Abstract
Abstract
The Central Limit Theorem (CLT) ensures asymptotic normality, but the convergence rate is highly affected by population skewness. Common rules of thumb (e.g., \(\:n\ge\:30\)) are often inadequate for asymmetric data. This study quantifies this relationship using a wide range of simulations from the Beta distribution family. We derive a conservative quadratic regression model, \(\:{n}_{cons}\approx\:\text{exp}\left(1.59+1.41\left|\gamma\:\right|+0.61{\left|\gamma\:\right|}^{2}\right)\), to predict the minimum sample size required for normality as a function of skewness (\(\:\left|\gamma\:\right|\)). Validation against hold-out distributions reveals that while the model is robust for continuous, moderate-kurtosis data, convergence is significantly delayed by discreteness (lattice effects) and severe excess kurtosis, requiring sample sizes beyond standard recommendations.