## 7.5 Correlation Coefficient

• The correlation coefficient $$\rho$$ measures the strength of association between $$X$$ and $$Y$$ in the population.
• $$\sigma^{2} = VAR(Y|X)$$ is the variance of $$Y$$ for a specific $$X$$.
• $$\sigma_{y}^{2} = VAR(Y)$$ is the variance of $$Y$$ for all $$X$$’s.

$\sigma^{2} = \sigma_{y}^{2}(1-\rho^{2})$ $\rho^{2} = \frac{\sigma_{y}^{2} - \sigma^{2}}{\sigma_{y}^{2}}$

• $$\rho^{2}$$ = reduction in variance of Y associated with knowledge of X/original variance of Y
• Coefficient of Determination: $$100\rho^{2}$$ = % of variance of Y associated with X or explained by X
• Caution: association vs. causation.