Pearson Correlation Coefficient

probability-theory

Definition

Correlation Coefficient

The correlation coefficient between two real-valued random variables is defined as:

$$\rho (X,Y)=\frac{\mathbb{C}ov(X,Y)}{\sqrt{\mathbb{V}ar(X)}\cdot \sqrt{\mathbb{V}ar(Y)}}$$

It is bounded, i.e.:

$$-1\le \rho (X,Y)\le 1$$

and it holds that:

$$\rho (X,Y)=\pm 1⟺Y=a\cdot X+b\wedge \mathrm{sign}(a)=\rho (X,Y)$$

It measures the strength and direction of linear relationship between two quantitative variables.

Covariance

The correlation coefficient is the covariance of $X,Y$, scaled to variance 1 or standardised:

$$\begin{array}{rl}\rho (X,Y)& =\mathbb{C}ov\left(\frac{X}{\sqrt{\mathbb{V}ar(X)}},\frac{Y}{\sqrt{\mathbb{V}ar(YY)}}\right)\\ & \mathbb{C}ov\left(\frac{X-\mathbb{E}[X]}{\sqrt{\mathbb{V}ar(X)}},\frac{Y-\mathbb{E}[Y]}{\sqrt{\mathbb{V}ar(Y)}}\right)\end{array}$$

Examples