Relationship between Correlation and Regression Coefficients

Regression and correlation are closely linked — regression coefficients and the correlation coefficient are computed from the very same covariance and standard-deviation values. The key relationship is:

The sign of $r$ is taken as the common sign of $b_{yx}$ and $b_{xy}$ (both positive $\Rightarrow$ $r$ positive; both negative $\Rightarrow$ $r$ negative).

Special cases:

• When $r = 0$: both $b_{yx} = 0$ and $b_{xy} = 0$ — the two regression lines are perpendicular (parallel to the axes, through the mean point), confirming there is no linear relationship to exploit for prediction.
• When $r = \pm 1$: the two regression lines coincide into a single line — every point lies exactly on it, so predicting $Y$ from $X$ or $X$ from $Y$ gives a perfectly consistent answer either way.
• For every other value of $r$ (i.e. $-1 < r < 0$ or $0 < r < 1$), the two lines are genuinely distinct, intersecting only at $(\bar{X}, \bar{Y})$. …