# Statistics - Central limit theorem (CLT)

The central limit theorem says that the averages of several samples obtained from the same population (ie a sampling distribution) following the central limit theorem rules will be distributed according to the normal distribution.

Therefore:

The central limit theorem began in 1733 when de Moivre approximated binomial probabilities using the integral of $exp(-x^2)$. The central limit theorem achieved its final form around 1935 in papers by Feller, Lévy, and Cramér.

The central limit theorem is a fundamental component of inferential statistics

## 3 - Rules

• The sample must contain a large number of observations (N>30)
• Each observation must be randomly generated (No relationship/dependencies between the observations)
• The shape of the distribution of sample means is always normal (not negatively or positively skewed, not uniform)

The population doesn't have to be normally distributed, as long as we get multiple samples of large enough size (N>30) then the sampling distribution will take on a normal distribution.