# Math Help - Centroid point proof

1. ## Centroid point proof

Hello everyone!

Does anybody know how to prove the fact that the centroid point (x_bar, y_bar) lies on the regression line?

Thank you!

2. Just plug in $\bar x$ in for the x and see if you get $\bar y$.
You saw my other proof earlier today, use that post.

3. Right, so I have

β0_hat + β1x_bar = y_bar – β1_hat*x_bar + β1_hat*x_bar =y_bar
It just seemed too good to be true

4. It looks right.
But again I do need to know the model.
However I think this is true for any model.

$\hat y=\hat\beta_0+\hat\beta_1 x$

is the fitted line, with $\hat\beta_0=\bar y-\hat\beta_1 \bar x$

giving us

$\hat y =\bar y-\hat\beta_1 \bar x +\hat\beta_1 x =\bar y+\hat\beta_1(x- \bar x)$.

NOW setting $x=\bar x$ in the line $\hat y =\bar y+\hat\beta_1(x- \bar x)$ completes this.

5. Matheagle,

thank you very very much for all your help!