In each league the better scorers would probably have only slightly higher SATs on average. In other words, there wouldn't be a high correlation between SAT scores and success in soccer within any league.
There would, however, be a much higher correlation between SAT scores and soccer success were the students randomly assigned to the teams in the five leagues.
(Similar remarks could be made about boxing, the different weight divisions washing out much of the correlation between greater weight and success at boxing.)
Of course, there are many dimensions of soccer ability that aren't measured by this imaginary SAT just as there are many, many dimensions of scholastic ability that aren't measured by the SAT. Concentrated work over an extended period is certainly one of the latter, the premium the SAT places on one morning's speedy work being especially difficult to defend.
The analogy between soccer and scholastics is not perfect, of course, but the point remains. Like the soccer "SAT," the scholastic SAT provides incomplete, but useful information to students and colleges. A rough measure of intellectual preparedness, the SAT shouldn't be made into a fetish, but neither should it be ignored.
Without it, colleges would undoubtedly place more emphasis on high school grades and extracurricular activities, measures that also have serious shortcomings — grade inflation and meaningless resume-puffing being the main ones.
The SAT is a flawed predictor, but it is also relatively objective and, among other virtues, sometimes provides a way for the bright, yet socially inept student to be recognized.
Professor of mathematics at Temple University, John Allen Paulos is the author of several best-selling books, including Innumeracy and A Mathematician Reads the Newspaper. His Who’s Counting? column on ABCNEWS.com appears on the first day of every month.