Variance Calculator: Population vs Sample + Stdev Guide
A variance calculator quantifies how spread out a set of numbers is — essential for quality control, lab data, finance risk, and stats homework. Use a free variance calculator to get population or sample variance (and standard deviation) without hand-cranking every squared deviation.
This guide covers the formulas, when to divide by n vs n−1, how stdev relates to variance, and a full worked example.
Compute population or sample variance
Free variance calculator — n vs n−1, stdev, clear worked results.
Population Variance Formula
For a complete population of n values with mean μ:
Use this when every member of the group you care about is in the dataset (for example, all exam scores in one closed class if that class is the population).
Sample Variance and Why n−1
For a sample used to estimate a larger population, with sample mean x̄:
Dividing by n−1 (Bessel’s correction) compensates for the fact that the sample mean is itself estimated from the data. Spreadsheet functions like Excel’s VAR.S use n−1; VAR.P uses n.
Quick rule of thumb
- Entire population in hand: divide by n (population variance)
- Sample to infer a larger group: divide by n−1 (sample variance)
- When a textbook or teacher says “sample variance,” they almost always mean n−1
Standard Deviation from Variance
Variance is in squared units (e.g., cm²). Standard deviation puts spread back into the original units:
Report stdev when you want an interpretable “typical distance from the mean.” Report variance when formulas (ANOVA, finance models) call for it directly.
Worked Example
Data: 2, 4, 4, 4, 5, 5, 7, 9 (n = 8). Mean = 40 ÷ 8 = 5.
- Squared deviations: 9, 1, 1, 1, 0, 0, 4, 16 → sum = 32
- Population variance: 32 ÷ 8 = 4 · σ = 2
- Sample variance: 32 ÷ 7 ≈ 4.571 · s ≈ 2.138
Paste the same list into the ToolMars variance calculator to confirm both results and avoid arithmetic slips on longer datasets.
Compute population or sample variance
Free variance calculator — n vs n−1, stdev, clear worked results.
Frequently Asked Questions
What is variance in statistics?
The average of squared deviations from the mean — a measure of spread. Zero means no variation.
When do you use population variance vs sample variance?
Population (÷ n) for a full population; sample (÷ n−1) when estimating from a sample.
How is standard deviation related to variance?
Stdev is the square root of variance and shares the original data units.
Why divide by n−1 for sample variance?
Bessel’s correction reduces bias when the sample mean is estimated from the same data.
Can a variance calculator also give standard deviation?
Yes — stdev = √variance for both population and sample versions.