Correlation coefficient
$$r = \frac{1}{n}\sum_{i=1}^n\frac{(x_i-\bar x)(y_i-\bar y)}{\sigma_x\cdot\sigma_y}$$
But for a given data point $x_i$ and predicted value $y_p$,
$$\frac{(y_p-\bar y)}{\sigma_y} = r\cdot\frac{(x_i-\bar x)}{\sigma_x}$$
This is very non intuitive.
If we consider $$\alpha = \frac{(x_i-\bar x)}{\sigma_x}$$
and $$\beta =\frac{(y_i-\bar y)}{\sigma_y} $$ How can the average of $\alpha.\beta$ be same as $\frac{\alpha}{\beta}$?