Central Limit Theorem (CLT)

The CLT is an essential theorem in probability and statistics about the distribution of the sample mean \(\bar{x}_n\) from any population with finite variance. The CLT states that sample means \(\bar{x}_n\) converges to a normal distribution as the sample size \(n\) grows.


Let \(x_1, x_2, \ldots, x_n\) a random sample from a population with mean \(\mu\) finite variance \(\sigma^2\), the sample mean

\[\bar{x}_n= \frac{1}{n} \sum_{i=1}^{n}x_i\]

has the next distribution

\[\sqrt{n} \left( \bar{x}_n - \mu \right) / \sigma \overset{d}{\rightarrow} N(0, 1), \quad n \to \infty \]

that is to say, that the distribution of \(\bar{x}_n\) can be written as

\[ \bar{x}_n \overset{aprox}{\sim} N(\mu, \sigma^2/n) \]

as \(n\) increases.


  • It guarantees a pattern of behavior of the sample means as long as the condition that \(n\) is large enough is met.

  • The mean of the distribution of \(\bar{x}\) coincides with the mean of the population \(\mu\).

  • The variability of the distribution of \(\bar{x}\) is given by \(\sigma^2/n\) and decreases as \(n\) increases.

  • As \(n\) increases, the sample means \(\bar{x}\) have a normal distribution regardless of the distribution of the population from which they were obtained.