Central Limit Theorem

The Central Limit Theorem (CLT) states that for i.i.d. samples $X_1,\dots,X_n$ with finite mean $\mu$ and variance $\sigma^2 < \infty$, the standardized sample mean converges in distribution to a standard normal: $\frac{\bar{X} - \mu}{\sigma/\sqrt{n}} \xrightarrow{d} \mathcal{N}(0,1)$, at rate $O(1/\sqrt{n})$; the underlying samples need not be normal.

normal approximation for probabilities and quantiles of $\bar{X}$ when its exact distribution is intractable — the theoretical basis of Z-tests, CI formulas and Bootstrap; interviews typically probe “why $\sqrt{n}$”, the convergence rate and when it breaks.

the LLN only asserts $\bar{X} \to \mu$ and Chebyshev gives only a crude $O(1/n)$ bound; the CLT gives distribution-level precision: fluctuations cancel in the sum, so the standard error decays as $1/\sqrt{n}$ — halving the error costs 4x the samples, and going from $n=100$ to $n=10000$ shrinks the error only 10x.

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