Variance Calculator

The variance of a set of numbers is the average of the squared distances from their mean. Squaring is what makes it work: without it, a value one above the mean and a value one below it would cancel, and every symmetric set would score zero spread no matter how far apart its extremes are.

The price of that trick is the units. Squaring a measurement in metres gives square metres, so the variance cannot be compared with the data or added to it. Its square root, the standard deviation, is back in the original units and is the form usually reported.

Find the mean of the list, then measure every value against it. A value above the mean gives a positive deviation, one below gives a negative one.

Square each deviation, which makes every contribution positive, and add them up. This total, the sum of squares, is what the mean has already been subtracted from.

Divide by n for a population, or by n − 1 for a sample. The mean was calculated from these same values, so it already fitted itself to them; the n − 1 divisor corrects for the one degree of freedom that consumed.

Take the square root to return to the original units. The result is the standard deviation, and roughly 95% of a normal distribution falls within two of them of the mean.

A variance of 4 and a standard deviation of 2 describe exactly the same spread. Report the standard deviation unless someone specifically asks for the variance, because it stays in the original units.

The coefficient of variation puts the spread in percentage terms, so a spread of 2 around a mean of 5 becomes 40%. This is the only fair way to compare spread across quantities measured in different units.

Zero variance means every value is identical. It is the one case where the standard deviation is unambiguously zero rather than a rounding artefact.

A large variance relative to the mean usually means outliers or a mixture of groups rather than natural variation, and the mean is a poor summary of such a list.

• The divisor. Switching from n to n − 1 always lowers the result, and the difference matters most for small lists where n − 1 is much smaller than n.
• Outliers. Squaring means one value ten standard deviations out contributes a hundred times as much as one at a single standard deviation.
• Sample size. The estimate of the population variance settles down roughly as 1/√n, so small lists give unstable results.
• Adding a constant. Shifting every value by the same amount changes the mean and the median but leaves the variance untouched.
• Scaling. Multiplying every value by c multiplies the variance by c², so a variance is not comparable across different units without converting.

• Summarising the spread of a measurement before comparing it with a mean or a tolerance.
• Checking whether a measurement process is stable, by watching the variance of repeated readings of the same thing.
• As the first step of a t-test or an analysis of variance, both of which are built on comparing variances.
• When a standard deviation is needed but only as an intermediate value, such as in a z-score or a confidence interval.
• Diagnosing data quality: a variance far larger than expected usually points to a unit mix-up, a stray decimal point or two populations mixed together.

• The variance depends entirely on the mean, so one extreme value moves both and can make the spread look smaller than the data really is.
• It says nothing about symmetry or normality. A symmetric and a heavily skewed list can share a variance exactly.
• Values from different units or scales cannot be combined; a variance of weights is meaningless alongside a variance of times.
• The sample divisor assumes the values are independent draws. Repeats of the same measurement are correlated and understate the true spread.
• The number is in squared units, so it is only directly comparable with another variance measured in the same units.