The central limit theorem (CLT) establishes that, when independent random variables are added, their properly normalized sum (and therefore their mean) tends toward a normal distribution even if the original variables themselves are not normally distributed.
Here we just take mean value of $n$ samples over $m$ trials using various different distributions.
using Distributions, Statistics, Plots
dists = [Normal(), Beta(), Gamma(), Poisson(), Exponential(), LogNormal()]
n = 50 # Num samples to obtain mean
m = 1000 # Number of sample means
means = [[mean([rand(dist) for i in 1:n]) for j in 1:m] for dist in dists]
histogram(means, layout=length(dists), leg=:none)
Here we see convergence to a normal distrubution