# The central limit theorem

## The gist of it

The core of the CLT is that each sample of random variables, when their number is large enough, will converge to a gaussian distribution.

## A more precise formulation

Refer to the page on independent variables
Let
${x_1, \ldots, x_n }$
be a sample of iid random variables extracted from a distribution whose expected value is
$\mu$
and whose standard deviation is
$\sigma$
, then, as
$n \to \infty$
,
$\sqrt{n} (s_n - \mu) \xrightarrow[]{\text{d}} \mathcal{N(0, \sigma^2)} \ ,$
that is, the difference between the sample average
$s_n$
and the population mean
$\mu$
, multiplied by
$\sqrt{n}$
converges in distribution to a normal with mean 0 and variance
$a = b$
.
Which means, the sample converges to a gaussian with mean
$\mu$
and variance
$\sigma^2$
.

1. 1.