Standard Deviation Calculator
Paste a list of numbers and this calculator returns the standard deviation and variance, in both the population and sample versions, plus the mean. Values can be separated by commas, spaces, or line breaks, so a spreadsheet column pastes straight in.
- Sample standard deviation (s)
- 1.8708
- Population standard deviation (σ)
- 1.7078
- Sample variance
- 3.5
- Population variance
- 2.9167
- Mean
- 5.5
- Count
- 6
Standard deviation measures how spread out the values are around the mean. A small SD means the data clusters tightly; a large one means it ranges widely. Two classes can both average 75 on a test, but the one with an SD of 4 had consistent scores while the one with an SD of 20 had both top marks and failures.
How it's calculated
σ = √( Σ(xᵢ − μ)² / N ) · s = √( Σ(xᵢ − x̄)² / (n − 1) )
The population formula (σ) divides by N and applies when your data is the entire group of interest. The sample formula (s) divides by n − 1, which corrects for the fact that a sample underestimates the true spread. Variance is simply the square of the standard deviation.
Worked example
For the values 4, 8, 6, 5, 3, 7: the mean is 5.5. The squared deviations sum to 17.5, so the population SD is √(17.5 ÷ 6) = 1.708 and the sample SD is √(17.5 ÷ 5) = 1.871. The sample version is always slightly larger.
Frequently asked questions
›Sample or population: which should I use?
Use population when your data covers everyone or everything you care about, such as all students in one class you are describing. Use sample when the data is a subset standing in for a larger group, which is the common case in statistics, surveys, and science.
›What does one standard deviation mean in practice?
For roughly bell-shaped data, about 68% of values fall within one SD of the mean, 95% within two, and 99.7% within three. This is the empirical rule, and it fails for heavily skewed data.
›What is variance used for if SD is easier to read?
Variance is the mathematically convenient form: variances of independent variables add, which SDs do not. SD is preferred for reporting because it is in the same units as the data.
›Can standard deviation be zero or negative?
It can be zero, exactly when every value is identical. It can never be negative, since it is the square root of a sum of squares.
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