Product moment correlation coefficient

Question

How to know if $pmcc>0$ or $pmcc<0$ if I am provided the two gradients of the regression lines? Since $pmcc=±√product~of~gradients$, how to deduce the sign of pmcc?

Answer 1

If you know the slope $\hat{b}$ of the regression line, then you know the sign of $\hat{\rho}=r_{XY}$. Note that, $$ r_{XY}= \frac{\sum(X_i - \bar{X})(Y_i - \bar{Y})}{\sqrt{ \sum(X_i - \bar{X})^2 \sum(Y_i - \bar{Y}) }}, $$ and $$ \hat{b}= \frac{\sum(X_i - \bar{X})(Y_i - \bar{Y})}{\sum(X_i - \bar{X})^2 }, $$ hence, $\sum(X_i - \bar{X})(Y_i - \bar{Y})$ determines the sign of the (estimated) slope and the (estimated) correlation coefficient.