What this tool calculates
This calculator summarizes a numeric list with its count and arithmetic mean, then reports population variance, sample variance, population standard deviation, and sample standard deviation. Presenting both denominator conventions prevents a result from being labeled simply “standard deviation” without stating its role.
Population measures describe the entered values as the complete population of interest. Sample measures use a correction intended for estimating population variance from a sample. With only one observation, sample variance and sample standard deviation are not defined.
Inputs
Enter finite numbers separated by commas, spaces, semicolons, or line breaks. Use one measurement definition and compatible unit. Repeated observations remain part of the list.
Empty input is rejected. The tool does not recognize missing-value markers, weights, frequency tables, or decimal commas. Clean and document the data rather than allowing invalid tokens to disappear unnoticed.
Method and formula
For n values with mean x̄, calculate squared deviations (xᵢ − x̄)².
population variance = Σ(xᵢ − x̄)² ÷ n
sample variance = Σ(xᵢ − x̄)² ÷ (n − 1)
Each standard deviation is the non-negative square root of its corresponding variance. Variance has squared units; standard deviation returns to the original measurement unit. The sample formula requires at least two observations.
Worked example
Enter 2, 4, 4, 4, 5, 5, 7, 9. There are 8 observations and their mean is 5.
The squared deviations sum to 32. Population variance is 32 ÷ 8 = 4, so population standard deviation is √4 = 2. Sample variance is 32 ÷ 7 ≈ 4.571429, giving sample standard deviation ≈ 2.13809.
The output therefore includes count 8, mean 5, population variance 4, population standard deviation 2, and sample standard deviation 2.13809.
How to interpret the result
Standard deviation measures spread around the arithmetic mean. A value of 2 in the example means deviations have a root-mean-square scale of 2 units under the population convention. It does not mean every observation is within 2 units or that the distribution is normal.
Choose the population column when the list is exactly the complete group being described. Choose the sample column when the observations are treated as a sample and the inferential assumptions are appropriate. State the convention in reports.
Accuracy and limitations
Standard deviation is sensitive to outliers and can be misleading for strongly skewed or multimodal data. The calculator does not assess normality, sampling independence, bias, uncertainty, confidence intervals, or whether the mean is a meaningful center.
Floating-point arithmetic and the direct two-pass-style computation can lose precision for extreme magnitudes. The tool is not an arbitrary-precision statistical package. Review collection design, visualize the distribution, and retain raw data before drawing conclusions.
Sources
- NIST/SEMATECH e-Handbook, Standard Deviation defines and interprets standard deviation as a scale measure.
- NIST/SEMATECH e-Handbook, Measures of Scale provides broader context for variance and alternative dispersion summaries.
Editorial record
Author: SoupCalc Editorial Team
Last reviewed: August 14, 2026
Review scope: Population and sample denominators, unit interpretation, one-value limit, distribution caveats, NIST references, and the eight-value example were checked.