Home Statistical Dictionary Central Limit Theorem

Central Limit Theorem

As sample size grows, the sampling distribution of the mean approaches a normal distribution, regardless of the shape of the original population.

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In Plain English

It doesn't matter if your underlying data is skewed, bimodal, or anything else -- if you take enough samples and average them, those averages pile up in a bell-curve shape. This is why the normal distribution shows up everywhere in statistics: it's not that nature is normal, it's that averages of almost anything become normal.

Definition

Let $X_{1}, \dots, X_{n}$ be i.i.d. random variables with mean $μ$ and finite variance $σ^{2}$ . As $n$ approaches infinity, the standardized sample mean converges in distribution to a standard normal distribution $N (0, 1)$ .

Formula

\frac{X ˉ - μ}{σ / n} \to N (0, 1) as n \to \infty

Properties

The original population can have any shape -- skewed, uniform, even bimodal.
Convergence is slower for more skewed or heavy-tailed populations.
It justifies treating sample means as approximately normal even when the underlying data isn't.

At a Glance

Requiresfinite mean and variance

Applies tothe sample mean, not individual observations

Rule of thumb

n \geq 30

is often enough for moderate skew

Powerst-tests, z-tests, most large-sample inference

Last updated July 30, 2026← Back to the Dictionary