> [!theorem] > > Let $X$ and $Y$ be [[Random Variable|random variables]]. Then the **correlation coefficient** of $X$ and $Y$ is the [[Correlation|correlation]] between their normalised versions: > $ > \rho_{XY} = \ev\braks{\paren{\frac{X - \mu_X}{\sigma_X}}\paren{\frac{Y - \mu_Y}{\sigma_Y}}} > = \frac{\ev(XY) - \mu_X\mu_Y}{\sigma_X\sigma_Y} > $ $ \begin{align*} \rho(X, Y) &= \frac{\cov{X, Y}}{\sigma_x \sigma_y} \\ r &= \frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}} = b_1 \frac{s_x}{s_y} \end{align*} $ The slope of a [[Linear Regression|linear regression]] model and the [[Covariance|covariance]] do not indicate the strength of the [[Correlation|correlation]]. The parameters must be [[Z-Score|standardised]] by dividing them with their standard deviation. This standardised slope $\rho$ (for [[Population|population]]) and $r$ (for [[Sample|sample]]) is equal to the [[Covariance|covariance]] between the two [[Random Variable|random variables]] divided by their [[Standard Deviation|standard deviations]], known as the **Pearson Linear Correlation Coefficient**. * The magnitude of $r$ represents the strength of the relationship. * The sign of $r$ represents the direction of the relationship. - The correlation coefficient is equal to the slope when $x$ and $y$ are equally spread out ($s_x = s_y$). - Interchanging $x$ and $y$ do not change the value of $r$. - $r \in [-1, 1]$, by the [[Cauchy-Schwarz Inequality]]. - Only applicable to *linear* relationship.