Standard Deviation Calculator

How to find the standard deviation

1. Enter your numbers in the box, separated by commas, spaces or new lines. You can paste a column straight from a spreadsheet.
2. Choose sample or population. Use Sample if the numbers are a subset of a bigger group, and Population if they are the whole group.
3. Read the result. The standard deviation is shown first, followed by the variance, the mean, the standard error and the coefficient of variation.
4. Check the working. The table lists each value, its deviation from the mean and the squared deviation, so you can follow every step.

A small standard deviation means the values are bunched close to the mean. A large one means they are spread out. The result is in the same units as your data.

Standard deviation formulas

Standard deviation measures how far the values in a data set typically sit from their mean.

Population: σ = √( Σ(x − μ)² ÷ N )

Sample: s = √( Σ(x − x̄)² ÷ (n − 1) )

• x = each value, μ or x̄ = the mean, N or n = the number of values
• Σ(x − mean)² = the sum of the squared deviations
• The variance is the same expression without the square root (σ² or s²)

Step by step

1. Find the mean by adding the values and dividing by how many there are.
2. Subtract the mean from every value to get its deviation.
3. Square each deviation.
4. Add the squared deviations.
5. Divide by N (population) or n − 1 (sample). This is the variance.
6. Take the square root. This is the standard deviation.

Worked example

Data: 2, 4, 4, 4, 5, 5, 7, 9

1. Mean = 40 ÷ 8 = 5
2. Deviations: −3, −1, −1, −1, 0, 0, 2, 4
3. Squared: 9, 1, 1, 1, 0, 0, 4, 16, which add up to 32
4. Population variance = 32 ÷ 8 = 4, so σ = 2
5. Sample variance = 32 ÷ 7 = 4.571, so s = 2.138

Sample or population?

| Use | When | |---|---| | Population (÷ N) | Your numbers are every member of the group, such as the test scores of all 30 students in one class | | Sample (÷ n − 1) | Your numbers are a subset used to estimate a larger group, such as 30 people surveyed out of a city |

Dividing by n − 1 (Bessel's correction) makes the sample variance an unbiased estimate of the population variance. If you are not sure, a sample is the usual choice.

Related measures

• Standard error of the mean: s ÷ √n. It shows how precisely the sample mean estimates the true mean.
• Coefficient of variation: standard deviation ÷ mean × 100. It compares spread between data sets with different units or sizes.