# Standard Deviation Calculator

- **URL:** https://codeasystem.com/calculators/statistics/standard-deviation-calculator/
- **Category:** statistics
- **Description:** Compute sample or population standard deviation and variance from any list of numbers, with the mean shown alongside.
- **Primary output:** Population standard deviation: 2

## Inputs
- Your numbers (name: `numbers`, type: textarea, example: e.g. 2, 4, 4, 4, 5, 5, 7, 9)
- Data type (name: `type`, type: select)

## Outputs
- Population standard deviation: 2
- Population variance: 4
- Mean: 5
- Count: 8

## Formula / methodology
```
σ = √(Σ(x − μ)² / N)   s = √(Σ(x − x̄)² / (n − 1))
```

Standard deviation measures how far values typically sit from their mean. Each deviation is squared so negatives do not cancel positives, averaged, and square-rooted to return to the original units.

Choose population when your data covers the entire group of interest (divide by N). Choose sample when it is a subset used to estimate a larger group (divide by n − 1, Bessel’s correction, which slightly enlarges the estimate to compensate for missing information).

## Assumptions & limitations
- Values are separated by commas, spaces or newlines.
- Sample mode needs at least two numbers.
- Squaring makes the metric sensitive to outliers by design.

## How to use
1. **Enter your data**; Paste values separated by commas, spaces or newlines.
2. **Pick sample or population**; Population if you have the whole group; sample if you are estimating from a subset.
3. **Interpret the size**; Small SD means values cluster near the mean; large SD means they scatter widely.

## Example
The dataset 2, 4, 4, 4, 5, 5, 7, 9 has mean 5 and population standard deviation exactly 2; deviations of −3, −1, −1, −1, 0, 0, 2, 4 averaging 2 in magnitude after squaring.

Result for these inputs:

```
Population standard deviation: 2
```

## About this calculator
### What standard deviation actually tells you

For roughly bell-shaped data, about 68% of values fall within one SD of the mean and 95% within two. This lets you turn an abstract spread into concrete expectations; e.g. test scores of 75 ± 8 mean most students scored between 67 and 83.

### Sample versus population in practice

If you measured every customer this year, that dataset is your population. If you surveyed 200 of them hoping to describe next year’s customers, treat it as a sample. With large n the difference becomes negligible (the divisor differs by just one).

- Divisor N for complete populations.
- Divisor n−1 (Bessel’s correction) for samples.
- Same units as your original data.

## FAQs
### When should I use n−1 instead of n?

Use n−1 when your data is a sample meant to estimate a larger population. Use n only when the data literally is the whole population.

### Can standard deviation be zero?

Yes; it means every value equals the mean, i.e. there is no spread at all.

### Why is SD preferred over variance?

SD is in the same units as the data (variance is squared units), making it directly comparable to individual values.

### How does outlier sensitivity show up?

Because deviations are squared, one extreme value inflates SD dramatically; inspect min/max rows before trusting the figure.

## Related calculators
- [Variance Calculator](https://codeasystem.com/calculators/statistics/variance-calculator/)
- [Mean, Median & Mode Calculator](https://codeasystem.com/calculators/statistics/mean-median-mode-calculator/)
- [Z-Score Calculator](https://codeasystem.com/calculators/statistics/z-score-calculator/)

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Last updated: 2026-08-23 · Version: 1.0.0 · [HTML version](https://codeasystem.com/calculators/statistics/standard-deviation-calculator/)