Friday, July 16, 2010

Do foul shooters choke in the last minute of close games?

Searching Google Scholar for studies about "choking," I came across an interesting one, a short, simple analysis of free-throw shooting in NBA games.

It's called "Choking and Excelling at the Free Throw Line," by Darrell A. Worthy, Arthur B. Markman, and W. Todd Maddox. (.pdf)

The authors looked at all free throws in the last minute of games in the three seasons from 2002-03 to 2004-05. They broke their sample down by score differential, and compared the success percentage to the players' career percentages.

They found that for most of the scores, the shooters converted fewer than expected. Here's the data as I read it off the graph (but see the PDF for yourself). The score differential is from the perspective of the shooting team, and the "%" column is actually percentage points.

-5 points: -3%
-4 points: -1%
-3 points: -1%
-2 points: -5% (significant at 5%)
-1 points: -7% (significant at 1%)
+0 points: +2%
+1 points: -5% (significant at 5%)
+2 points: +0%
+3 points: -1% ("also signficant")
+4 points: +1%
+5 points: -1%

The authors conclude that choking occurs, especially when down by 1 point.

It may not be obvious at first glance from the chart, but there's a tendency to "choke" all the way down: there are 8 negatives and only 3 positives (and the negatives are generally more extreme than the positives). Do players actually shoot worse in the last minute of close games?


I couldn't think of any statistical reason the results might be misleading ... but in an e-mail to me, Guy came up with a good one.

Suppose that career shooting percentage is not always a good indicator of a player's percentage that game. Maybe it varies throughout a career, somehow -- higher at peak age and lower elsewhere, or, even, increasing throughout a career. (It doesn't matter to the argument *how* it varies, just that it does.)

You might expect that the differences would just cancel out. However, the overestimated shooters would be appear in each category more than the underestimated shooters. Why? Because they would miss the first shot more often, and take a second shot *within the same score category*.

As an example, suppose two players have 75% career percentages, but, on this day, A is a 100% shooter and B is a 50% shooter. Suppose they each go to the line twice with the game tied. On their first shot, A makes two and B makes one. So far, their percentage is 75%, as expected. Perfect.

But, only B gets to take a second shot with the game still tied. He does that once, the one time in two he missed the first shot. And he makes it half the time.

So, on average, you have these guys taking five shots, and making 3.5 of them. That's 70 percent -- 5 percentage points less than the career average would suggest.

Now, the numbers I used here are not very realistic -- nobody's a 100% shooter, and hardly anyone is a 50% shooter. What if I change it to 80% and 70%?

Then, following the same logic, and if my arithmetic is correct, those two players combined would make 74.8% of their shots instead of 75%. It's still something, but not nearly enough to explain the results. Still, I really like Guy's explanation.


So, there you go: it does look like, for those three seasons, players shot worse in the last minute than expected. Can anyone think of an explanation, other than "choking" and luck, for why that might be the case? Has anyone done this kind of analysis for other seasons to confirm these results?

UPDATE: Maybe it's just fatigue! See comments.

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At Friday, July 16, 2010 2:47:00 PM, Anonymous Mike said...


At Friday, July 16, 2010 2:52:00 PM, Anonymous Mike said...

Also... player free throw percentages vary over one's career, right? I'm talking true talent, not just a year of observed stats.

Let's say Shaq has been at times during his career, a 50% free throw shooter and a 70% free throw shooter, and everything in between. And let's say opposing teams/coaches have a good idea of his true talent each year, based on his stats that year, the immediate year prior, etc.

Wouldn't you be more likely to foul him at the end of a game in seasons in which he is closer to the 50% of the spectrum versus the 70% end of the spectrum?

So if Shaq was, for his career, a 60% shooter, and in the three seasons they're examining, his true talent was 50%, 60%, and 70%... wouldn't there be a larger sample of free throws coming from his 50% season, since it's more valuable to foul him then than it would be in the 70% year?

At Friday, July 16, 2010 2:56:00 PM, Anonymous JB H said...

I expected the second half of your post to be about how it's obviously due to players tiring :)

I don't know why the authors only looked at close games. Free throws late in blow outs seem like a necessary control group. Although the level of fatigue in the population of FT shooters goes down the less close the game is.

At Friday, July 16, 2010 3:30:00 PM, Blogger Phil Birnbaum said...

Er, yes, fatigue indeed. I should have thought of that. Oops.

Mike's explanation is probably part of it too, but ... I'm thinking fatigue.

At Friday, July 16, 2010 10:00:00 PM, Blogger Patrick said...

Why don't they just compare players' FT% in the last minute of close games to the last minute of non-close games? Isn't that the counterfactual?

At Friday, July 16, 2010 10:09:00 PM, Blogger Phil Birnbaum said...

Agreed. That would certainly help settle the question ...

At Saturday, July 17, 2010 9:29:00 AM, Anonymous Anonymous said...

Two thoughts:

1) In the last minutes of a close contest, the "flow of the game" is interrupted by timeouts and the clock-stopping fouls themselves. This change of flow does not happen in blowouts. (Also, in blowouts, it is likely that both teams have bench warmers on the floor. The career foul shooting percentages of bench warmers could easily be meaningless.)

2) In a close game, the leading team, generally, does not want to foul; it is the team that is behind that is doing the fouling. One would think that the sample size of foul shots by teams behind would be significantly smaller than the sample of teams leading.

At Sunday, July 18, 2010 12:23:00 AM, Anonymous Jim A said...

If fatigue were a factor, wouldn't we see FT% decrease quarter-by-quarter? That would seem easy enough to check.

I think we should be receptive to the possibility that it is more difficult to perform in high-pressure situations. This is not exactly the same as what we typically call "choking" because most sports skills involve head-to-head battles. For example, any pressure felt by the batter in baseball would be offset by equal pressure felt by the pitcher. Plus, people tend to look for individual chokers rather than a universal effect.

Another example of this is field goal kicking in football, which the defense has little control over. A study by Bilder and Loughlin in Chance (see link) found that lead-changing and late-game attempts have lower success rates.

At Monday, July 19, 2010 12:52:00 AM, Blogger Phil Birnbaum said...

Thanks, Jim, I've printed out the paper and will take a look at it.

BTW, is it possible that some kicks are more difficult because of the defense? Is it possible that late in the game, they might send more guys in to try to block the kick, or some such? Or is there no defensive tradeoff on FG attempts, so that they're doing their maximum to block every kick regardless?

Don't know if that makes sense, or if such a thing is possible. Maybe Brian Burke knows?

At Monday, July 19, 2010 11:22:00 AM, Anonymous Jim A said...

I suppose it's theoretically possible, but the stats I've seen show no repeatable ability among FG defenses to block kicks (or force misses). It's pretty much all luck.

Plus, there's not much evidence that there are meaningful differences in long-snapping, holding, or protect-blocking skills, at least at the NFL level. Sure, you'll see a random breakdown occasionally, but these players are pretty easily replaceable. FG kicking is almost entirely due to the individual skill of the placekicker (plus distance, weather, altitude, etc.).

At Monday, July 19, 2010 11:30:00 AM, Anonymous Anonymous said...

They don't shoot "one and one" in teh NBA. The first explanation/comment is not valid. Liek the original post though. Validates conventional wisdom. Its hard to put the game away or to tie.

At Thursday, July 22, 2010 1:13:00 AM, Anonymous Anonymous said...

Aren't we just ignoring the obvious reason: that pressure situations actually rattle the nerves? Have we really gotten to the point that we don't think that professional basketball players are human enough to be influenced by the pressures of close-game situations? Even the study itself seems to be built around a hypothesis of "a player is more susceptible to missed free throws in pressure situations", which the statistics validate. I don't think it is necessary to put too much stock in fatigue or 1-and-1 variables; guys just choke.

At Thursday, July 22, 2010 12:57:00 PM, Anonymous Anonymous said...

With FG kicking, there's the occasional added pressure by "icing" (the opposing team calls unnecessary timeouts to try and cause the kicker to lose focus, confidence, etc).

At Wednesday, August 04, 2010 8:19:00 PM, Anonymous Anonymous said...

I do not agree with Guy's scenario. I think the number of foul shots made with the score tied would be 4 not 3.5. A would make 2 for 2 and B would make 2 of 3 for 80% With the score +1 - 2 of 3 shots would be made.
Nevertheless anyone playing sports (golf, BB, bowling etc.)knows it is much harder to perform under pressure.Thanks

At Thursday, September 09, 2010 6:15:00 PM, Blogger Unknown said...

What is the explanation for why people don't choke when the game is tied? Not even the fatigue explanation explains that. The authors argue that it is because the shooter is in an approach-promotion focus because he can secure the win, whereas at -1 and 1 the shooter is in an avoidance-prevention focus because he is trying to prevent losing or prevent leaving the game open to a last second win. Thus, the motivational incentive to make the shot differs based on the score differential. Players may best avoid the pressure if they can view the situation as an opportunity to win rather than as an attempt to avoid failure.


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