Standard Deviation Calculator
Paste your data separated by commas, spaces or new lines: you'll get the standard deviation and variance, both population and sample, along with the formulas used.
Separate values with commas, spaces, semicolons or new lines. Use a period for decimals (e.g. 7.5).
σ = √( Σ(xᵢ − μ)² ⁄ N ) — population: divide by N
s = √( Σ(xᵢ − x̄)² ⁄ (n − 1) ) — sample: divide by n − 1
Population or sample: why you get two results
If your data is the entire population (say, the test scores of every student in a class), you divide by N and get the population standard deviation, σ. If your data is only a sample drawn from a larger group (a survey of 100 people used to estimate a whole city), you divide by n − 1 and get the sample standard deviation, s. Bessel's correction (the famous n − 1) compensates for the fact that a sample tends to underestimate the true variability: that's why s always comes out slightly larger than σ, though the difference becomes negligible once you have a lot of data points.
How to interpret the standard deviation
The standard deviation measures how far values typically stray from the mean, and it is expressed in the same unit as your data: if you measure inches, σ is in inches too (variance, by contrast, is in squared units). A low value means the data is clustered tightly around the mean; a high value means it is widely spread out.
For bell-shaped distributions the empirical rule applies: roughly 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three. It's a quick way to judge whether a single data point is "normal" or surprising.