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[ap-stat] RE: Coefficient of Determination
Posted:
Sep 29, 2005 5:06 PM
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Leslie,
In our scatterplot, not all the y-values are the same. Why do the
y-values 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 y-values {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 y-values
will
fall exactly on our regression line. The r^2 value is comparing in a
sense,
the variation of the y-values of the data and the variation in the yhat
values on the regression line. The variation of the yhat-values can
attributed to the linear relationship between x and y and the variation
in
x. The variance of the y-values is larger than the variance in the
yhat-values. 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 y-values. The variance of the yhat-values is attributed to the
linear relationship between x and y. We changed the x-values so the
y-values 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 y-values). 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 yhat-values). 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: [ap-stat] 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 least-squares
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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