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# 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.

# Symbolically

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.

# Implications

• 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.