# Sampling distribution and standard error

A sampling distribution shows how a statistic, such as the sample mean, varies from sample to sample. Its standard deviation is the standard error: SE = sigma / square root of n.

Statistics (university) · Part 05: Sampling and estimation · Card 3 of 8 · Last updated 2026-10-19

## Analogy

If many people each tasted a different spoonful, their verdicts would vary. The standard error measures that variation.

## Example

**Quadruple the sample**

sigma = 12
n = 36: SE = 12 / 6 = 2
n = 144: SE = 12 / 12 = 1
4 times the sample halves the SE

## In practice

A bigger sample gives a smaller standard error, so the estimate is more precise.

## Practice

### Flashcards

- **Formula for standard error of the mean?** sigma / square root of n.

### 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

### What is the difference between standard deviation and standard error?

Standard deviation measures how spread out individual values are. Standard error measures how spread out a statistic, such as a sample mean, is across repeated samples. The standard error is smaller, equal to the standard deviation divided by the square root of n.

### Why do I need four times the data to halve the error?

Standard error shrinks with the square root of n, not with n itself. To halve it you must make the square root twice as big, which means multiplying n by four. Precision gets expensive: each extra bit of accuracy costs more data.

## Related concepts

- [Central limit theorem](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-5/statistics-university-part-5-concept-4)
- [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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