Population Standard Deviation Calculator

The population standard deviation (σ) measures how far values typically sit from their mean when your data covers the whole group you are describing: every student in one class, every machine on a line, all twelve months of a year. Paste the numbers to get σ, the variance, the mean and a table of deviations, with the sample standard deviation alongside so you can see how the two differ.

How to use this tool

1. Paste or type your values, separated by commas, spaces or new lines.
2. Check that the count N matches the number of values you meant to enter.
3. Read σ and the variance σ², then the table of deviations from the mean.
4. If your numbers are only a sample of a larger group, use the sample standard deviation instead.

Formula

σ = √( Σ(xᵢ − μ)² / N )

μ is the mean of all N values and Σ(xᵢ − μ)² is the sum of squared deviations. The variance σ² is that sum divided by N. The sample standard deviation divides by N − 1 instead (Bessel’s correction), which makes it slightly larger, most noticeably for small data sets.

Example

For the eight values 2, 4, 4, 4, 5, 5, 7, 9, the mean is 40 / 8 = 5. The squared deviations are 9, 1, 1, 1, 0, 0, 4 and 16, which add up to 32. The variance is 32 / 8 = 4, so σ = √4 = 2. Treated as a sample, the answer would be √(32 / 7) ≈ 2.13809.