Variance Calculator

Sample variance versus population variance

Population variance (σ²) divides the sum of squared deviations by n — the total count — and describes the exact spread of a complete, closed dataset. Sample variance (s²) divides by n − 1 instead. The reduction, known as Bessel's correction, compensates for the fact that a sample mean is calculated from the same data, which causes the raw squared deviations to slightly underestimate the true population spread.

Use sample variance whenever your data is a subset drawn from a larger population; use population variance only when you have every member of the group.

Variance and standard deviation formulas

Standard deviation is simply the square root of variance. Variance is expressed in squared units (e.g. metres² if your data is in metres), which makes it less intuitive for direct interpretation. Taking the square root returns the spread to the original units, giving standard deviation.

Step-by-step worked example

For the dataset 2, 4, 4, 4, 5, 5, 7, 9: the mean is (2+4+4+4+5+5+7+9) ÷ 8 = 40 ÷ 8 = 5. The squared deviations are (2−5)²=9, (4−5)²=1, (4−5)²=1, (4−5)²=1, (5−5)²=0, (5−5)²=0, (7−5)²=4, (9−5)²=16, totalling 32.

Population variance = 32 ÷ 8 = 4; population standard deviation = 2. Sample variance = 32 ÷ 7 ≈ 4.57; sample standard deviation ≈ 2.14.

When variance is zero or very large

Variance of zero means every value in the dataset is identical — there is no spread at all. Very large variance relative to the mean suggests outliers or a wide range. Because variance squares the deviations, a single extreme value contributes disproportionately to the total.

For datasets with suspected outliers, the interquartile range or mean absolute deviation may give a more robust picture of typical spread.

Common uses of variance

Variance appears in finance as a measure of investment risk (portfolio variance combines individual asset variances and covariances), in quality control to monitor process consistency, in experimental science to partition sources of variation (ANOVA), and in machine learning to describe model uncertainty.