Standard Deviation Calculator

Standard deviation measures how spread out a set of numbers is around its mean. A small standard deviation means values cluster close to the average; a large one means they scatter widely — two datasets can share the exact same mean and look completely different once you see how far individual values stray from it.

Sample vs population — the one choice that matters

Population standard deviation (σ, dividing by N) is correct when your numbers are literally every member of the group you're studying — every student in one specific class, every transaction in a closed dataset. Sample standard deviation (s, dividing by N−1) is correct when your numbers are a sample drawn from a larger population you're trying to estimate — a survey of 200 customers standing in for all customers.

The N−1 divisor, known as Bessel's correction, exists because a sample's own mean is calculated from the same data, which slightly understates the true spread — dividing by a smaller number corrects that bias. The difference is small for large samples and can matter noticeably for small ones, which is why the two are kept separate rather than blended into one 'close enough' answer.

Reading the result

For roughly bell-shaped data, about 68% of values fall within one standard deviation of the mean, and about 95% fall within two — the basis of quality control charts, test-score scaling, and most 'is this value normal or unusual' judgments in statistics.