Standard Deviation and Variance Calculator

How it works

The standard deviation measures how far values typically sit from their mean. Each value's distance from the mean is squared (so distances below and above do not cancel), the squares are averaged to give the variance, and the square root brings the result back to the units of the data.

The one choice you make is the divisor. If your values are the whole group you care about, divide by the count N: that is the population standard deviation. If they are a sample and you want to estimate the spread of the larger population they came from, divide by n − 1 instead (Bessel's correction). Dividing by n − 1 makes the result a little larger and corrects the tendency of a sample to underestimate the spread, because the sample mean is always closer to the sample values than the true mean is.

The standard error of the mean, s ÷ √n, is how much the sample mean itself would vary from sample to sample; it shrinks as you collect more data. The coefficient of variation expresses the standard deviation as a percentage of the mean so that spreads of data on different scales can be compared.

Formula

x̄   = Σx ÷ n
SS  = Σ(x − x̄)²
s²  = SS ÷ (n − 1)     sample variance        s = √s²
σ²  = SS ÷ N           population variance    σ = √σ²
SE  = s ÷ √n
CV  = s ÷ x̄ × 100

Example

Twelve quiz scores: 4, 7, 2, 5, 3, 9, 4, 7, 5, 3, 4, 7. They sum to 60, so the mean is 5. The squared deviations from 5 are 1, 4, 9, 0, 4, 16, 1, 4, 0, 4, 1, 4, which sum to 48.

As a sample, the variance is 48 ÷ 11 = 4.3636 and the standard deviation is √4.3636 = 2.0889. The standard error is 2.0889 ÷ √12 = 0.6030 and the coefficient of variation is 2.0889 ÷ 5 × 100 = 41.78%. As a population, the variance is 48 ÷ 12 = 4 and the standard deviation is exactly 2.

Assumptions and limitations

• Choose sample when the values are a sample from a larger population whose spread you want to estimate; choose population when the values are the entire group of interest.
• The coefficient of variation is only meaningful for data on a ratio scale with a positive mean, and is undefined (shown as —) when the mean is zero.
• The standard error assumes the values are independent observations.
• Every entry must be a number. Text such as units in the list stops the calculation and names the offending entry.

Frequently asked questions