A fundamental theorem of statistics connects the distribution of a growing sample from a given distribution with the normal distribution. Formally, let X₁, X₂, X₃,…be a sequence of independent, identically distributed random variables with mean μ‎ and finite variance σ‎², and let

Then, as n increases, the distribution of Z_n converges in distribution to the standard normal distribution with mean 0 and variance 1.

It implies that for a large sample the sample mean has approximately the normal distribution with mean μ‎ and variance σ‎²/n.

http://onlinestatbook.com/stat_sim/sampling_dist/index.html An applet that lets you define the population and then take multiple samples of different sizes to explore the behaviour of the Central Limit Theorem.