# Centroid point proof

• May 30th 2009, 08:34 PM
StillRussian
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!
• May 30th 2009, 09:48 PM
matheagle
Just plug in $\displaystyle \bar x$ in for the x and see if you get $\displaystyle \bar y$.
You saw my other proof earlier today, use that post.
• May 31st 2009, 10:10 AM
StillRussian
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 (Smile)
• May 31st 2009, 08:38 PM
matheagle
It looks right.
But again I do need to know the model.
However I think this is true for any model.

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

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

giving us

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

NOW setting $\displaystyle x=\bar x$ in the line $\displaystyle \hat y =\bar y+\hat\beta_1(x- \bar x)$ completes this.
• Jun 1st 2009, 04:50 AM
StillRussian
Matheagle,

thank you very very much for all your help!