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Topic: [ap-stat] why does the regression line always pass through x bar y bar?
Replies: 4   Last Post: Sep 30, 2010 12:05 AM

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Andrew Bressette

Posts: 16
Registered: 7/27/10
re:[ap-stat] why does the regression line always pass through x bar y bar?
Posted: Sep 30, 2010 12:05 AM
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An excellent question and in a rather confusing lesson I attempted with my students I think we discussed a very reasonable answer.

Imagine you have a bunch of data points that you would represent as ordered pairs: (x1,y1) (x2,y2) so on and so forth. We want to standardize these points and therefore take the z-score of each coordinate and then replot these z-scores. (Zx1,Zy1) (Zx2,Zy2) so on and so forth.

Our new plot is now "centered" on the coordinate plane. More importantly (xbar,ybar) is now (0,0), this comes naturally from the definition of a zscore. Z=(x-xbar)/s If x=xbar naturally our zscore becomes 0.

Now, I'm an AP Statistics n00b and I would personally stop here and say the least-squared line of z-scores always passes through the origin but that may seem a little like "hand-waving" to your students so you can continue.

The equation for a linear model is given by yhat=bnot+b1x

bnot=(ybar)-(b1)(xbar) where b1 is the slope. TADA!!! with our standardized scatter-plot both xbar and ybar are 0 and therefore the y-intercept of our standardized regression line is 0. Therefore, our regression line must pass through (0,0) AKA (xbar,ybar).

Please forgive the notation. bnot = "b sub-zero"

I really hopes this help a bit. I'm using BVD 3rd and the two equations I use near the end are derived in the text but once again... *shudders*
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