The Central Limit Theorem (CLT) offers asymptotic results on the convergence of sample means to normality, but its finite sample properties are relatively less clear. This paper reports the findings of a simulation study on the convergence process of CLT for various continuous distributions, such as Normal, Exponential, Uniform, Gamma, Beta, and Cauchy. By conducting Monte Carlo simulations in R and employing the Shapiro–Wilk test for normality, we have found distribution-wise “threshold” and “optimum” sample sizes for which approximate normality is achieved.

One of the most important results of this study is that the convergence process to normality is not monotonic for finite samples. Even after achieving normality at a threshold sample size, statistically significant deviations are often found at intermediate sample sizes. This “post-threshold instability” is found to occur for all studied distributions, including the Normal distribution itself, for which non-negligible failure rates are found for specified ranges of sample size.

We also investigate the possible causes of these anomalies, focusing on the importance of outliers, simulation methods, and the limitations of normality tests. The empirical findings indicate that the behavior of outliers and the properties of the tails of distributions are of prime importance, especially for skewed and heavy-tailed distributions. In the context of the Cauchy distribution, the traditional CLT is invalid because of the lack of finite variance. Nevertheless, the use of the sample median and its asymptotic distribution offers a promising alternative.

Moreover, we illustrate that the number of iterations is not necessarily a factor that improves the accuracy of convergence with certainty, contrary to general beliefs in Monte Carlo simulation. The effect of the size of the iteration is also shown to be dependent on the distribution.