Variance Calculator

Variance measures how spread out a set of values is around their mean — the larger the variance, the more scattered the data. This variance calculator returns both the sample variance and the population variance, so you can pick the result that matches your situation. If you’ve wondered how to calculate variance by hand, the worked example below shows every step.

How to use this calculator

1. Enter your numbers in the values box, separated by commas or spaces (for example, 2, 4, 6, 8, 10).
2. Read the Sample Variance if your data is a sample drawn from a larger population — the standard choice for coursework, surveys, and research.
3. Read the Population Variance if your numbers represent every member of the group, with nothing left out.

The key difference: sample variance divides the sum of squared deviations by n − 1 (Bessel’s correction), while population variance divides by n.

Worked example

Take the five values 2, 4, 6, 8, 10.

mean (x̄) = (2 + 4 + 6 + 8 + 10) / 5 = 30 / 5 = 6

Subtract the mean from each value and square the result:

(2  − 6)² = 16
(4  − 6)² =  4
(6  − 6)² =  0
(8  − 6)² =  4
(10 − 6)² = 16
            ──
Σ(xi − x̄)² = 40

Divide by n for population or by n − 1 for sample:

population variance (σ²) = 40 / 5 =  8.000
sample variance     (s²) = 40 / 4 = 10.000

The formal definition and notation for variance is documented in the NIST/SEMATECH e-Handbook on standard deviation and variance.

Frequently asked questions

What is variance?

Variance is the average of the squared deviations from the mean. Squaring the differences prevents negative and positive gaps from cancelling out, producing a single number that describes how spread out the data is.

When should I use sample variance versus population variance?

Use sample variance (divide by n − 1) when your data is a sample drawn from a larger group — the usual case for surveys, experiments, and statistics homework. Use population variance (divide by n) only when every member of the group is included in your data set.

Why does sample variance divide by n − 1?

Dividing by n − 1 is Bessel’s correction. A sample tends to underestimate the true spread of the population it comes from; reducing the denominator by one corrects that downward bias, making the sample variance an unbiased estimator of the population variance.

What is the difference between variance and standard deviation?

Standard deviation is the square root of variance. Both measure spread, but standard deviation is expressed in the same units as the original data, which makes it easier to interpret. Variance, in squared units, is more useful when combining or decomposing spread across multiple variables.