The meaning of the slope of the regression line

Question

We know that the equation of the regression line of y to x is

$y = a + bx$

Where

$b=\frac{S_{xy}}{S_{xx}}$

Now ,$b$

Is the slope of the line ,

My question is :- Does the constant $b$ represents the intensity of the correlation between the tow variables like the the product moment correlation coefficient $r$?

I think they are completely different because $r\in[-1,1] $ but $b$ is not bounded .

If it is correct what does the constant $b$ say about the correlation between $x,y$?

Answer 1

You are right, they are not the same. You can look at correlation as a standardized slope between the $x$ and $y$, since correlation is covariance divided by the respective standard deviations:

\begin{align} r_{xy} &=\frac{Cov(x,y)}{\sigma_{x}\sigma_{y}} \end{align}

The constant $b$ doesn't tell us anything directly about the correlation. You can have a small value of $b$, with $y$ and $x$ highly correlated, and vice-versa. What $b$ does tell us, is how each increase in $x$ corresponds to a scaled increase in $y$, as $b$ is that scaling factor.