Standard Deviation Calculator
Standard deviation tells you how spread out your numbers are. Paste your data, pick sample or population, and read off the answer with the steps.
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The formula
Population: s = sqrt( sum of (x - mean)^2 / n ). Sample: s = sqrt( sum of (x - mean)^2 / (n - 1) ).
What standard deviation actually tells you
It is the typical distance between each data point and the mean. A small standard deviation means the values bunch tightly around the average. A large one means they are spread out.
Two classes can both average 70 on a test, but if one has a standard deviation of 3 and the other 15, the second class had far more variation between strong and weak students.
Sample or population?
Use <strong>population</strong> when your numbers are the whole group you care about, for example every employee in a small company. Use <strong>sample</strong> when they are a subset used to estimate a bigger group, such as 50 customers out of thousands.
The sample version divides by n - 1 instead of n. That small change corrects for the fact that a sample tends to underestimate how spread out the full group is. If you are unsure, sample is the safer choice.
A worked example
Data: 4, 8, 6, 5, 3, 7. The mean is 5.5. The squared differences from the mean are 2.25, 6.25, 0.25, 0.25, 6.25 and 2.25, which add to 17.5. For a sample, divide by 5 to get a variance of 3.5, and the standard deviation is the square root, about 1.87.
Rule of thumb for bell-shaped data
If the data is roughly bell-shaped, about 68% of values fall within one standard deviation of the mean, 95% within two and 99.7% within three.
Examples to check
Type these into the calculator above to see how it behaves. Every result below was produced by the same calculator code that runs on this page.
| What you enter | What you get |
|---|---|
| Numbers (separated by commas, spaces or new lines): 4 8 6 5 3 7; Sample (divide by n - 1) | Sample standard deviation: 1.87082869 |
| Numbers (separated by commas, spaces or new lines): 2 4 4 4 5 5 7 9; Population (divide by n) | Population standard deviation: 2 |
| Numbers (separated by commas, spaces or new lines): 10 12 23 23 16 23 21 16; Sample (divide by n - 1) | Sample standard deviation: 5.23722937 |
Standard Deviation Calculator: frequently asked questions
What is the difference between variance and standard deviation?
Variance is the average squared distance from the mean. Standard deviation is its square root, which puts the number back into the same units as your data, so it is easier to interpret.
Can standard deviation be negative?
No. It is a square root of a sum of squares, so it is zero or positive. It is zero only when every value is identical.
How many numbers do I need?
At least two. With one value there is nothing to compare, and a sample standard deviation would divide by zero.
How accurate is this calculator?
It uses standard double-precision arithmetic, accurate to about 15 significant digits. Results are rounded only for display.
Is what I enter in this calculator saved or sent anywhere?
No. The calculation runs in your browser, so what you enter is not sent to our servers. We only count, anonymously, whether a tool ran successfully.
Is this calculator free to use?
Yes. It is free, needs no sign-up and has no usage limit.
Last reviewed: October 3, 2026