Do variances always add?

Only when the covariance contributions sum appropriately to zero, with uncorrelated terms as a common sufficient condition.

AI Glossary / Statistics and probability / Intermediate

Variance Sum Law

Definition

The variance sum law gives the variance of a sum or difference of random variables, including their covariance. For two variables with finite variances, Var(X + Y) = Var(X) + Var(Y) + 2 Cov(X,Y). For a difference, the covariance term changes sign. Variances add directly when the variables are uncorrelated, including the special case of independence. Ignoring covariance can substantially misstate uncertainty when quantities move together or in opposite directions.

What it means

The variability of a total depends both on how much its parts vary and on whether their deviations align. If two quantities tend to be high together, their sum fluctuates more than it would under zero covariance. Their difference can fluctuate less because shared movements cancel. If they tend to move oppositely, the pattern reverses. This explains why uncertainty in a combined score, paired difference or aggregate demand cannot generally be found from separate standard deviations alone.

How it works

Start with the centered sum and expand its square. The squared terms give individual variances, and the cross-product terms give covariances. For constants a and b, Var(aX + bY) = a squared times Var(X) plus b squared times Var(Y) plus 2ab Cov(X,Y). A constant added to the expression has no effect on variance. For several variables, include every pairwise covariance, or write the result as w transpose Sigma w using the covariance matrix Sigma and coefficient vector w. These identities require the relevant second moments to exist.

Three practical examples

Example 1: A service team combines demand from two channels that rise together during busy periods. Example 2: An AI evaluation compares paired scores from two models on the same tasks, where shared task difficulty creates covariance. Example 3: A weighted performance index combines several correlated measurements. The law applies to all these combinations, but estimating the component variances and covariances reliably is a separate statistical task.

Think of it this way

Think of two people carrying a platform. If both rise and fall together, the platform's overall height changes strongly while their height difference stays small. If they move in opposite directions, the difference becomes more variable. Covariance records whether movements align, so it changes the spread of sums and differences in opposite ways.

Worked example

Suppose X has variance nine, Y has variance sixteen and their covariance is six. Their standard deviations are three and four, so the correlation is 6/(3 times 4) = 0.5. The variance of X + Y is 9 + 16 + 2 times 6 = 37. The variance of X minus Y is 9 + 16 minus 2 times 6 = 13. For 2X minus Y, the variance is 4 times 9 plus 16 minus 4 times 6 = 28. Omitting covariance would give 25 for both the simple sum and difference, overestimating one and underestimating the other. The covariance is feasible because its absolute value does not exceed the product of the standard deviations. All values are illustrative population quantities, not sample estimates.

Illustrative case study

In this fictional case study, Chestnut Evaluation compares two AI systems on identical tasks. An analyst initially treats their average scores as independent when estimating uncertainty in the difference. The team instead retains the paired task results and estimates variation in task-level differences. Shared difficulty means both systems often score high or low on the same tasks, so covariance matters. The report explains the paired design and avoids assuming that separate score variability fully determines comparison uncertainty.

Common misconceptions and mistakes

Common mistakes include adding standard deviations, subtracting variances when taking a difference, and assuming independence merely because two variables have different names. For a difference, the individual variance terms remain positive; only the covariance contribution changes sign. Zero covariance is sufficient for the two-variable addition rule, but it does not generally imply independence. Another mistake is using correlations without converting them to covariance through the standard deviations and consistent units.

Business use

Businesses use the law to combine uncertainty in forecasts, measurements, paired comparisons and weighted scores. Preserve paired or joint data so covariances can be estimated rather than guessed. Check whether dependence changes across time or operating conditions. If assumptions about covariance are uncertain, compare plausible scenarios instead of presenting one unexplained total. For nonlinear combinations, this exact linear identity may be used within an approximation, but the approximation and its limitations should be stated.

Check your understanding

What is Var(X + Y) in the example?

Answer: Thirty-seven.

What is Var(X minus Y)?

Answer: Thirteen.

Why is the difference less variable than the sum?

Answer: Positive covariance cancels some shared movement in the difference.

What is Var(2X minus Y)?

Answer: Twenty-eight, because the coefficients affect both squared variance terms and the covariance term.

Related terms

Correlation
Standard deviation
Covariance matrix
Related terms include covariance
paired design and uncertainty propagation

Learn next

Learn next how the law produces the variance of a sample mean and why correlation changes the benefit of averaging. Compare independent observations with repeated measurements sharing a common effect. Then study first-order uncertainty propagation for nonlinear functions, where derivatives determine the coefficients in an approximate variance formula.

Sources

Sources: University of Vermont, Statistical Methods for Psychology glossary, Variance Sum Law, https://www.uvm.edu/~statdhtx/methods8/Glossary/Glossary.html; Stanford CS109, Joint Statistics lecture, https://web.stanford.edu/class/archive/cs/cs109/cs109.1244/lectures/13_joint_statistics_annotated.pdf. These primary teaching sources explain variance and covariance combinations. The three linear-combination calculations are original and independently checked; the paired AI comparison case is fictional.

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