Correlation coefficient

Definition

Let $X$ and $Y$ be real random variables on one probability space with finite and positive variances. The correlation coefficient of $X$ and $Y$ is $$\rho_{X, Y} = \operatorname{Corr}(X, Y) = \frac{\operatorname{Cov}(X, Y)}{\sigma_X \, \sigma_Y},$$ where $\operatorname{Cov}$ is the covariance and $\sigma_X$, $\sigma_Y$ are the standard deviations.

Remarks

By the Cauchy–Schwarz inequality, $-1 \le \rho_{X, Y} \le 1$. $|\rho_{X, Y}| = 1$ if and only if $Y = aX + b$ almost surely for some reals $a \ne 0$ and $b$. The sign of $\rho$ is the sign of $a$. The coefficient does not depend on units: $\rho_{aX + b, \, cY + d} = \rho_{X, Y}$ if $a c > 0$. For a sample of pairs $(x_i, y_i)$, the sample correlation coefficient $r$ uses sample means and sample sums of squares instead of expectations.

Explanations

L1 human

The correlation coefficient tells how close the points $(X, Y)$ are to a straight line. A value near $1$ means a strong increasing linear relation, and a value near $-1$ means a strong decreasing one. A value near $0$ means that there is no linear relation.