The Correlation Coefficient for Two Random Variables

Defines the correlation coefficient $\rho(X,Y) = \operatorname{Cov}(X,Y) / (\sigma_X \sigma_Y)$ for two random variables, and shows how to compute it directly from variances and covariance, from raw moments, and how to recover the covariance from a given correlation.

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Tutorial

Introduction

The correlation coefficient of two random variables $X$ and $Y,$ denoted $\rho(X,Y)$ or $\rho_{X,Y},$ is defined as the covariance divided by the product of the standard deviations:

\rho(X,Y) = \dfrac{\operatorname{Cov}(X,Y)}{\sigma_X \, \sigma_Y}

where $\sigma_X = \sqrt{\operatorname{Var}(X)}$ and $\sigma_Y = \sqrt{\operatorname{Var}(Y)}.$

The covariance tells us the direction of the linear association between $X$ and $Y,$ but its magnitude depends on the units of the two variables. Dividing by $\sigma_X \sigma_Y$ removes those units and rescales the result so that it always lies in $[-1, 1].$ Values close to $\pm 1$ indicate a strong linear relationship, and values close to $0$ indicate a weak one.

For example, if $\operatorname{Cov}(X,Y) = 6,$ $\operatorname{Var}(X) = 9,$ and $\operatorname{Var}(Y) = 16,$ then $\sigma_X = 3$ and $\sigma_Y = 4,$ so

\rho(X,Y) = \dfrac{6}{3 \cdot 4} = \dfrac{1}{2}.