On 22 Aug 2003, Eric Bohlman wrote in part: > But mathematically, that's nonsensical; zero correlation implies a > roughly circular pattern of points
While a circular pattern implies zero correlation, the converse is not true. A horizontal (or vertical) line also implies zero correlation. So, for that matter, does the symmetrical center section of a parabola. and any number of other patterns: think of all the patterns described by a set of orthogonal polynomials. "Zero correlation" of itself does not imply ANYthing about the pattern described by the bivariate space, except that the sum of products of deviations from the means of the two variables is zero -- which can happen in an infinite number of ways, arising from an infinite number of possible patterns. > (I like Stephen Jay Gould's heuristic that correlation measures the > "skinniness" of a scatterplot), Which is true enough for the usual rather amorphous scatterplot that the term "scatterplot" conjures up in one's mind, AND where the "skinniness" in question is tilted (NOT horizontal, NOT vertical). > and it's rather obvious that if you simply interchange axes, there's > no difference between the two situations. Precisely. > I think this is a case where we're encountering what Sir Francis > Bacon called an "idol," a prejudice of thought that distorts our > view of reality. We want to believe that correlation is defined for > any pair of random variables, but in fact it's meaningless if one of > those RVs has all its density concentrated at a single point. We > somehow expect that because we call it a random "variable" it has to > *vary*, but the definition of a random variable is merely "a > function whose domain is a probability space" and constants meet > that definition. More than that, though: correlation is defined whether the variables in question are random or not. ----------------------------------------------------------------------- Donald F. Burrill [EMAIL PROTECTED] 56 Sebbins Pond Drive, Bedford, NH 03110 (603) 626-0816 . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
