Standard Deviation Calculator

How to calculate standard deviation

1. Enter your numbers, separated by spaces or commas.
2. Choose whether the data is a sample or the whole population.
3. Read the standard deviation, with both types shown.
4. See the variance and mean below it.

Sample or population: the key choice

The most common confusion in statistics is which standard deviation to use, so this tool shows both and lets you pick. Use the population standard deviation when your numbers are the entire group you care about. Use the sample standard deviation, the usual case, when your numbers are a sample drawn from a larger group. The only difference is the divisor: population divides by N, while sample divides by n minus 1. Dividing by n minus 1 corrects for the fact that a sample tends to underestimate the spread of the whole group.

What standard deviation measures

Standard deviation measures how spread out numbers are around their mean. A small value means the data clusters close to the average; a large value means it is widely scattered. Std dev is the square root of the variance, and this doubles as a variance calculator since it shows both. Because it is in the same units as the data, standard deviation is easier to interpret than variance.

A worked example

Take the sample 2, 4, 4, 4, 5, 5, 7, 9. The mean is 5. The squared distances from the mean add up to 32. For the population standard deviation, divide 32 by 8 to get a variance of 4, so the standard deviation is 2. For the sample standard deviation, divide 32 by 7 (n minus 1) to get about 4.57, so the standard deviation is about 2.14. The tool shows both.

FAQ

What is the difference between sample and population standard deviation?

The population version divides the squared distances by N; the sample version divides by n minus 1. Use the sample version when your data is a subset of a larger group, which is most of the time.

How do I calculate standard deviation?

Find the mean, then the squared distance of each value from it, add those up, divide by N (or n minus 1 for a sample), and take the square root. The tool shows every step's result.

What is variance?

Variance is the average of the squared distances from the mean. Standard deviation is its square root, which puts the spread back in the original units.

Why does the sample formula divide by n minus 1?

A sample tends to sit slightly closer to its own mean than to the true population mean, so it underestimates the spread. Dividing by n minus 1 corrects for that bias.

Is my information private?

Yes. Everything is calculated in your browser and nothing you enter is uploaded or saved.