Vincent Granville explains the central limit theorem:

The theorem discussed here is the central limit theorem. It states that if you average a large number of well behaved observations or errors, eventually, once normalized appropriately, it has a standard normal distribution. Despite the fact that we are dealing here with a more advanced and exciting version of this theorem (discussing the Liapounov condition), this article is very applied, and can be understood by high school students.

In short, we are dealing here with a not-so-well-behaved framework, and we show that even in that case, the limiting distribution of the “average” can be normal (Gaussian.). More precisely, we show when it is and when it is not normal, based on simulations and non-standard (but easy to understand) statistical tests.

Read on for more details.