📉 Standard Deviation Calculator

How the Standard Deviation Calculator Works

Standard deviation measures how spread out numbers are around their mean. A small standard deviation means values sit close to the average; a large one means they are widely scattered.

There are two versions, and choosing the right one matters:

• Sample standard deviation (s) — use it when your numbers are a sample drawn from a bigger group (for example, 30 surveyed customers out of thousands). It divides by n − 1 (Bessel's correction).
• Population standard deviation (σ) — use it when your numbers are the entire group you care about (for example, the scores of every student in one class). It divides by n.

The table below the result shows each value's deviation from the mean and its square, so you can follow — or check — the calculation by hand.

Formula

Mean  x̄ = Σx / n
Population variance  σ² = Σ(x − x̄)² / n        σ = √σ²
Sample variance      s² = Σ(x − x̄)² / (n − 1)  s = √s²
Standard error of the mean = s / √n

How to Use It

1. Enter or paste your numbers (commas, spaces, or new lines).
2. Read both the sample and the population standard deviation.
3. Open the step table to see each deviation and squared deviation.

Frequently Asked Questions

Should I use sample or population standard deviation?

Use the sample standard deviation (dividing by n − 1) when your data is a subset of a larger group you want to draw conclusions about — this is the usual case in research and surveys. Use the population version (dividing by n) only when your data includes every member of the group.

What is the difference between variance and standard deviation?

Variance is the average squared distance from the mean; standard deviation is its square root. Standard deviation is easier to interpret because it is in the same units as the original data.

What does a standard deviation of zero mean?

It means every value in the data set is identical, so there is no spread at all around the mean.

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Formula and content last reviewed: September 2026. All calculations run in your browser — nothing you enter is sent to or stored on our servers.