Standard Deviation Calculator
Calculate the standard deviation, variance, and mean of a list of numbers.
Try an example
2.0656
About this calculator
How This Standard Deviation Calculator Works
Enter a list of numbers, and the calculator returns the mean, variance, and standard deviation — both the sample and population versions — showing how spread out the values are around their average.
Worked Example
For the values 4, 8, 6, 5, 3, 8: the mean is 5.67, and the sample standard deviation is 2.07 (population standard deviation: 1.89). The sample variance is 4.27.
Sample vs. Population Standard Deviation
Use population standard deviation when your numbers are the entire group you care about — every student in one class, every day this month. Use sample standard deviation when your numbers are a subset used to estimate a larger, unmeasured group — a survey sample standing in for a full population. The sample version divides by (n − 1) instead of n, which slightly inflates the result (Bessel's correction) to compensate for the fact that a sample tends to underestimate the true spread of the full population it's drawn from.
The Formula
σ = √[Σ(x − mean)² ÷ n] for population, or divide by (n − 1) instead of n for the sample version. Each value's distance from the mean is squared (so negative and positive distances don't cancel out), averaged, and then square-rooted to bring the units back to the original scale.
Worked example
For 4, 8, 6, 5, 3, 8: the mean is 5.6667, and the sample standard deviation is about 2.07 — a measure of how spread out the values are around that mean.
Frequently asked questions
What's the difference between sample and population standard deviation?
Population standard deviation treats your numbers as the entire group you care about; sample standard deviation treats them as a sample used to estimate a larger population, and divides by one less (n − 1) to correct for that. Use sample unless you're certain you have every value in the group.
What does a small vs. large standard deviation mean?
A small standard deviation means values cluster tightly around the mean; a large one means they're spread out widely — two data sets can have the same mean but very different standard deviations.
How is variance related to standard deviation?
Variance is the standard deviation squared — standard deviation is just the square root of variance, converted back to the same units as your original data, which is why it's usually the more intuitive number to read.