How to calculate standard deviation
- Find the mean of the numbers.
- Subtract the mean from each number and square the result.
- Add up the squares (the sum of squares).
- Divide by n − 1 for a sample, or by n for a whole population. This is the variance.
- Take the square root.
Example: for 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5 and the sum of squares is 32. The population standard deviation is √(32 ÷ 8) = 2, and the sample standard deviation is √(32 ÷ 7) = 2.138.
Sample or population?
Use population only when you have every member of the group you care about, like the scores of every student in one class. Use sample when your numbers are a subset used to estimate a larger group, which is most real data. Dividing by n − 1 (Bessel's correction) makes up for a sample usually underestimating the spread.
In roughly bell-shaped data, about 68% of values fall within one standard deviation of the mean and about 95% within two.
Next steps
For the mean, median and mode of the same list, use the average calculator.
Frequently asked questions
What does standard deviation tell you?
How spread out the numbers are from the mean. A small value means they cluster tightly; a large value means they vary a lot.
Why divide by n − 1 for a sample?
A sample's values sit closer to its own mean than to the true population mean, which makes the spread look smaller. Dividing by n − 1 corrects for that.
What is variance?
The square of the standard deviation: the average squared distance from the mean. It's in squared units, which is why the standard deviation is usually reported instead.
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