Variance

Variance measures how spread out a set of values is by averaging the squared distance of each value from the mean. Squaring the differences means values far from the mean contribute disproportionately, and it also guarantees every term is positive so that above- and below-average deviations cannot cancel out.

The formula is variance = sum((x - mean)^2) / n for a population, or division by n - 1 for a sample, an adjustment called Bessel's correction that keeps the sample variance from systematically underestimating the population value. Because variance is expressed in squared units, squared dollars, squared seconds, it is rarely reported on its own; its square root, standard deviation, converts the measure back to the original scale and is what most people actually interpret.

Variance is the building block behind much of inferential statistics: it appears directly in the formula for standard error, in the assumptions behind the central limit theorem, and in techniques like ANOVA that partition total variance into components attributable to different factors. The main pitfall is comparing variances across variables measured in different units or scales, where the squaring makes the numbers hard to interpret side by side; standard deviation or a normalized measure is usually the better comparison tool.