Standard Deviation Calculator
Paste numeric observations to compare population and sample spread using transparent two-pass variance formulas.
Sample versus population changes the result even with identical inputs
For the values 2, 4 and 6, the mean is 4 and the sum of squared deviations is eight. Population variance is 8 ÷ 3 ≈ 2.67, giving standard deviation about 1.63. Sample variance is 8 ÷ (3 − 1) = 4, giving standard deviation 2. Both follow the same input data but answer different questions.
Use population mode when the data cover the complete population you intend to describe. Use the usual sample estimator when they are observations used to estimate variability in a wider population, with at least two observations. Data entry units, duplicates and missing values still need checking.
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Formula
population variance = Σ(x − mean)² ÷ n; sample variance = Σ(x − mean)² ÷ (n − 1)
One data set can have two legitimate standard deviations
- For 2, 4, 4, 4, 5, 5, 7 and 9, the mean is 5 and squared deviations sum to 32.
- Population variance is 32 ÷ 8 = 4, giving standard deviation 2. Sample variance is 32 ÷ 7, giving sample standard deviation about 2.1381.
What the result tells you
The denominators answer different statistical questions. Use the population or sample result appropriate to the analysis.
Before you use the result
Standard deviation describes spread in the original units and is sensitive to unusual observations. It is not the standard error of the mean.
References for this method
Useful for
- Statistics homework
- Data spread checks
- Population vs sample variance
Questions
Why are sample and population standard deviation different?
Sample variance divides by n−1 to estimate population variability from a sample, while population variance divides by n.
Can the calculator use negative values?
Yes. Any finite numeric observations are accepted.