How to use this calculator
- Enter or choose numbers.
- Enter or choose your data is.
- The standard deviation updates as you type. Use Copy result or Share to keep it.
Formula
Sample: s = √( Σ(xᵢ − x̄)² ÷ (n − 1) ); population: σ = √( Σ(xᵢ − μ)² ÷ n )
- xᵢ
- each value
- x̄ (μ)
- the mean
- n
- number of values
Worked example
- Mean = 40 ÷ 8 = 5.
- Squared deviations: 9, 1, 1, 1, 0, 0, 4, 16; their sum is 32.
- Population variance = 32 ÷ 8 = 4, so σ = √4 = 2.
- As a sample, variance = 32 ÷ 7 ≈ 4.571 and s ≈ 2.138.
The population standard deviation is 2.
What the result means
For roughly bell-shaped data, about 68% of values lie within one standard deviation of the mean and about 95% within two.
Assumptions and limits
- Choose “a sample” when your numbers are drawn from a larger group (the usual case); this divides by n − 1 (Bessel’s correction).
- Standard deviation is sensitive to outliers.
Questions
Why divide by n − 1 for a sample?
A sample’s values sit closer to their own mean than to the true population mean, so dividing by n would underestimate the spread. n − 1 corrects that bias in the variance.
What is the difference between variance and standard deviation?
Variance is the average squared deviation; standard deviation is its square root, so it is in the same units as the data.