The Central Limit Theorem is a statistical concept that states that the sum of independent samples from any distribution with finite mean and variance converges to the normal distribution as the sample size goes to infinity, regardless of the shape of the original distribution of the variables.
This theorem is fundamental in statistics and allows researchers to make inferences about a population based on the distribution of sample means, even if the underlying population is not normally distributed.
Consider a set of independent, identically distributed random variables \(X_1, X_2, \ldots, X_n\) with a common mean \(\mu\) and a finite variance \(\sigma^2\), and let \(\overline{X_n}\) denote the sample mean of that sample. Then: \[ \lim_{{n \to \infty}} \frac{{\bar{X}_n - \mu}}{{\sigma_{\bar{X}_n}}} = Z, \quad \text{where } \sigma_{\bar{X}_n} = \frac{\sigma}{\sqrt{n}} \]
As observed in the simulation, the distribution of the population mean becomes increasingly similar to a normal distribution as the number of samples grows larger. The top-left graph displays the original distribution, while the top-right, bottom-left, and bottom-right graphs depict CLT simulations with 1000, 50, and 30 samples, respectively.