This variance calculator finds the variance of any data set — both population variance (σ²) and sample variance (s²) — with the mean, sum of squared differences and standard deviation shown step by step. Paste your numbers below.
What Is Variance?
Variance measures how far a set of numbers is spread out from their mean — it is the average squared deviation. Squaring does two things: it makes all deviations positive, and it punishes big outliers more than small ones. Variance is the working currency of statistics: risk in finance, error in engineering, and group differences in research are all expressed through it. Its square root is the standard deviation — if you need the full step table with each deviation listed, use our Standard Deviation Calculator. Both live in the Statistics Calculators collection.
The Variance Formulas
- Population variance: σ² = Σ(x − μ)² ÷ N
- Sample variance: s² = Σ(x − x̄)² ÷ (n − 1)
Worked example (verified): test scores 85, 90, 78, 92, 88. Mean = 86.6. Squared deviations: 2.56 + 11.56 + 73.96 + 29.16 + 1.96 = 119.20. Population variance = 119.20 ÷ 5 = 23.84; sample variance = 119.20 ÷ 4 = 29.80. The population standard deviation is √23.84 ≈ 4.88 points.
Why Two Variances? (n vs. n − 1)
If your data is the entire group you care about, divide by N. If it is a sample standing in for a bigger group, divide by n − 1 — Bessel's correction. A sample's own mean sits closer to the sample's values than the true population mean does, so raw squared deviations underestimate the real spread; dividing by the smaller n − 1 inflates the estimate back to unbiased. Research, surveys and quality sampling use the sample version almost by default.
Variance, Standard Deviation, and Units
Variance is measured in squared units (points², dollars²), which are hard to interpret directly — that is why standard deviation, back in the original units, is the number usually reported. But variance has the mathematical superpower: variances of independent quantities add, standard deviations do not. That additivity is why variance powers ANOVA, portfolio theory and error propagation. Once you have a mean and SD, the Z-Score Calculator locates any single value in the distribution.
Related Tools
- Standard Deviation Calculator — full deviation table
- Z-Score Calculator — position in a distribution
- Statistics Calculators — the full collection
Frequently Asked Questions
What is the difference between variance and standard deviation?
Standard deviation is the square root of variance. Variance is in squared units; standard deviation is in the data's original units and is easier to interpret.
Can variance be negative?
No — it is an average of squares, so the minimum is 0 (all values identical).
Which variance does Excel calculate?
Both: VAR.P for population, VAR.S for sample. This calculator matches them and shows the working.
What is a high variance?
Only relative to the mean and context. A variance of 24 on test scores (SD ≈ 5 points) is modest; the same variance on daily temperatures would be large. Compare SD to the mean for a quick sense of scale.
Why square the deviations at all?
Because unsquared deviations always sum to zero around the mean. Squaring preserves the size of every deviation and makes the mathematics of combining sources of variation work.
Is this variance calculator free?
Yes — free, instant, no sign-up.