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Central Limit Theorem (CLT)

The sum (or average) of many independent, identically distributed random variables with finite mean and variance tends toward a Normal distribution as sample size grows.

Key points

Aspect	Explanation
Requirements	Independent (or weakly dependent) variables, finite mean and variance
Convergence	Distribution of the standardized sum/mean → standard normal as \(n \to \infty\)
Implication	Sampling distributions of means approximate Normal even if original data are non-normal
Use in practice	Justifies confidence intervals and many hypothesis tests

Practical example

Step	Description
Sample	Draw \(n\) observations from any distribution with mean \(\mu\) and variance \(\sigma^2\)
Compute	Sample mean \(\bar{X}\)
Standardize	\(Z = \dfrac{\bar{X}-\mu}{\sigma/\sqrt{n}}\)
Result	For large \(n\), \(Z \overset{approx}{\sim} \mathcal{N}(0,1)\)