Variance Calculator
Calculate population and sample variance from a list of numbers. Free variance calculator showing the mean, sum of squared deviations and standard deviation.
Updated 2026-06-14 · Free · No sign-up · Runs privately in your browser
Show the steps
How the Variance Calculator Works
Enter a list of numbers and this calculator returns both the sample variance (s²) and the population variance (σ²), along with the mean and the sample standard deviation. Values can be separated by commas, spaces or new lines.
The Formulas
Population variance: σ² = Σ(x − μ)² ÷ N Sample variance: s² = Σ(x − x̄)² ÷ (N − 1)
Both start from the sum of squared deviations — the total of each value’s squared distance from the mean. The only difference is the divisor: N for a full population, N − 1 for a sample.
Worked Example
For the data 10, 12, 23, 23, 16, 23, 21, 16 (N = 8):
- Mean x̄ = 144 ÷ 8 = 18
- Squared deviations: 64, 36, 25, 25, 4, 25, 9, 4 → sum = 192
- Population variance = 192 ÷ 8 = 24
- Sample variance = 192 ÷ 7 ≈ 27.43
- Sample standard deviation = √27.43 ≈ 5.24
Population vs Sample at a Glance
| Quantity | Divisor | Use when |
|---|---|---|
| Population variance σ² | N | You have every member of the group |
| Sample variance s² | N − 1 | You have a sample of a larger group |
Why Variance Matters
Variance underpins most of inferential statistics. It feeds into the standard deviation, confidence intervals, t-tests and ANOVA, and it quantifies risk in finance and quality control. Because variance is in squared units, analysts usually report the standard deviation alongside it for an intuitive sense of spread.
Frequently asked questions
What is variance in statistics?+
Variance measures how spread out a set of numbers is around their mean. It is the average of the squared distances from each value to the mean. A small variance means values cluster close to the average, while a large variance means they are widely scattered.
What is the difference between sample and population variance?+
Population variance divides the sum of squared deviations by N, the full number of values, and is used when your data is the entire population. Sample variance divides by N − 1 instead, which corrects for the bias you get when estimating a population's spread from a sample. Use sample variance for survey or experimental data.
Why does sample variance divide by n minus 1?+
Dividing by N − 1 (Bessel's correction) compensates for the fact that a sample's mean is itself estimated from the data, which slightly understates the true spread. Using N − 1 makes the sample variance an unbiased estimator of the population variance, so it is the standard choice for samples.
How is variance related to standard deviation?+
Standard deviation is simply the square root of the variance. Variance is in squared units, which are hard to interpret, so taking the square root returns the spread to the same units as the original data. This calculator shows both.
How do I calculate variance step by step?+
Find the mean, subtract it from each value to get deviations, square each deviation, add them up, then divide by N for population variance or N − 1 for sample variance. The calculator shows the sum of squared deviations so you can follow every step.