Variance measures spread: the average of the squared differences from the mean. Square the distance of every value from the mean, add them up, and divide — by N for a whole population or by n − 1 for a sample. The square root of variance is the standard deviation, which brings the figure back into the original units.
Reviewed: June 20, 2026 · Author: Naveen P N, Founder — AI Calculator · Verified against: the population and sample variance formulas, recomputed in code.
The variance formulas
First compute the mean. Then for each value, take its difference from the mean and square it; sum those squares to get the sum of squared deviations. Divide by N for a population or by n − 1 for a sample. Dividing by n − 1 — Bessel's correction — compensates for using the sample mean and gives an unbiased estimate of the true population variance.
Worked example — 2, 4, 4, 4, 5, 5, 7, 9
The squared deviations from the mean of 5 add to 32. As a population that's a variance of 4 (SD 2); as a sample it's 4.571 (SD 2.138). The sample figures are a little larger because they divide by 7 rather than 8.
Frequently Asked Questions
Mean of squared deviations: population σ² = Σ(x−μ)²/N; sample s² = Σ(x−μ)²/(n−1).
Population divides by N (whole group); sample divides by n−1 (estimate from a sample). Sample is a bit larger.
SD = √variance. Set 2,4,4,4,5,5,7,9: population variance 4, SD 2. SD is in original units.
Bessel's correction removes bias from using the sample mean. 32/7 = 4.571 vs population 32/8 = 4.
Values are more spread out. Zero variance means all values are identical. Outliers inflate it strongly.