Standard deviation calculator
Sample and population standard deviation, variance and mean of a list, with every step.
About this calculator
This standard deviation calculator measures how spread out a list of numbers is. Paste your data, separated by spaces, commas or new lines.
It gives both the sample and the population standard deviation, the variance and the mean, and shows every step of the working.
How it works
Find the mean, subtract it from each value and square the result. Add up the squares and divide by n − 1 for a sample, or by n for a population, to get the variance. The standard deviation is its square root.
Last reviewed:
Worked example
- Numbers, separated by spaces, commas or new lines: 10 12 23 23 16 23 21 16
- The numbers are: A sample
Sample standard deviation (s): 5.23722936566
Common questions
What is the difference between sample and population standard deviation?
The sample formula divides by n − 1 instead of n, which corrects for estimating the mean from the same data. Use it unless you have every member of the group.
What does the standard deviation tell me?
How spread out the numbers are. In a normal distribution, about 68% of values lie within one standard deviation of the mean and 95% within two.
What is variance?
The square of the standard deviation: the average squared distance from the mean. It is in squared units, which is why the standard deviation is easier to read.
Sources and further reading
The story behind it
Where the idea came from: Karl Pearson coined the term standard deviation in 1893, in a lecture on evolution.
Before that, scientists measured spread with the probable error. The standard deviation won out because it works neatly with the normal distribution, the bell curve that describes so much natural variation.
Why you can trust this calculator
- It runs in your browser. Nothing you type into it is sent anywhere.
- The method is explained on this page, so you can check it yourself.
- Free to use, with no sign-up.
- Results are estimates, so please double-check anything important.
- Spotted a mistake? Tell us and we'll look into it.
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