Where does the correlation come from in the regression coefficient equation for simple regression

Question

In simple linear regression. $\beta = \frac{Cov(x,y)}{s_x^2}$. This is often written as $\beta = r_{xy}(\frac{s_y}{s_x})$

Where does the correlation come from in this equation? From my understanding

$ r_{xy} = \frac{\Sigma(x_i - \bar x)(y_i-\bar y)}{\sqrt{\Sigma(x_i - \bar x)^2\Sigma(y_i-\bar y)^2}} $

I know you can expand the first $\beta$ equation to yield

$\frac{\frac {\Sigma(x_i - \bar x)(y_i - \bar y)}{N-1}}{\Sigma(x_i-\bar x)(x_i- \bar x)} $

But I don't see how you can obtain $r_{xy}$ from an equation in which $\Sigma(y_i - \bar y)^2$ isn't represented. Let along obtain that along with $\frac{s_y}{s_x}$

Answer 1

There are several problems in your question -- here's two of them:

1) You're conflating sample and population quantities. This will lead to confusion.

2) your 'expanded' equation for "$\beta$"* is wrong.

* (which isn't an equation for $\beta$, it's for $\hat\beta$)

On to the question itself:

Did you notice that both $s_y$ and $r$ contain a term in $\sqrt{\sum_i(y_i - \bar y)^2}$ and the two cancel?