Strength of a relationship

The correlation coefficient summarises the strength of the relationship between two variables.

Properties of correlation

For any random variables, \(X\) and \(Y\),

\[ -1 \;\le\; \Corr(X,Y) \;\le\; +1 \]

(Proved in full version)

The next result shows that the correlation coefficient between two variables can only be ±1 if the variables are linearly related.

Linear relationships and correlation

\[ \left|\Corr(X,Y)\right| = 1 \quad\text{if and only if} \quad Y = a + bX \quad\text{for some constants } a \text{ and } b. \]

(Proved in full version)