Determining The Relative Normality Convergence Speed Across Different Skewness and Tail Heaviness of Population Distribution

This study investigates how the different distributions with varying characteristics and sample sizes converge to the normal distribution at different speeds. Using uniform, exponential, lognormal, and Pareto distributions, we find that the approximation error decreases as the sample size increases, which is consistent with the Central Limit Theorem. However, the rate of convergence differs depending on the shape of the distribution. Distributions with greater skewness and heavier tails tend to converge more slowly. The empirical analysis using IPUMS data shows similar patterns. As the sample size increases, the approximation error decreases, while the scaled error remains relatively stable. Overall, the results suggest that although normal convergence generally occurs, the speed of convergence varies depending on distributional characteristics and may not strictly follow the theoretical rate.