The classical theory of sampling is based on the following fundamental theorem.

### Tip

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:

We draw samples from populations. This means that we know the populations. This is not the case in practice, but we show that the population can have any distribution as long as the variance is not infinite.

We draw many samples from the population. Note that in practice, only one sample is drawn. For simulation purposes, we assume that we can draw many samples.

For the purpose of looking at our defined...