Standard Deviation Calculator
Calculate the standard deviation and mean of a set of numbers. Enter the values separated by commas to see the population and sample standard deviation.
How it works
Standard deviation measures how spread out the numbers are from the mean. Population deviation divides by N; sample deviation divides by N−1.
σ = √(Σ(x − μ)² ÷ N)Worked examples
Population standard deviation of 10, 12, 23, 23, 16
The mean is 16.8 and the population standard deviation is about 5.42.
The same numbers treated as a sample
Treating the same five numbers as a sample drawn from a larger group gives about 6.06 instead of the 5.42 above - larger, because the sample form divides by one fewer value. Which one applies depends on whether your numbers are the whole group or a sample of it.
Why the sample form is larger
The same numbers treated as a sample give a larger figure than they do as a population, because the sample form divides by one fewer value. On five values the gap is about 12 per cent, and it narrows as the set grows.
What this assumes, and where it does not apply
What it assumes
- Population standard deviation divides by n; sample standard deviation divides by n-1.
- Values are treated as one set of numbers, with no weighting.
- Population and sample standard deviation are both shown because which one applies is a fact about the data, not a preference.
Where the result does not apply
- Does not test whether a difference between two sets is statistically significant.
- Sensitive to outliers: a single extreme value raises the result sharply, which is information rather than an error.
- Does not decide for you whether a set is a population or a sample. That depends on how the numbers were collected.
Frequently asked questions
What is standard deviation?
It measures how much a set of numbers varies from their average — a higher value means more spread.
Population vs sample?
Use population when you have all values; use sample when your numbers are a sample of a larger group (divides by n−1).
What does a high standard deviation mean?
The values are more spread out from the mean; a low one means they cluster close to the average.
When should I use the population standard deviation?
When the numbers are the whole group you care about. Use the sample form when they are a sample drawn from a larger group you are trying to describe.
What does a standard deviation of 5.42 mean?
It means the values sit about 5.42 away from the mean on average. It is in the same units as the data, so it only means something next to the mean it belongs to.
Maintained by David M at Delta Tech Solutions · Figures last checked 10 October 2026 · Next check due 10 April 2027
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