Covariance describes whether two random variables tend to move together, but its magnitude depends on their units. Correlation removes that scale dependence by dividing covariance by the two standard deviations.

For random variables with nonzero standard deviations,

$$\rho_{X,Y}

\frac{\operatorname{Cov}(X,Y)}{\sigma_X\sigma_Y}.$$

Range and interpretation

Correlation always satisfies

$$-1\le\rho_{X,Y}\le1.$$

A positive value indicates positive linear association, a negative value indicates negative linear association, and values near zero indicate weak linear association.

The extreme values $1$ and $-1$ occur when one variable is an exact positive or negative linear function of the other.

Invariance to units

Changing measurement units by positive scaling does not change correlation. Measuring height in metres or centimetres therefore gives the same correlation with another variable.

This is why correlations can be compared across variables with different units more easily than covariances.

Zero correlation is not independence

A relationship can be strongly nonlinear while having zero correlation. For example, a symmetric relationship between $X$ and $X^2$ may have no linear association even though one variable is completely determined by the other.

Independence implies zero correlation when the relevant moments exist, but zero correlation does not generally imply independence.

Correlation is not causation

A large magnitude of correlation describes association in a probability distribution. It does not establish why the variables are associated or whether changing one would cause the other to change.