## Roots of Unity

Mathematics: learning it, doing it, celebrating it.

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The opening tip of the 2012 NCAA women's basketball championship game, played April 3, 2012. My Baylor Lady Bears, led by #42 Brittney Griner and #0 Odyssey Sims, defeated Notre Dame 80-61. Image: flickr user Han Shot First.

March Madness always sneaks up on me. I mean, I know that March has started because my dad's birthday and my wedding anniversary are right at the beginning of the month, but I always end up scrambling to make my NCAA basketball tournament picks the day before games start. Mathematician Jordan Ellenberg has taken the strategy of choosing the team with the best math department, but for some reason I'm not too confident that a 14-seed will win it all.

As you're working on your bracket, remember what CBS sports commentator Gregg Doyel wrote: "For teams with a realistic chance at winning multiple games in the NCAA tournament,…the worst seed to have is the No. 8 or the No. 9. That's statistical certainty." But is it? In a paper in the Journal of Quantitative Analysis in Sports (I believe the full text is available with free registration), statisticians Tracy L. Morris and Faryal H. Bokhari tested that bold assertion.

This assertion isn't unreasonable. In the NCAA tournament, the field of 64 teams is divided into four subsections of 16 teams each, seeded 1-16. In the first round, team 1 plays team 16, team 2 plays team 15, and so on. So teams 8 and 9, the "dreaded middle seeds," as Morris and Bokhari call them, play each other first. So they have "easier" first-round competitors than the 10th seeds and beyond, but in the second round, they're stuck with the winner of the 1-16 match, which is almost always the 1 seed.

In order to study the situation rigorously, Morris and Bokhari first had to clarify the question of what it means for a seed to be "better." "Obviously, the worst seed to have is 16, since no 16 seed has ever defeated a 1 seed in the men's tournament and only one 16 seed has defeated a 1 seed in the women's tournament," they wrote in their paper. "We assume what Doyel meant to argue is that it would be better to receive a slightly lower seed than to be seeded 8 or 9. Specifically, we want to know whether it is statistically better to receive a 10, 11, or 12 seed than an 8 or 9 seed."

Their analysis found no significant difference between the number of wins achieved by the 8 and 9 seeds versus the 10, 11, and 12 seeds, but then they narrowed their focus to teams that won at least one game. If a team can make it out of the first round, is a 10, 11, or 12 seeds better than an 8 or 9 seed? For this question, they found that the answer was yes, at least in the men's tournament. In this case, 10 seeds have more wins, and more 10, 11, and 12 seeds make it to the Sweet Sixteen than 8 and 9 seeds. In the women's tournament, Morris and Bokhari found that an 11-seed that made it out of the first round had an advantage over an 8 or 9 seed that made it out of the first round, but that was the only statistically significant result they found.

As a side note, I ran across another interesting March Madness paper in the Journal of Analytics in Sports when I was doing research for this post. In it, Kathy L. Gray and Neil C. Schwertman develop their own measure of success in the tournament, basing it on points scored rather than wins. They then compare the selection and seeding as predicted by two well-known computer models to that chosen by the NCAA and find that the NCAA seeding does a better result of predicting tournament success than the computer models do. I thought this was interesting because, as Morris and Bokhari's paper suggests, seeding might have an effect on the outcome, so the NCAA selection and seeding might be a somewhat self-fulfilling prophecy. Without the ability to play the tournament multiple times, there isn't a good way to remove this confounding factor from the analysis.

I actually prefer the women's tournament to the men's. The Baylor men's team was suspended under upsetting circumstances during my last two years of college, but our women's team won the title the same day I found out I had gotten into Rice, the graduate program I ended up attending. Brittney Griner's transcendence these past four years has done nothing to dampen my enthusiasm for the Lady Bears!

When I filled out my women's bracket, I thought it would be nice to have a little friendly competition, so I started a group called Roots of Unity on the ESPN tournament challenge website. I'd love for you to fill out a bracket (or 263) and join me. Women's games start March 23, so you have until then to fill out your bracket. I made two, one of which is my "serious" entry and one of which is a "no upsets" bracket so we can see how the rankings compare to the actual outcome. There are no prizes for winning, but it should be fun.

The views expressed are those of the author and are not necessarily those of Scientific American.