Standard deviation calculator

What standard deviation tells you

Standard deviation measures how far values typically are from their average. A small standard deviation means the values cluster near the mean; a large one means they are spread out. In a normal distribution, about 68% of values fall within one standard deviation of the mean and 95% within two.

Sample or population?

Use population when you have every member of the group you care about (all employees' salaries, all items in a batch). Use sample when your numbers are a subset used to estimate a larger group (a survey, a set of measurements). The sample formula divides by n − 1 instead of n, which corrects the tendency of a sample to underestimate the spread. Spreadsheets call these STDEV.P and STDEV.S.

Worked example

For 2, 4, 4, 4, 5, 5, 7, 9: the mean is 5. The squared deviations are 9, 1, 1, 1, 0, 0, 4 and 16, which add up to 32. As a population, variance = 32 ÷ 8 = 4 and σ = 2. As a sample, variance = 32 ÷ 7 ≈ 4.571 and s ≈ 2.138.

The page computes the mean, then sums the squared differences between each value and the mean (two passes, which stays accurate even when all values are large, such as 1,000,000,001 and 1,000,000,002). It divides by n for a population or by n − 1 for a sample to get the variance, and takes the square root for the standard deviation. The coefficient of variation is the standard deviation as a percentage of the mean. Everything runs in your browser.