pearson (DP) omnibus tests. Power comparison were obtained via Monte Carlo simulation of sample data generated from four alternative distributions like Uniform, Logistic, Student t and Gamma distribution. Our simulation results show that for Uniform distribution, DP test has better power compared to JB test. For Logistic, Student t and Gamma distributions, we find JB normality test to be powerful compared to DP test. We further apply the moment‐based normality tests empirically on the Indian stock market indices (NSE Nifty 50 and BSE Sensex) for different frequencies (daily, weekly, monthly and quarterly) during the period from 2010 to 2020. We find that daily returns of Indian stock indices are non‐normal whereas weekly, monthly and quarterly returns are normally distributed.