The classical theory of sampling is based on the following fundamental theorem.
When the distribution of any population has finite variance, then the distribution of the arithmetic mean of random samples is approximately normal, if the sample size is sufficiently large.
The proof of this theorem is usually about 3-6 pages (using advanced mathematics on measure theory). Rather than doing this mathematical exercise, the "proof" is done by simulation, which also helps to understand the central limit theorem and thus the basics of statistics.
The following setup is necessary: