#### by Darci L. Kracht

This column appeared in the Daily Kent Stater on Thursday March 21, 2002 under the headline "Basketball can make math more fun."

In conference play last season, Kent State's Trevor Huffman outshot teammate Bryan Bedford both from the two-point range and from the three-point range.

True or false: This means that Huffman outshot Bedford from the field overall.

I posed this question to my Math 11008 (Explorations in Mathematics) students last spring on their final exam. To my surprise, all 19 students in the class answered "false." And guess what? They were right! In fact, data from Kent State athletics' Web page shows that Huffman's shooting percentage was higher than Bedford's for both two-point and three-point field goals, yet his overall field goal average was lower than Bedford's.

What's going on here? Let's look at the numbers to find out. The first row of the table has the data for two-point field goals. We see that Huffman made 57 out of 127 attempts for an average of 57/127 = 0.449, while Bedford made 13 of his 30 attempts for an average of 0.433. The second row of the table contains the same statistics for shots taken from the three-point range. Here, Huffman made 35 of 100 for an average of 0.350 while Bedford missed his only attempt for an average of 0.000. Thus, Huffman outshot Bedford in both categories.

 Trevor Huffman Bryan Bedford Made Attempted Average Made Attempted Average Two-pointers 57 127 0.449 13 30 0.433 Three-pointers 35 100 0.350 0 1 0.000

Now combine the categories to find the players' overall field goal shooting percentages. Adding down the columns of the first table, we see that Huffman made 57+35=92 out of 127+100=227 attempts, giving him an average of 0.405. On the other hand, since Bedford made 13 of his 31 attempts, his average is 0.419, beating Huffman's!

 Trevor Huffman Bryan Bedford Made Attempted Average Made Attempted Average All field goals 92 227 0.405 13 31 0.419

This is an example of an unusual statistical phenomenon known as Simpson's Paradox, after the statistician who described it in 1951. It can occur whenever data from two or more distinct categories are combined into a single category. Simpson's Paradox is more likely to occur if there is some "lurking" variable that makes the subcategories rather different from one another.

There are a couple of lurking variables in this example. First is the level of difficulty of the shots. Shooting from beyond the three-point arc is considerably more difficult than shooting closer to the hoop, as shooting percentages reflect. Combine this with the fact that guards (like Huffman) attempt three-pointers much more frequently than do forwards (like Bedford). Indeed, 100 of Huffman's 227 attempts (44%) were from the three-point range compared to only 1 out of 31 (3%) for Bedford. Consequently, the more difficult shots had a much greater impact on Huffman's overall percentage than Bedford's.

Finally, we should ask just how useful these statistics are. In other words, how accurately does a player's average reflect the probability that he will make the next shot? With 100 or more attempts in each category, an extra hit or miss would change Huffman's average by at most one percentage point. Therefore, his shooting percentage is probably a fairly accurate predictor of future performance--- not so for Bedford, as can be seen most dramatically in the statistic for three-point field goals. With only one three-pointer attempted, his accuracy in this category must be perfect (100%) or dismal (0%). Such a statistic is clearly of little predictive value. Indeed, Bedford has proven to be one of the more accurate shooters on the team this season.

I leave as an exercise for the reader to find and interpret examples of Simpson's Paradox in this season's statistics. (Quiz on April 2nd!)

Darci Kracht is an instructor in the Department of Mathematical Sciences. Her Web page ( www.math.kent.edu/~darci/) contains more examples of Simpson's Paradox in men's and women's basketball.