The central limit theorem explains why averages of many independent measurements look approximately bell-shaped, even when the individual measurements do not.
The statement
Take independent, identically distributed random variables with finite mean μ and finite variance σ². The standardized sample mean, (mean − μ) divided by σ/√n, converges in distribution to a standard normal distribution as n grows.
Reading it in practice
The spread of a sample mean shrinks like 1/√n. To cut the typical error in half, you need four times as much data. That scaling shows up again and again in machine learning, from evaluating a model on a test set to estimating gradients from mini-batches.
Why it justifies confidence intervals
Because sample means are approximately normal, we can build intervals such as mean ± 1.96 × (standard error) with about 95% coverage for large samples.
When to be careful
- Heavy tails: distributions with infinite variance break the theorem.
- Dependence: correlated data reduces the effective sample size.
- Small samples: the normal approximation may be poor, especially for skewed data.
Related limits
The law of large numbers says the sample mean converges to the true mean. The central limit theorem describes the shape and scale of the error. Concentration inequalities give non-asymptotic bounds when you need guarantees for finite samples.
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Adapted from Mathematics for AI and Machine Learning, published by Choral Software. Article text and original examples © 2026 Choral Software. All rights reserved.