Foul when up three? You make the call

You remember the game. Memphis vs. Kansas -- the 2008 national title. With 10.8 seconds remaining in regulation, Derrick Rose made the second of two free throws to put Memphis up by three. Kansas inbounded the ball, and Memphis coach John Calipari did something that would haunt him for the rest of his career: He didn't foul. You know what happens next: Mario Chalmers sinks Mario's Miracle, tying the game and sending it to overtime, where Kansas eventually outlasted the Tigers for Bill Self's first national title.

Calipari has long been criticized for that decision. Calipari has said his team was trying to foul but didn't get it done; he's even criticized himself for allowing Kansas to shoot that three. You have to assume he feels pretty rotten about it.

That's because the conventional wisdom -- forged over the past few years -- says that teams leading by three at the end of games should foul. The thinking is simple: It's harder for a team to make one free throw, miss it, and then grab a rebound and score than it is for that team to come down the court and make a quick, albeit challenged, three point shot.

But is that really true? In what the Harvard Sports Analysis Collective calls the "first comprehensive empirical study of the issue for college basketball," the HSAC found that teams in Memphis' situation have just as good a chance of winning if they don't foul as if they do. I'll let author John Ezekowitz explain:

In the 2009-2010 season, I found 443 instances where a team held the ball down three points during their last possession of a period (either the end of the 2nd half or an overtime period). In 391 of those cases, the team leading did not foul. In 52 cases, the team chose to foul. [...]

Of the 52 teams that committed a foul, six lost the game for a winning percentage of 88.46%. Of the 391 teams that did not foul, 33 lost the game for a winning percentage of 91.56%. Both a two sample t-test of proportion and a Chi-squared test fail to reject the null hypothesis that there is a difference in winning percentage between the two strategies. In this sample, teams that did not foul won slightly more often. For the less statistically inclined, this means that there is no significant difference between the two strategies.

Even if you have no idea what a two-sample t-test of proportion is (I don't know if I got that far in the Statistics 101 class I hurriedly got out of the way during one collegiate summer session; if I did, I wouldn't remember it), John's results are clear. Generally speaking, teams do not gain a significant statistical advantage from fouling while up three. In John's data, those teams actually lost slightly more often than teams that did not foul.

There are a few problems with using the words "generally speaking" about this dilemma. For one, not every team's timeframe for preventing that game-tying three -- or that game-tying tip-in -- is the same. The skill of the opposing team also factors in; if the team you're trying to hold off has a particularly dangerous three-point shooter, maybe fouling is a slightly more appropriate strategy. This stuff all matters.

Still, this is an awfully enlightening start. We see this all too often in sports: A strategy sprouts, and whether or not that strategy is right, it becomes codified and accepted as conventional wisdom. In this case, though, we see the conventional wisdom might not matter. If you're up three with a few seconds left, you're usually going to win the game. No matter what you do.

That lack of efficacy might not make most college hoops coaches comfortable, but at the very least, John Calipari should rest easy. Mario's Miracle was just a really, really good shot.

(Hat tip: Rush The Court)