Variance
Squared dispersion measure.
What does Variance mean?
Variance is a statistical measure that quantifies how far a set of numbers is spread out from their mean (average). It is calculated as the average of the squared differences from the mean. A high variance indicates that data points are spread widely, while a low variance means they are clustered close together. Variance is the foundation of many statistical techniques and is the square of the standard deviation.
How to calculate Variance
To calculate the population variance, find the mean of all values, subtract the mean from each value and square the result, then compute the average of those squared differences. The formula is: σ² = Σ(xᵢ − μ)² / N. For sample variance, divide by (N − 1) instead of N to apply Bessel's correction: s² = Σ(xᵢ − x̄)² / (N − 1). For example, for the values 10, 20, 30, 40, 50, the mean is 30, and the population variance is 200.
FAQ
Population variance (σ²) uses the entire population and divides by N. Sample variance (s²) uses a subset of the population and divides by (N − 1) to correct for the bias that arises when estimating population variance from a sample. This adjustment, called Bessel's correction, ensures the sample variance is an unbiased estimator of the population variance.
Variance is the square of standard deviation, and standard deviation is the square root of variance. Variance is expressed in squared units of the original data, while standard deviation is in the same units as the data. For example, if data is measured in meters, variance is in meters squared (m²) and standard deviation is in meters (m).
Despite the squared units making it harder to interpret directly, variance has useful mathematical properties. It is additive for independent random variables, plays a central role in ANOVA and regression analysis, and is easier to work with algebraically than standard deviation. In practice, standard deviation is often preferred for interpretation, while variance is used in formulas and proofs.
Variance can be zero when all values in the data set are identical — there is no spread at all. Variance can never be negative because it is calculated as the average of squared differences, and squaring always produces a non-negative result.
Use standard deviation when you want an intuitive measure of spread in the same units as your data — for example, reporting that exam scores vary by about 10 points. Use variance when performing statistical calculations that require it, such as ANOVA, portfolio theory in finance, or combining variability from independent sources.
Related calculators
- Standard Deviation— Spread of values.
- Mean— Average value.
- Median— Middle value.
- Mode— Most frequent value.