# Central limit theorem

The central limit theorem says that sample means from almost any population are close to normal when n is large enough (often 30 or more). The mean of the sample means is mu and their SD is sigma / square root of n.

Statistics (university) · Part 05: Sampling and estimation · Card 4 of 8 · Last updated 2026-10-19

## Analogy

Even if single dice rolls are flat, the average of many rolls forms a bell shape.

## Example

**P(sample mean > 53)**

Population mean 50, SD 10, n = 25
SE = 10 / 5 = 2
z = (53 - 50) / 2 = 1.5
P(Z > 1.5) = 1 - 0.9332 = 0.0668

## In practice

It is why z and t methods work for data that is not itself normal.

## Practice

### Flashcards

- **What does the central limit theorem say?** Sample means are approximately normal for large n, whatever the population shape.

### Interview questions

**Explain the central limit theorem and why it justifies using normal methods for sample means.**

- Sample means become approximately normal for large n
- Holds for non-normal populations
- Mean is mu, SD is sigma / square root of n
- Does not change the shape of the raw data

## Questions people ask

### How large must n be for the central limit theorem?

A common rule is 30 or more, but it depends on the shape. Near-symmetric data may need only 10 or so, while heavily skewed data may need hundreds. If the population is already normal, any sample size gives a normal sample mean.

### Does the central limit theorem make the data normal?

No. It says the distribution of the sample mean becomes normal. The individual values keep their original shape, skewed or not. A histogram of raw data stays lopsided, while a histogram of many sample means looks like a bell, which is the part the theorem describes.

## Related concepts

- [Sampling distribution and standard error](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-5/statistics-university-part-5-concept-3)
- [Population and sample](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-5/statistics-university-part-5-concept-1)
- [Law of large numbers](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-5/statistics-university-part-5-concept-2)

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