Standard Deviation Calculator
Calculate sample or population standard deviation, variance, and mean from a list of numbers.
How it works
Standard deviation formula
σ = √( Σ(xᵢ − μ)² / n ) (population) s = √( Σ(xᵢ − x̄)² / (n − 1) ) (sample)
Standard deviation measures how spread out values are around the mean. Each value's distance from the mean is squared (so negative and positive distances both count), averaged, and square-rooted back to the original units. Samples divide by n − 1 (Bessel's correction) to avoid underestimating the spread of the larger population.
Worked example
For 4, 8, 6, 5, 3, 7: the mean is 5.5. Squared distances sum to 17.5. Sample variance = 17.5 / 5 = 3.5, so the sample standard deviation is √3.5 ≈ 1.8708.
Frequently asked questions
Sample or population — which should I choose?
Population if your data includes every member of the group you care about (e.g., all students in one class, analyzing only that class). Sample if your data is a subset used to estimate a larger group — this is the more common case.
What does a standard deviation value actually mean?
In roughly normal data, about 68% of values fall within 1 standard deviation of the mean and about 95% within 2. A small SD means values cluster tightly around the mean; a large SD means they are spread out.
What is the relationship between variance and standard deviation?
Standard deviation is the square root of variance. Variance is in squared units (harder to interpret); standard deviation is in the same units as your data.