Standard deviation calculator

Find whether the numbers in a list stay close together or spread far apart.

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Formula and steps included

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Optional learning Learn how this math works See the formula, definitions, and step-by-step method.

Formula

Population and sample standard deviation formulas

Population: σ = √(Σ(x − μ)² / N); Sample: s = √(Σ(x − x̄)² / (n − 1))

Standard deviation measures how far the numbers usually are from their average. Squaring each difference keeps positive and negative differences from canceling each other out.

Choose Whole group when your list has every member you care about. Choose Part of a group when your list is a smaller sample used to learn about a larger group.

How to use the formula

Choose how much of the group you have
Pick Whole group for every member or Part of a group for a smaller sample, then enter the numbers.
Find and compare with the average
Find the average, subtract it from each number, square every difference, and add those squared differences.
Divide, then take the square root
Divide by the number of values for a whole group, or one less for a sample. This gives variance, meaning the average squared difference. Take its square root for standard deviation.

Worked examples

Standard deviation examples

Population standard deviation

For the whole group 2, 4, 4, 4, 5, 5, 7, and 9, the average is 5. The average squared difference (variance) is 4, so the standard deviation is 2.

Sample standard deviation

For the sample 10, 12, and 14, the average is 12. The squared differences total 8, so the sample standard deviation is √(8 / 2) = 2.

Common questions

Standard deviation FAQ

Should I choose population or sample standard deviation?

Choose population when your data contains every member of the group you want to describe. Choose sample when the data is only part of a larger population you want to estimate.

What does a standard deviation of zero mean?

It means every value in the list is identical, so no value differs from the mean.

What units does standard deviation use?

Standard deviation uses the same units as the original numbers. Variance means the average of the squared differences from the average, so its units are squared.

Is my data list uploaded or saved?

Your number list stays in your browser and is not saved by default or uploaded. Use Save on this device to keep it here, or download a backup file.