Sample Size Requirements for the Central Limit Theorem for Skewed Distributions: A Simulation Study

Abstract

The Central Limit Theorem (CLT) plays a foundational role in statistical inference, often serving as the rationale for assuming a normal approximation of the sample mean. Yet, the pace at which this assumption becomes valid is influenced by the shape of the parent distribution, especially its skewness. This research quantifies the minimum number of observations required for the mean of samples drawn from skewed, non-normal distributions specifically Gamma, Poisson, Binomial, and Beta to achieve a satisfactory normal approximation. We implemented a Monte Carlo simulation and applied both the Shapiro-Wilk and Kolmogorov-Smirnov tests to assess the adequacy of the normal approximation. Results indicate a nonlinear association between the degree of skewness and the sample size required for acceptable normal approximation. For distributions with mild asymmetry (|skewness| < 0.5), 20 samples often suffice, whereas more heavily skewed distributions (|skewness| ≥ 2.5) may necessitate sample sizes beyond 100. These findings call into question the blanket use of the "n ≥ 30" heuristic and suggest more tailored guidelines are necessary for accurate inference. A graphical overview summarizes these results across the examined distributional families, offering clear guidance for applied researchers working with non-normal data.