Central Limit Theorem

• Take any random variable $X$ with finite variance.
• Add $N$ samples of this variable $X_1+X_2+\cdots +X_N$
• The distribution of this sum looks more and more like a bell curve as $N\to \infty$

→ A collection of iid random variables with finite variance have a sample mean that is distributed normally:

$$\bar{X}=\frac{1}{n}\sum _i^n{X}_i\sim \mathcal{N}(\mu ,\frac{{\sigma }^2}{n})$$

$X_i$ can be different measurements of the same random variable, different random variables, non-normally distributed ones, etc.

Another way of writing it, describing the sum of the random variables $S_n={\sum }_i^n{X}_i$

$$S_n\sim \mathcal{N}(n\mu ,n{\sigma }^2)$$

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Visually:
If we roll the dice a couple of times… add those distributions together and normalize them (zero mean, unit std), the distribution of values takes on the shape of a bell curve:

References

probability
But what is the Central Limit Theorem?