Variance Calculator

Variance measures spread. Subtract the mean from every value, square each difference, add them up, and divide — by n − 1 for a sample, by n for a full population. Squaring stops positive and negative deviations from cancelling, but it also changes the units: variance in a data set measured in inches is in square inches, which is why the standard deviation, its square root, is usually what gets reported.

Seven values summing to 42, so the mean is 6. The deviations are −2, 2, 0, −1, −3, 3, 1, and their squares are 4, 4, 0, 1, 9, 9, 1 — a sum of squared deviations of 28. As a sample, variance is 28 ÷ 6 = 4.667 and the standard deviation is 2.160. Treated as a whole population, it's 28 ÷ 7 = 4 exactly, with a standard deviation of 2.

A sample's own mean is always the point that minimises its squared deviations, so the spread around it is systematically smaller than the spread around the true population mean. Dividing by n − 1 rather than n inflates the estimate just enough to correct that bias. The difference is large in small samples — 17% here with n = 7 — and negligible past a few hundred observations. If your numbers are the entire population, use n.

The question is not how many values you have but what they represent. Test scores for every student in a class you are describing are a population: use n. The same scores treated as a sample of what the teaching method produces are a sample: use n − 1. When in doubt in research, use the sample formula — nearly all statistical software defaults to it, and it is the conservative choice.

Because deviations are squared, variance is dominated by outliers: one value far from the mean can inflate it beyond recognition. It also says nothing about shape — a uniform spread, a bimodal split, and a normal bell can all share the same variance. And it is scale-dependent, so comparing variance across data sets with different units or magnitudes is meaningless; the coefficient of variation, standard deviation over mean, is the comparable measure.