What does covariance mean?
Covariance is a statistical measure of how two variables change together. A positive covariance indicates that when one variable tends to rise above its mean, the other tends to as well; a negative covariance indicates that one tends to fall while the other rises; a value near zero suggests no consistent linear tendency either way. Unlike correlation, which standardizes this relationship onto a scale from -1 to 1, covariance retains the units of the underlying data, so its size depends on the scales being compared. This makes covariance especially useful in fields like finance, where it underpins portfolio theory by quantifying how different assets move relative to one another, and in machine learning, where covariance matrices summarize relationships across many variables. Beyond statistics, mathematicians and physicists use 'covariance' to describe laws or equations that keep the same form under changes of coordinates — a principle central to Einstein's theory of relativity.
A measure of the degree to which two variables change together, equal to the expected value of the product of their deviations from their respective means.
"The sample covariance between height and weight was clearly positive."
Often encountered as an element of a covariance matrix when many variables are analysed at once.
In mathematics and physics, the property of an equation or law that it retains the same form under a specified change of coordinates or reference frame.
"Maxwell's equations exhibit covariance under Lorentz transformations."
Distinct from the statistical sense; found chiefly in relativity, tensor analysis, and differential geometry.
The plural typically refers to multiple pairwise covariance values, most often the off-diagonal entries of a covariance matrix rather than separate abstract concepts.
"The covariance matrix stores every pairwise covariance among the ten asset returns."
Covariance is the mathematical reason diversification works: in modern portfolio theory, combining assets whose returns have low or negative covariance is what actually reduces risk.
Reviewed by Deb Chak, Editor. AI-assisted content curated by RJS Tech Solutions LLP.
Etymology of covariance
Covariance derives from Latin roots: the prefix co- ('together with') combined with variance, from Latin variare, 'to change'. The term emerged within probability theory in the nineteenth century alongside related notions of joint variation, building on earlier statistical work by figures such as Gauss and Bravais. Its sibling concepts include variant and variance, which share the same Latin root variare, and the adjective covariant, used both in statistics and in mathematics to describe quantities that transform in a corresponding way.
Related word forms
How covariance is actually used
Covariance is technical register, used almost exclusively in statistics, finance, machine learning, physics, and related quantitative fields; it is rare in everyday speech. In statistics it appears most often in plural contexts such as covariance matrices. The physics sense — covariance of equations under coordinate transformation — is distinct from the statistical one and is encountered mainly in relativity and tensor analysis.
Easily confused with covariance
Correlation is a standardized measure of the relationship between two variables bounded between -1 and 1, while covariance is unstandardized and its magnitude depends on the units of the variables.
Variance measures how a single variable spreads around its own mean, whereas covariance measures how two variables vary jointly.