Central Limit Theorem

Standardized sums of independent terms approach a normal, whatever their shape.

The idea

Theorem (Central Limit Theorem).

Let $X_{1}, X_{2}, \ldots$ be independent and identically distributed with mean $\mu$ and finite variance $\sigma^{2}$, and write $S_{n} = X_{1} + \cdots + X_{n}$. Then as $n$ grows, the distribution of $\frac{S_{n} - n\mu}{\sigma\sqrt{n}}$ approaches the standard normal $\mathcal{N}(0, 1)$.

The fraction makes the two corrections without which $S_{n}$ has no limiting shape. Subtracting $n\mu$ recenters the sum, which drifts as terms accumulate; dividing by $\sigma\sqrt{n}$, the standard deviation of $S_{n}$, holds its spread fixed. The theorem says what shape the standardized sum then settles into.

The hypothesis never mentions the distribution of a single $X_{i}$. It may take two values or be spread over an interval, symmetric or lopsided; only $\mu$ and $\sigma^{2}$ survive into the limit. The figure shows this with dice: one die is flat, the standardized sum of two dice is a triangle, and the sum of five already tracks the standard normal curve. Any quantity assembled from many small independent contributions inherits this shape, which is why the normal curve appears in measurements across unrelated fields.

Ways to work on it

• Walkthrough. Mean and variance of a sum, standardizing, and the normal limit.
• Practice. Standardize a sum to a z-score.
• Hardest. Normal-approximate a tail probability for a sum.

Not sure where to start? Take the ten-question placement test.