# Free online standard deviation calculator

> Standard deviation measures how spread out values are around the mean. This calculator gives population σ and sample s, plus variance, mean, and count.

Calculate population and sample standard deviation, variance, and mean from a list of numbers. Enter values separated by commas, spaces, or new lines.

**Interactive version:** https://onlinecalculator.me/math/standard-deviation/
**Category:** Math

## How to use

1. Type or paste your numbers into the text area — commas, spaces, semicolons, and new lines all work as separators.
2. Results update as you type.
3. Use "Share with my numbers" to share the data set via URL.

## Formulas

**Population standard deviation:**
```
σ = √( Σ(xᵢ − μ)² / n )
```

**Sample standard deviation:**
```
s = √( Σ(xᵢ − x̄)² / (n − 1) )
```

Where:
- **μ** (or **x̄**) = mean of all values
- **n** = count of values
- **Σ(xᵢ − μ)²** = sum of squared deviations from the mean

## Worked example

Data: 2, 4, 4, 4, 5, 5, 7, 9 (n = 8)

1. Mean = 40 / 8 = **5**
2. Squared deviations: (2−5)²=9, (4−5)²=1 × 3, (5−5)²=0 × 2, (7−5)²=4, (9−5)²=16
3. Sum of squared deviations = 9 + 3 + 0 + 4 + 16 = **32**
4. Population variance σ² = 32 / 8 = **4** → σ = **2**
5. Sample variance s² = 32 / 7 ≈ **4.5714** → s ≈ **2.1381**

## Notes

- Sample standard deviation requires at least 2 values; with only 1 value the result is 0.
- Results are rounded to six decimal places.
- Share URLs encode your data in the `d` query parameter.

## Frequently asked questions

### What is the difference between population and sample standard deviation?

Population standard deviation (σ) divides the sum of squared differences by n. Sample standard deviation (s) divides by n − 1, which corrects for the bias introduced when estimating from a subset of a larger population.

### When should I use sample vs. population standard deviation?

Use population (σ) when you have data for every member of the group. Use sample (s) when your data is a subset and you want to estimate the spread of the full population.

### What is variance?

Variance is the square of the standard deviation. It measures the average squared distance from the mean. Population variance = σ², sample variance = s².

### Why does the sample standard deviation divide by n − 1?

Dividing by n − 1 (Bessel's correction) gives an unbiased estimate of the population variance. Without it, a sample tends to underestimate the true spread.

### What does a high standard deviation mean?

A high standard deviation means values are spread widely around the mean. A low standard deviation means values cluster closely around the mean.

### How do I enter my numbers?

Type or paste values separated by commas, spaces, semicolons, or new lines. Any mix of delimiters works.

## Sources

- [Standard deviation](https://en.wikipedia.org/wiki/Standard_deviation) — Wikipedia
- [Standard Deviation](https://mathworld.wolfram.com/StandardDeviation.html) — Wolfram MathWorld

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