The Central Limit Theorem and Sampling

> According to the central limit theorem, the average value of the data sample will be closer to the average value of the whole population and will be approximately normal, as the sample size increases.

This is a pretty loose statement of the central limit theorem. Sample averages converge to the population average by the strong law of large numbers (almost surely, under some mild conditions). The central limit theorem is a statement about the differences between sample averages and population mean. Multiply the differences (sample avg. - pop mean) by sqrt(sample size) and let sample size go to infinity to converge to a Normal distribution. (under some stronger conditions).

This is 1st year undergraduate statistics and has nothing to do with data science. The graphics are cute though. I'm sorry for being so blunt.

> this is true regardless of the distribution of population

Nope. Not true for all the distributions. For example, stock prices cannot be used in central limit theorem.

Distributions must have second moment, in other words, have finite variance.

It is easy to find this material. This link adds nothing to the vast amount of tutorials in probability.