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Topic:
[apstat] Coefficient of Determination
Replies:
2
Last Post:
Sep 29, 2005 8:51 PM




[apstat] RE: Coefficient of Determination
Posted:
Sep 29, 2005 5:06 PM


Leslie,
In our scatterplot, not all the yvalues are the same. Why do the yvalues vary?
Well, if y is linearly related to x, and I change x, then y also changes. Part of the variation we see in y can be explained by the fact that y is related to x and we changed x.
If we look at the ordered pairs (1,2), (3,6), (4,8), and (5, 10), we see that y = 2x. There is some variation in the yvalues {2, 6, 8, 10}. We can measure this variation and say that the variance of y is 11.666. If we didn't change the x's, what would have been the variance of y? If y = 2x and there is no variation in the x's, there will be no variation in the y's, so all of the variation in y is attributed to the variation in x.
Thus, r^2 = 1.
Now, in the real world we don't really expect that all of the yvalues will fall exactly on our regression line. The r^2 value is comparing in a sense, the variation of the yvalues of the data and the variation in the yhat values on the regression line. The variation of the yhatvalues can attributed to the linear relationship between x and y and the variation in x. The variance of the yvalues is larger than the variance in the yhatvalues. The extra variation comes from the residuals. The value of r^2 is the ratio of the variance of the fitted yhat values to the variance of the yvalues. The variance of the yhatvalues is attributed to the linear relationship between x and y. We changed the xvalues so the yvalues had to also change.
Try this. Pick a set of ordered pairs and put them in L1 and L2. Find the standard deviation sy for L2. Store this as A (so A^2 is the variance of the yvalues). Now fit a regression line, store this in Y1. Now let L3 = Y1(L1). These are your fitted values. Find the standard deviation for L3. Store this in B (so B^2 is the variance of the yhatvalues). Now, compare r^2 to B^2/A^2.
Dan
Daniel J. Teague NC School of Science and Mathematics 1219 Broad Street Durham, NC 27705 teague@ncssm.edu
Original Message From: Leslie Free [mailto:lfree@ncusd203.org] Sent: Thursday, September 29, 2005 2:22 PM To: for Teachers of AP Statistics Subject: [apstat] Coefficient of Determination
In my 3rd year of teaching this course, I'm still trying to wrap my brain around the concept of the coefficient of determination. I know all about the mantra we're supposed to teach our kids and that it's the fraction of variation in the y that's explained by leastsquares regression of y on x, but it still doesn't make any sense to me. Can someone please explain it to me in plain English??
Thanks, Leslie
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