The variance measures the mean squared distance of a random variable from its expected value. For a real square-integrable random variable ,The square prevents positive and negative deviations from cancelling. It also gives the variance squared units: if measures a length in metres, its variance has units of square metres. Its nonnegative square root is the standard deviation, which has the same units as .
For example, let take , and with equal probabilities. Its expected value is zero, but its variance is , and its standard deviation is . The constant random variable zero has the same expected value and zero variance: a mean alone does not describe the spread. The variance is zero exactly when almost surely.
The variance is useful because squared deviations interact neatly with linear combinations. The covariance supplies the cross terms when combining different random variables, and a covariance matrix organizes those terms for many variables. The characterization of variance as the minimum mean squared error of a constant also explains why the expected value is the best constant prediction under squared error.
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