What is the difference between covariance and correlation?
Answer
Both measure how two variables move together. Covariance is the raw co-movement, expressed in the product of the two variables’ units: positive when the variables tend to sit above their means together, negative when one runs high while the other runs low. Correlation (Pearson’s 
) is covariance divided by both standard deviations, which strips the units and bounds the result to ![[-1, 1]](https://s0.wp.com/latex.php?latex=%5B-1%2C+1%5D&bg=ffffff&fg=000&s=2&c=20201002)
, so strengths become comparable across datasets. Neither proves causation, and both capture only linear association: a perfect parabola scores zero correlation despite being fully determined.
(1) Units: covariance keeps the product of the units (dollars times kilograms); correlation is unitless and bounded, so rescaling a variable changes the covariance but leaves the correlation untouched.
(2) Sign vs Strength: covariance tells you the direction of co-movement; only correlation tells you a comparable strength, since 
vs 
means the same thing for any pair of variables.
(3) Linear Only: both are second-moment statistics of linear association. Zero correlation does not imply independence: 
on symmetric data has 
while 
fully determines 
.
Figure 1: Correlation reads linear co-movement: positive (top left), negative (top right), zero with independence (bottom left), and the trap case, zero correlation with complete dependence on a parabola (bottom right). The first three have matching covariance signs; the fourth shows why a second-moment statistic can miss structure.
Mathematical Formulation:
![\mathrm{Cov}(X, Y) = \mathbb{E}\big[(X - \mu_X)(Y - \mu_Y)\big]](https://s0.wp.com/latex.php?latex=%5Cmathrm%7BCov%7D%28X%2C+Y%29+%3D+%5Cmathbb%7BE%7D%5Cbig%5B%28X+-+%5Cmu_X%29%28Y+-+%5Cmu_Y%29%5Cbig%5D&bg=ffffff&fg=000&s=3&c=20201002)
Where:
are random variables with means 
and standard deviations 
.![\mathbb{E}[\cdot]](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNCIgaGVpZ2h0PSIxNyIgdmlld0JveD0iMCAwIDI0IDE3Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
is expectation over the joint distribution; in practice both quantities are estimated by sample averages.
lies in ![[-1, 1]](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI0MyIgaGVpZ2h0PSIxNyIgdmlld0JveD0iMCAwIDQzIDE3Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
by the Cauchy-Schwarz inequality, with 
meaning an exact linear relationship.
are random variables with means 
and standard deviations 
.![\mathbb{E}[\cdot]](https://s0.wp.com/latex.php?latex=%5Cmathbb%7BE%7D%5B%5Ccdot%5D&bg=ffffff&fg=000&s=2&c=20201002)
is expectation over the joint distribution; in practice both quantities are estimated by sample averages.
lies in 
meaning an exact linear relationship.| Feature | Covariance | Correlation |
|---|---|---|
| Definition | Mean product of centered values | Covariance divided by both  |
| Units | Product of the variables’ units | Unitless |
| Range | Unbounded | |
| Scale Invariance | No: rescaling a variable rescales it | Yes: unchanged by a positive affine rescaling (a negative scale flips the sign) |
| Typical Use | Portfolio variance, covariance matrices, PCA | Comparing association strength, feature screening |
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