Central Limit Theorem

Compact study note.

Summary

The central limit theorem explains why normalized sums and averages are often approximately normal, even when individual observations are not normal.^[1]

Prerequisites

• IID Random Variables
• Normal Distribution

Notation and Assumptions

For IID $X_i$ with mean $\mu$ and variance $0<{\sigma }^2<\mathrm{\infty }$ , $\frac{\sum _{i=1}^n{X}_i-n\mu }{\sigma \sqrt{n}}$ converges in distribution to $N(0,1)$ .

Essential Result

Equivalently, ${\overline{X}}_n$ is approximately $N(\mu ,{\sigma }^2/n)$ for large $n$ under the theorem assumptions.

Small Example

For many independent Bernoulli trials with success probability $p$ , the sample proportion is approximately normal with mean $p$ and variance $p(1-p)/n$ when $n$ is large enough.

Common Mistakes

• Applying the CLT to strongly dependent data without justification.
• Forgetting that the approximation is about normalized sums, not raw observations.

Connections

• Law of Large Numbers
• Normal Distribution
• Characteristic Function

References