# Standard deviation

The standard deviation measures how far values typically sit from the mean. A small value means the data is bunched up. A large value means it is spread out. It uses the same units as the data.

Statistics (university) · Part 01: Describing data · Card 5 of 9 · Last updated 2026-10-19

## Analogy

Two archers both average the centre of the target. The one whose arrows land closer together has the smaller standard deviation.

## Example

**Population SD**

Data: 2, 4, 4, 4, 5, 5, 7, 9
Mean = 40 / 8 = 5
Squared gaps: 9,1,1,1,0,0,4,16
Sum = 32; variance = 32/8 = 4
SD = square root of 4 = 2

## In practice

Quality control checks that the standard deviation of bottle fill volume is small, so bottles are consistent.

## Practice

### Interview questions

**A sample is 2, 4, 6. Calculate the sample standard deviation and explain the n minus 1.**

- Mean is 4
- Squared gaps sum to 8
- Variance 8 / 2 = 4
- SD is 2
- n minus 1 corrects bias

## Questions people ask

### How do you calculate standard deviation step by step?

Find the mean. Subtract it from each value and square the result. Add the squares and divide by the count (or by n minus 1 for a sample). That is the variance. Take its square root to get the standard deviation, back in the original units.

### Why do we square the deviations instead of just adding them?

Deviations from the mean always add up to zero, because positives cancel negatives. Squaring makes every term positive, so spread cannot cancel out. It also punishes large gaps more heavily. The square root at the end returns the answer to the original units.

## Related concepts

- [Sample variance](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-1/statistics-university-part-1-concept-6)
- [Quartiles and IQR](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-1/statistics-university-part-1-concept-7)

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