Explanation
What standard deviation measures
Standard deviation measures typical spread around an arithmetic mean in the original units. Squaring deviations prevents opposite signs from cancelling, and taking a square root restores the units.
Population or sample?
Choose population when your values are the entire group you want to describe, such as every test score in one class. Choose sample when they are a sample used to estimate the spread of a larger group. Enter values separated by semicolons; a sample needs at least two.
σ = √(Σ(xᵢ − μ)² ÷ n)
| σ | population standard deviation |
|---|---|
| xᵢ | a value in the list |
| μ | population mean |
| n | number of values |
s = √(Σ(xᵢ − x̄)² ÷ (n − 1))
| s | sample standard deviation |
|---|---|
| xᵢ | a value in the list |
| x̄ | sample mean |
| n | number of values |
A population with mean 5
| Values, separated by semicolons | 2;4;4;4;5;5;7;9 |
|---|---|
| Result | 2 |
Mean 5 of 8 values; variance = sum of squared deviations ÷ 8 = 4; σ = √4 = 2
1, 3 and 5 as a sample
| Values, separated by semicolons | 1;3;5 |
|---|---|
| Result | 2 |
Mean 3 of 3 values; variance = sum of squared deviations ÷ 2 (n − 1) = 4; s = √4 = 2
Identical values have no spread
| Values, separated by semicolons | 7;7;7 |
|---|---|
| Result | 0 |
Mean 7 of 3 values; variance = sum of squared deviations ÷ 3 = 0; σ = √0 = 0
Diagram of this result
A dot plot of the values with the mean and a shaded band one standard deviation either side.
Each value is a dot on a number line; equal values stack. A marker shows the mean and a shaded band covers one standard deviation below and above it. The exact numbers are stated in the answer and the steps.
Common mistakes
Do not confuse variance with standard deviation: variance has squared units. Do not divide by n for the usual sample variance estimate; its divisor is n − 1. A small spread does not prove accuracy if every measurement shares a bias.
Assumptions and limits
The calculator describes unweighted numerical observations and assumes no measurement error model. Sample variance uses Bessel’s correction; sample standard deviation itself is not an unbiased estimate in general. It does not establish normality or a confidence interval. Numeric observations use up to 30 significant input digits and 50 internal digits; no presentation rounding enters the calculation.
Sources
NIST Engineering Statistics Handbook, measures of scale: https://www.itl.nist.gov/div898/handbook/eda/section3/eda356.htm
Why divide by one fewer observation for samples?
Estimating the mean from the same observations uses one degree of freedom. Dividing the squared deviations by n − 1 gives the conventional unbiased variance estimate for independent observations from one population.