Standard Deviation Calculator

Population vs. Sample Standard Deviation

Standard deviation measures how spread out a set of numbers is around its mean. There are two standard formulas, and the correct one depends on whether your data is the entire population or a sample drawn from a larger population. Population standard deviation divides the sum of squared deviations by n (the number of values). Sample standard deviation divides by n − 1 instead — a correction known as Bessel's correction, which compensates for the fact that a sample's mean is itself an estimate and tends to slightly understate the true spread if divided by n. Use population when your data set is the complete group you care about (e.g. test scores for every student in a class); use sample when your data is a subset used to estimate a larger population's variability (e.g. a survey of 50 customers out of thousands).

The Formula, Step by Step

For a data set of values x₁, x₂, ..., xₙ with mean x̄: first find each value's deviation from the mean (xᵢ − x̄), then square each deviation (this makes negative and positive deviations both count as positive spread and penalizes larger gaps more heavily), then sum the squared deviations. Variance is that sum divided by n (population) or n − 1 (sample). Standard deviation is simply the square root of variance, which brings the units back in line with the original data instead of squared units. The table above shows every one of these intermediate steps for your entered data set so you can verify the arithmetic by hand.

Related Tools

If you need other summary statistics — mean, median, mode, range, and more — alongside standard deviation, try the full statistics calculator. To turn this same data into a confidence interval or margin of error for a larger population, see the confidence interval calculator.