Standard Deviation Calculator
Sample and population standard deviation and variance of a list of numbers, with the mean, sum of squares and count — the working shown.
Find the mean, take each value's difference from it, square the differences and average them: that is the variance, and its square root is the standard deviation.
How the standard deviation calculator works
Find the mean, take each value's difference from it, square the differences and average them: that is the variance, and its square root is the standard deviation. Divide by n for a whole population, by n − 1 for a sample (Bessel's correction) — the sample version is the one you usually want.
Formula: s = √(Σ(x − x̄)² / (n − 1)); σ = √(Σ(x − μ)² / n)
Worked examples
| Inputs | Sample standard deviation (s) | Note |
|---|---|---|
| 4, 8, 15, 16, 23, 42 | 13.490738 | s ≈ 13.49, σ ≈ 12.32 |
| 2, 4, 4, 4, 5, 5, 7, 9 | 2.13809 | σ = 2 exactly |
FAQFrequently asked questions
Sample or population — which should I use?
Population (÷ n) when your list is everything you care about; sample (÷ n − 1) when it is a sample of a larger group, which is the usual case.
What does a standard deviation of 2 mean?
Typical values sit about 2 units from the mean; for roughly normal data about 68% lie within one SD and 95% within two.
What is variance?
The standard deviation squared — the average squared distance from the mean.
Why n − 1?
Using the sample mean makes the deviations slightly too small; dividing by n − 1 corrects the bias.
What is the coefficient of variation?
SD ÷ mean, as a percentage — a scale-free measure of spread for comparing datasets.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- National curriculum in England — Mathematics — the terms and methods taught in UK schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.