StatReason

Variance Calculator

Find how much a set of numbers spreads out around its mean, measured in squared units.

What this answers

Variance answers "how much does this data spread out around its mean, on average?" Unlike standard deviation, which reports that spread in your data's original units, variance reports it in squared units, which makes it awkward to interpret directly but essential as a building block for many other statistical formulas, from standard deviation itself to ANOVA and regression.

How it is calculated

Variance is the average squared distance of each value from the mean. This calculator supports both the sample convention (dividing by n minus 1, which corrects for the fact that a sample's own mean understates the true population variance slightly) and the population convention (dividing by n, used only when your data genuinely is the entire population, not a sample of it). The convention you select is always stated directly in the result.

Worked example

For the values 1, 2, and 3 under the sample convention: the mean is 2, the squared distances from the mean are 1, 0, and 1, summing to 2, divided by n minus 1 (which is 2), giving a variance of exactly 1. Note that the standard deviation of this same data is also 1, since the square root of 1 is 1, a coincidence specific to this example, not a general rule.

Assumption audit

Calculated from your data: the count of values, and whether the sample convention has at least the 2 values it requires.
Evidence to review: whether your data contains extreme values, since variance squares every deviation from the mean, giving extreme values disproportionate weight compared to a rank-based spread measure.
You must verify: which convention actually applies to your situation, sample versus population, since the software cannot know whether your list represents a sample or a complete population.

Common mistakes

Reporting variance directly to a non-technical audience is a common misstep: a variance of 100 for a dataset measured in dollars is 100 squared dollars, a unit with no intuitive meaning. Taking the square root to get the standard deviation, back in dollars, is almost always what should actually be communicated, with variance kept as the intermediate step used to get there or to feed into further calculations like ANOVA.

Limitations

Because variance is expressed in squared units, it is rarely reported on its own for direct interpretation; standard deviation, its square root, is almost always the more useful number to communicate for describing spread in real terms.