The Fundamental Statistics Theorem Revisited

In this article, we revisit the most fundamental statistics theorem, talking in layman terms. We investigate a special but interesting and useful case, that is not discussed in textbooks, data camps, or data science classes. This article is part of a series about off-the-beaten-path data science and mathematics, offering a fresh, original and simple perspective on a number of topics. Previous articles in this series can be found here and also here.

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. Content

1. A special case of the Central Limit Theorem

2. Generating data, testing, and conclusions

• Simulations
• Analysis and results
• The Liapounov connection

3. Generalizations

• Generalization to correlated observations
• Generalization to non-random (static) observations
• Other interesting stuff related to the Central Limit Theorem

Appendix: source code and chart