## 3.6 Three versions of $$b_1$$

We’ve just shown that you can think of $$b_1$$ as a weighted average of the response values – weighted by where the $$x$$ values are relative to $$\bar{x}$$:

$b_1 = \sum_{i} \left( \frac{x_i-\bar{x}}{S_{xx}} \right) y_i$ The original version of $$b_1$$ that we mentioned looked different: $b_1 = r\frac{s_y}{s_x}$

In that formulation, we can think of $$b_1$$ as the correlation between $$X$$ and $$Y$$, but “scaled to match” $$X$$ and $$Y$$ by taking their standard deviations into account.

There’s a third way to write $$b_1$$ as well: $b_1 = S_{xy}/S_{xx}$

This one makes it look like $$b_1$$ is related to the covariance of $$X$$ and $$Y$$, but scaled relative to the amount of variance in $$X$$.

I know I said I didn’t really care about $$b_0$$, but in case you’re wondering (and can’t find this in your old stats notes), one way to write the least-squares estimate of $$b_0$$ is: $b_0 = \bar{y} - b_1 \bar{x}$ This is mildly interesting as well: the intercept relates to the actual means of $$X$$ and $$Y$$, as well as the relationship between them.

You may also recognize this as related to the old “point-slope form” for defining a line – what is the point we know the line must pass through?

They’re all true! You can get from any of these formulations to the others with a bit of algebra. What they all have in common is the idea of the slope coefficient reflecting both the relationship between $$X$$ and $$Y$$, and the amount of variation/scale of $$X$$ and $$Y$$. (That’s why $$b_1$$ changes if you change the units of your variables!) Depending on the situation, you may find it helpful to think about the slope in any of these ways :)