Standard deviation calculator
Math & numbersHow spread out are these numbers?
Sample or population standard deviation and variance of a list of numbers, with the mean and the standard error.
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Separate with spaces, line breaks or semicolons.
Standard deviation
2.138
Sample standard deviation of 8 numbers with mean 5: a typical value lies about 2.138 from the mean.
More about the spread
- Variance
- 4.571
- Mean
- 5
- How many numbers
- 8
- Standard error of the mean
- 0.7559
- Coefficient of variation
- 42.76%
- Population standard deviation
- 2
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Sample or population standard deviation and variance of a list of numbers, with the mean and the standard error.
How to use
- Type or paste your numbers, separated by spaces, line breaks or semicolons.
- Choose whether the numbers are a sample of a larger group or the whole population.
- Read the standard deviation as the answer, with the variance, mean and standard error below it.
How it works
The standard deviation tells how far the numbers typically sit from their mean. A small value means they cluster tightly; a large one means they are spread out. The variance is the same idea squared, and the standard error says how precisely the mean of a sample estimates the mean of the whole group.
Take the mean x̄, square each value's distance from it, add the squares and divide: by n for a whole population, by n − 1 for a sample (Bessel's correction, because a sample's own mean sits a little too close to its values). The square root of that is the standard deviation. For 2, 4, 4, 4, 5, 5, 7 and 9 the mean is 5 and the squared distances add up to 32, so the population SD is √(32 ÷ 8) = 2 and the sample SD is √(32 ÷ 7) ≈ 2.138. Definitions follow the NIST/SEMATECH e-Handbook of Statistical Methods.
Use the sample setting for measurements or survey answers from part of a group, and the population setting only when you have every member. The standard deviation is in the same unit as the data, so it is easy to read, but a few extreme values pull it up a lot. Entries that are not numbers are left out and listed so you can fix them.
Worked example
What is the sample standard deviation of 2; 4; 6?
- Mean = 4.
- Sample variance = ((2 − 4)² + (4 − 4)² + (6 − 4)²) ÷ (3 − 1) = 4.
- Standard deviation = √4 = 2.
The sample standard deviation is 2.
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Frequently asked questions
Should I use the sample or the population standard deviation?
Use the sample one when your numbers are part of a larger group you want to describe, which is almost always. Use the population one only when the list is the entire group.
What is the difference between variance and standard deviation?
The variance is the mean squared distance from the mean; the standard deviation is its square root, so it is in the same unit as your numbers.
What does the standard error tell me?
It is the sample standard deviation divided by the square root of the count: how much the mean would vary if you took another sample of the same size.
What is a good coefficient of variation?
It is the standard deviation as a percentage of the mean, useful for comparing spread across different scales. What counts as low depends on the field; it is meaningless when the mean is near zero.
Can the standard deviation be negative?
No. It is zero when all the values are equal and positive when they differ. Negative input values are allowed.
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