Standard Deviation Calculator

Enter a list of numbers to get the sample and population standard deviation, plus the variance and mean.

This result is an estimate only. Double-check important figures before relying on them.

How it is calculated

Mean = sum of the numbers ÷ count
Sum of squares = the sum of (each number − mean)²
Population standard deviation = √(sum of squares ÷ n)
Sample standard deviation = √(sum of squares ÷ (n − 1))
Sample variance = sum of squares ÷ (n − 1)

For the average and the middle value, use the mean, median and mode calculator. For values with different importance, use the weighted average calculator.

Worked example

The numbers 2, 4, 4, 4, 5, 5, 7, 9:

Mean: 40 ÷ 8 = 5
Sum of squares: 9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 32
Population standard deviation: √(32 ÷ 8) = 2
Sample standard deviation: √(32 ÷ 7) = 2.1381

Why your number may differ

Some textbooks and calculators show only one version, so check which one your course uses. Rounding can change the last digit. Very small lists give unreliable estimates.

Frequently asked questions

What is standard deviation?

It measures how spread out the numbers are around their mean. A small value means they are close together, and a large value means they are widely spread.

Sample or population?

Use the population value when your list is every item you care about. Use the sample value when the list is only a sample from a larger group, which is the usual case.

Why does the sample version divide by n − 1?

Dividing by one less corrects the bias that comes from estimating the mean from the same sample. It makes the sample variance a better estimate for the whole group.