Standard Deviation Calculator
Paste your data to get the standard deviation, variance and mean, for a sample or a whole population.
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Standard Deviation Calculator by TheCalcTools
How the Standard Deviation Calculator works
Standard deviation measures how spread out the values are around the mean. A small value means the data sit close to the average; a large value means they are widely scattered.
Work out the mean, subtract it from each value, square the differences and add them up. Divide by n − 1 for a sample or by n for a whole population, then take the square root.
Use the sample version (n − 1, Bessel’s correction) when your data are a sample used to estimate a larger population, which is almost always the case in surveys and experiments.
The formula
Worked example
Using the default values shown in the calculator:
| Your data | 12, 15, 15, 18, 20, 22, 25, 25, 25, 30 |
|---|---|
| Your data is | A sample (divide by n − 1) |
- The mean of the 10 values is 20.7.
- The squared deviations add up to 279.1. Divided by n − 1 = 9 gives a variance of 32.4556.
- The square root gives a sample standard deviation of 5.6970.
Result:
| Standard deviation | 5.6970 |
|---|---|
| Variance | 32.4556 |
| Mean | 20.7000 |
| Count | 10 |
| Standard error of the mean | 1.8015 |
| Coefficient of variation | 27.52% |
Frequently asked questions
Sample or population standard deviation?
Use the population formula only when you have every member of the group you care about, such as the scores of all students in one class. If the data are a sample from something bigger, use the sample formula.
Why divide by n − 1?
The sample mean is closer to the sample values than the true population mean is, so dividing by n underestimates the spread. Dividing by n − 1 corrects that bias in the variance.
What is a good standard deviation?
There is no universal good value. Compare it with the mean (the coefficient of variation) or with the standard deviation of similar data. About 68% of normally distributed data lie within one standard deviation of the mean.
Sources and further reading
- NIST/SEMATECH e-Handbook: Measures of scale
- Wikipedia: Standard deviation
- Wikipedia: Bessel's correction
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This calculator gives estimates for information only. Check important results independently before you rely on them. See our disclaimer.