Variance Calculator

What is variance?

Variance is the average squared distance between each value and the mean. It summarises how widely a data set is spread: a variance of zero means every value is the same, and the larger it gets, the more the values scatter.

Squaring the distances serves two purposes. It stops negative and positive deviations from cancelling each other, and it gives extra weight to values far from the mean. The cost is that variance is in squared units (dollars², seconds², cm²), which is why the standard deviation, its square root, is usually quoted when describing data to people. Variance, however, is the quantity that adds up nicely: variances of independent variables add, which underlies many statistical tests.

How to use the calculator

1. Keep Continuous data selected for measurements and paste the values into the Numbers box. Commas, spaces, tabs and new lines all work as separators.
2. Choose Sample if the numbers are a sample from a larger population (divides by n − 1) or Population if they are all the values there are (divides by n).
3. Press Calculate. The table shows the variance, standard deviation, mean, standard error of the mean and the count of numbers. At least two values are needed.

For a yes/no outcome select Proportion data, give the sample size, and enter the event proportion, a percentage rate or a count of events. The calculator returns the variance, the standard deviation and the standard error.

Variance formulas

Population: σ² = Σ (xᵢ − μ)² ÷ N

Sample: s² = Σ (xᵢ − x̄)² ÷ (n − 1)

Proportion: variance = p (1 − p)

The n − 1 in the sample formula is Bessel's correction. Because the sample mean is computed from the same data, the deviations around it come out slightly too small on average, and dividing by n − 1 instead of n corrects for it. With a proportion each observation is 1 (event) or 0 (no event), so the variance depends only on p and is largest, 0.25, at p = 0.5.

Worked example

Take five measurements: 12, 15, 11, 18 and 14. Their mean is 70 ÷ 5 = 14, and the deviations are −2, 1, −3, 4 and 0. Squared, these are 4, 1, 9, 16 and 0, which sum to 30.

The calculator reports the sample variance as 7.50. For a proportion, a survey in which 35% of 500 respondents agreed has a variance of 0.35 × 0.65 = 0.2275, a standard deviation of 0.4770 and a standard error of 0.0213.

Useful properties of variance

A few rules make variance easy to work with once you know them:

• Adding the same constant to every value leaves the variance unchanged, because the spread does not move.
• Multiplying every value by c multiplies the variance by c². Converting from metres to centimetres (c = 100) multiplies the variance by 10,000.
• For independent variables the variance of a sum is the sum of the variances: Var(X + Y) = Var(X) + Var(Y). Standard deviations do not add this way.
• The variance of a sample mean is the variance of a single observation divided by n, which is why averages are steadier than single readings.

Reading the output is straightforward: the variance row is the main answer, the standard deviation row is its square root in the original units, and the standard error is that standard deviation divided by √n.

Where variance is used

• Hypothesis tests: t-tests, ANOVA and the standard error of a difference are all built from variances. See the p-value calculator.
• Confidence intervals: the width of an interval depends on the variance of the estimate; see the confidence interval calculator.
• Finance: the variance of returns is the basic measure of an asset's risk, and portfolio variance combines asset variances with their covariances.
• Quality control and engineering: the variance of a process shows how consistent it is, and reducing it is the core idea of Six Sigma.
• Experiment design: the variance of the metric you are testing drives how large the sample has to be.

When you compare spreads across different scales, divide by the mean instead: the coefficient of variation (standard deviation ÷ mean) is unitless. See the coefficient of variation calculator.