Pearson's Correlation Coefficient

Topics: Measure of Association - Statistics

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(definition)

Let $(x_1,y_1),\dots ,(x_n,y_n)$ be observations of the variables $(X,Y)$.

Pearson’s correlation coefficient between the variables $(X,Y)$ is defined as:

$$r_{XY}=\frac{S_{XY}}{S_X{S}_Y}$$

…where $S_{XY}$ is the covariance of $(X,Y)$, and $S_X,S_Y$ are the standard deviations of $X$ and $Y$, respectively.

Properties §

• $r_{XY}\in [-1,1]$
• If $x\perp \perp y$, $r_{XY}=0$
• If $r_{XY}<0$, there’s an inverse relation between the variables (i.e. when one variable grows, the other decreases)