Standard deviation formula
Sample: s = √( Σ(x − x̄)² ÷ (n − 1) ) · Population: σ = √( Σ(x − μ)² ÷ n )
Standard deviation measures how spread out numbers are from their mean. A small value means the numbers sit close to the average; a large value means they are spread out.
Step by step
- Find the mean: (4 + 8 + 6 + 5 + 3 + 7) ÷ 6 = 5.5.
- Subtract the mean from each value and square it: 2.25, 6.25, 0.25, 0.25, 6.25, 2.25.
- Add the squares: 17.5.
- Divide by n − 1 = 5 (sample) to get the variance: 3.5.
- Take the square root: s ≈ 1.871. (Population: √(17.5 ÷ 6) ≈ 1.708.)
Sample or population?
Dividing by n − 1 (Bessel’s correction) makes the sample result a fairer estimate of the whole group’s spread. In Excel, STDEV.S is the sample version and STDEV.P the population version.
What the number tells you
For roughly bell-shaped data, about 68% of values fall within one standard deviation of the mean and about 95% within two. That rule doesn’t hold for skewed data.
Worked example
For the numbers 2, 4, 4, 4, 5, 5, 7, 9 the mean is 40 ÷ 8 = 5. The squared differences from the mean are 9, 1, 1, 1, 0, 0, 4 and 16, which add up to 32.
- Population SD: √(32 ÷ 8) = √4 = 2.
- Sample SD: √(32 ÷ 7) = √4.571 ≈ 2.138.
Use the sample version when your numbers are a sample from a bigger group, which is the case in most homework and research.
Related calculators
- Mean, Median & Mode Calculator: Average, median, mode and range.
- Percentage Calculator: X% of Y, X is what % of Y, and more.
- Rounding Calculator: Round to decimals, tens or significant figures.
Questions people ask
What is the difference between variance and standard deviation?
Variance is the average squared distance from the mean; standard deviation is its square root, which puts it back in the original units.
Can standard deviation be negative?
No. It is a square root of a sum of squares, so it is zero or positive. Zero means every value is the same.
Formula checked October 2026 · How we calculate