Calculate the population and sample standard deviation of a list of numbers, together with the mean and the count.
Calculations happen locally. We do not send your values to our server.
For 2, 4, 4, 4, 5, 5, 7 and 9 the mean is 5, the population standard deviation is 2 and the sample standard deviation is about 2.138. For 50, 60, 70, 80 and 90 the two results are about 14.142 and 15.811.
The tool finds the mean, adds up the squared distance of every value from it, then divides by the count for the population variance or by the count minus one for the sample variance. The square root of each variance is the standard deviation, so the results match STDEV.P and STDEV.S in spreadsheets.
Use the population figure when the list contains every member you care about, and the sample figure when the numbers are a subset used to estimate a larger group. Identical values such as 10, 10 and 10 give 0, and a single value has a population result of 0 with no defined sample result.
Paste the values and read both results. For 2, 4, 4, 4, 5, 5, 7 and 9 the mean is 5, the population standard deviation is 2 and the sample standard deviation is about 2.138. Use the population figure when the list contains every member you care about, such as all eight scores in a small class. Use the sample figure when the numbers are a subset used to estimate a larger group, such as five batteries tested from a production run.
The population version divides the sum of squared distances by N; the sample version divides by N − 1, which corrects the tendency of a small sample to understate variation. The difference shrinks as the list grows. For 8 and 12 the mean is 10, the population SD is 2 and the sample SD is about 2.828. For 50, 60, 70, 80 and 90 the results are about 14.142 and 15.811. A single value has a population SD of 0 and no defined sample SD. In spreadsheets these match STDEV.P and STDEV.S.
Standard deviation is in the same unit as the data, so a spread of 14.142 points on test scores means values typically sit roughly that far from the mean of 70. Identical values give 0, as 10, 10 and 10 do. Because distances are squared, one extreme value can raise the result sharply, so check the data for typing errors before interpreting it. For roughly bell-shaped data, about two thirds of values often fall within one standard deviation of the mean, but skewed data does not follow that rule.
| Input | Result |
|---|---|
2, 4, 4, 4, 5, 5, 7, 9 | σ 2 · s 2.138 |
50, 60, 70, 80, 90 | σ 14.142 · s 15.811 |
8, 12 | σ 2 · s 2.828 |
10, 10, 10 | σ 0 · s 0 |
5 | σ 0 · s — |
Because it divides by one less than the count, which corrects the tendency of a small sample to understate variation. For 8 and 12 the population standard deviation is 2 while the sample value is about 2.828, and the gap between the two shrinks as the list gets longer.
It uses the same unit as your data and shows how far values typically sit from the mean. Scores of 50, 60, 70, 80 and 90 have a mean of 70 and a population standard deviation of about 14.142, so values usually lie roughly that far from 70. For bell-shaped data about two thirds fall within one standard deviation, but skewed data does not follow that rule.
It can never be negative, because every distance is squared before the square root is taken. A single mistyped value can raise it sharply, though, since squaring gives extreme values a large weight. Check the minimum and maximum in the Average Calculator before trusting a surprisingly large standard deviation.
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