Standard Deviation Calculator

Standard deviation and variance for a list of numbers, sample and population.

How to calculate standard deviation

1. Find the mean: add the numbers and divide by how many there are.
2. Subtract the mean from each number.
3. Square each of those differences.
4. Add the squares together.
5. Divide by n for a population, or by n − 1 for a sample. This is the variance.
6. Take the square root of the variance. This is the standard deviation.

Worked examples

• 2, 4, 4, 4, 5, 5, 7, 9: the mean is 5 and the squared differences add up to 32. Population variance is 32 ÷ 8 = 4, so the population standard deviation is 2. Sample variance is 32 ÷ 7 = 4.5714, so the sample standard deviation is 2.1381.
• 10, 20, 30, 40, 50: the mean is 30 and the squared differences add up to 1,000. Population standard deviation is 14.1421 and sample standard deviation is 15.8114.

Frequently asked questions

What is the difference between sample and population standard deviation?

Use the population version when your numbers include every member of the group you care about. Use the sample version when they are only a subset, which is the most common case. The sample version divides by n minus 1, so it is slightly larger.

What does a high or low standard deviation mean?

A low standard deviation means the numbers are close to the mean. A high one means they are spread out. It is measured in the same units as your data.

What is variance?

Variance is the average of the squared differences from the mean. The standard deviation is its square root, which puts the result back in the original units.

Can the 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 all the numbers are identical.