The formula for the correlation coefficient, r, in the AP Statistics hand out is given as .
The student is then expected to see that the value of r is always in the interval between -1 and 1 inclusive. The fact that this is not even remotely obvious to the new student is indicated by the frequency with which this question is asked by AP Statistics teachers on the AP electronic discussion group. Because my stats course is still a mathematics course, I would like for my students to develop some understanding of the reason, rather than just accepting it is true because “the teacher said so.” Toward that end, and because I believe vectors are undernourished in the current curriculum, I tend to explain the reasoning using the dot product of vectors. Because of the number of equations involved, I have stored the paper (in docx format) here.
One reader also requested the link to the blog on Standard Deviation as Distance..