Sunday, March 18, 2012

Why does free throw percentage fluctuate so much?

Typically, we assume that a player's talent doesn't vary much from year to year, except perhaps because of injury and aging. When an established hitter drops, say, from .320 to .280, while in the prime of his career, we usually assume most of it is luck. It might be just randomness of outcome, perhaps in combination with random circumstances, like differences the opposition he faced. On occasion, we might consider the possibility that the player changed his approach or technique.

If you were to choose a skill that doesn't depend much on circumstances or technique, it might be NBA free throws. Every shot is from exactly the same place, and it's something the player has been doing since childhood. You wouldn't expect a player's foul shooting talent to jump around a lot.

But ... it seems that it does.

I took the 50 NBA shooters with the most attempts in 2004-2005, and checked how they did the next season, 2005-2006. I expected they'd be roughly the same. Specifically, if I expected that if I converted each difference to a Z-score, they'd form a bell curve with mean 0 and SD about 1. I say "about" 1 rather than exactly 1, because there are probably slight changes due to injuries, aging, etc. There's also a selective sampling issue, since I chose the players with the most attempts. (E-mail me for the spreadsheet -- I had to do it manually.)

But, the SD of the difference was much more than 1.00. It was 1.87.

The biggest changes were almost all declines. Desmond Mason went from .802 to .682 (4.8 SD). Tyson Chandler dropped from .673 to .503 (4.7 SD). Drew Gooden, Stephon Marbury, Jalen Rose, Mehmet Okur, and Pau Gasol also had big declines, 2.5 SD or more.

Only Yao Ming (.783 to .853, 2.7 SD) and Mike Bibby (2.5 SD) had improvements above 2.5.

Looking at these players' careers, you see that there's definitely some luck involved: many of them went from "too high" relative to the rest of their careers, to "too low". A couple of them changed (roughly) permanently -- Yao, for instance.

The scale for free throws isn't that much different than the scale for batting averages. For Mason, imagine a player who hits .302 in 420 AB one season, and then hits .182 in 524 AB the next. That can happen, but it doesn't happen that often (even taking into account that MLB players are unlikely to get 524 AB while hitting .182). Furthermore, whatever reasons you can think of that it would happen in MLB -- injury, conditioning, age -- you'd think wouldn't really apply to foul shooting.

None of the changes are that extreme, taken alone: the issue is that there are *too many of them* for it just to be binomial randomness. A Z-score of 3.0 or more should happen only one in 300 times, by chance. In this sample, it happened 4 times out of 50.

Let me make one more adjustment, for a "season effect". It turns out the overall league average FT% dropped by .011 between those two seasons. I don't know if that was cause or effect -- but, in any case, if I adjust every player by that .011, the scale is now balanced between improvers and decliners. But the overall SD stays the same (actually, it increases slightly, from 1.87 to 1.88). Also, the three biggest changes are still all decliners, and those three (Mason, Chandler, and Gooden) are at 4.4, 4.4, and 3.7 SD, respectively.

So, what's going on? Do players' free throw talents fluctuate that much? If so, what does that say about more complicated talents?


P.S. This post was inspired by a discussion about correlation at Sport Skeptic, here.

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At Tuesday, March 20, 2012 7:26:00 AM, Anonymous aweb said...

Free throws are the rare NBA activity where players can stop and think about what they are doing. It's very possible to get the free equivalent of the the "yips" for an extended period, where you just can't do it right anymore. Throw in minor injuries and what I assume are a lot of jammed fingers and hand/wrist problems in the course of a year too.

Free throws aren't really comparable to pretty much anything else a typical NBA player gets to do. They are more like 5 foot putts for golfers, or the serve for tennis players. Except if you suddenly find yourself unable to do those activities at a pro level, you aren't a pro anymore in short order - NBA players are only marginally selected for free throw ability.

I can't imagine a reasonable argument for adjusting individuals to league-wide changes. There is no "free throw environment" like baseball has with umpires, ball liveliness, weather, etc. They are taking the same shot you do in the local gym, with the same size ball. I could see arena effects I guess, if new places opened that were harder to shoot in (I seem to recall some arenas are just naturally harder to shoot in, although I would expect this to have the least effect on free throws).

At Tuesday, March 20, 2012 2:55:00 PM, Blogger Don Coffin said...

Isn't the real expectation that the year-to-year correlation of FT% will be higher than the year-to-year correlation of FG%? FWIW, using the same two years you did, I found 120 players with 500+ FGA in both 2005/2005 and 2005/2006. The correlation of FG% was actually much lower than I expected, only 0.219. I didnn't do exactly the same thing you did, obviously, but I'll bet the correlation of FT% is higher than .219.

At Tuesday, March 20, 2012 3:17:00 PM, Blogger Phil Birnbaum said...

Hi, Doc,

Yup, the correlation for FT% is much higher than .219, but that's probably just because there's so much more variation in individual FT% talent. You got guys at 80%, and guys at 50% ... but for field goals, everyone's much closer than that.

At Tuesday, March 20, 2012 3:51:00 PM, Anonymous Mike said...

And because variance in FG% corrects itself over the course of the season.

If I'm a 50% FG guy and I'm shooting 55% this year, I'm going to take more shots going forward. If I'm a 50% FG guy and I'm shooting 45% this year, my coach tells me to pass more and look for higher quality chances for myself.

At Tuesday, March 20, 2012 5:05:00 PM, Blogger Don Coffin said...

Actually, the range in FG% isn't much narrower than you're citing for FT%. The range is from about 38% to about 60%, althought the variance is fairly small. In both years, the mean is about 45% with a standard deviation of about 4.5%.

At Tuesday, March 20, 2012 5:11:00 PM, Blogger Don Coffin said...

FWIW, the year-to-year correlations are:

FGA: 68%
FGM: 67%
FG%: 22%

Make of that what you will...

At Tuesday, March 20, 2012 5:14:00 PM, Blogger Phil Birnbaum said...

Looking at 2006-07 ... the top 10 in FGA, leaving off the best and worst (I'm too lazy to calculate SD), range from .431 to .476 in FG%

For FTA, the range is .615 to .844.

I bet if I wasn't too lazy, the SD difference would be similar. :)

Hang on, I have at least one number. For the top 50 in FTA, the SD of FT% was around .0875. I can calculate FG% later when I have more time ...

At Wednesday, March 21, 2012 5:19:00 PM, Blogger Don Coffin said...

I played around a little more with the data, and, FWIW, here's what I found.

I looked at the 2004/2005 and 2005/2006 FT data as well, using all players with 200+ FTA in both years (there were 87 of them).

Mean FT% 2004/2005: 77.3%
Mean FT% 2005/2006: 75.7%

(I'd call that no significant difference).

StdDev FT% 2004/2005: 9.0%
StdDev FT% 2005/2006: 9.7%

Coefficient of Variation of FT%
2004/2005: 11.6%
2005/2006: 12.8%

Coefficient of Variation of FG%
2004/2005: 8.7%
2005/2006: 9.3%

So there is in fact considerably more variation in FT% than in FG%.

Correlation, 2004/2005 with 2005./2006
FT%: 0.875
FG%: 0.222

And there's considerable more consistency, year-to-year in FG%.

The year-to-year correlations for FGA and FGM are both abopu 0.67. The year-to-year correlations between FTA and FTM are 0.76 and 0.72. So there's considerable consistency, year-to-year in attempts.

There is some sample selection bias here, because, by including only players with large numbers of attempts in both years, I have probably eliminated the least consistent (and least talented) shooters.

If anyone wants my files to play with let me know and we'll figure out how to get them to you.

At Wednesday, March 21, 2012 5:21:00 PM, Blogger Don Coffin said...

"And there's considerable more consistency, year-to-year in FG%."

That should be, obviously, "And there's considerable more consistency, year-to-year in FT%."

At Wednesday, March 21, 2012 6:19:00 PM, Blogger Phil Birnbaum said...


That was a lot of work ... thanks!

What I'd be interested in is the SD of difference in talent from year to year, in FG%.

To calculate that, we'd need the SD of the difference in FG% between years (that is, the SD of each player's (FG%1 - FG%2)), and the average FGA.

Or, even better: can you compute the Z-score of the difference in FG% for each player? That is, assuming it's random binomial.

So if player X was 40% in 500 attempts year 1, but 38% in 400 attempts year 2, his binomial SD is .0327, so his Z-score is .02/.0327 = 0.61.

If we know the SD of all those Z-scores, we can figure out how talent fluctuates year to year, and compare that to FT%.

Um ... sorry I'm asking for more, when you've done all that work. :)

At Wednesday, March 21, 2012 7:43:00 PM, Blogger Don Coffin said...

Actually, I'm pretty sure that even a binomial distribution won't fit this stuff all that well. It's clearly non-normal (no one can be as far above the mean FT% as the worst FT shooters are below it), generally because the distribution is bounded. (Less of a problem for FG%, though.) If I had to guess, I'd say it's a poisson...

I can send you my spreadsheets and you can do the work (grin).

At Wednesday, March 21, 2012 11:22:00 PM, Blogger Phil Birnbaum said...

Sure, I'm happy to do the work! I hate typing in data, but I don't mind calculations.

My e-mail is my last name at sympatico dot ca.

Thanks, Doc!

At Friday, March 23, 2012 3:58:00 PM, Blogger Don Coffin said...

I'll send the files on Monday; they're on my office machine.

At Sunday, March 25, 2012 7:05:00 PM, Anonymous public liability insurance cost said...

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At Tuesday, March 27, 2012 6:36:00 PM, Anonymous bsball said...

I'm intrigued by this question and the results you are showing. I downloaded some data from for the two years you are reporting on and I get something in the ballpark of what you are showing.

My expectation was that in general basketball players get better at FTs. It's an old player skill, like walks in baseball. So I was surprised to see more extreme declines than improvements.

I dug into the data some and came up with the following:

1. Is it possible that there is some bias in the study by picking the top 50 players (in FTA) in year 1? If you look at FTA in year 2 vs year 1 the group declines by about 50 FTA per player.

2. I noticed that NYK has 4 players on the list and all 4 declined - and I have 3 as >2.5 Z score. In 2005-06 the Knicks went 23-59 (.280). Could it be that the players stopped trying?

At Tuesday, March 27, 2012 6:59:00 PM, Blogger Phil Birnbaum said...


1. The decline is because I deliberately chose the 50 players with the most FTA the first year. You'd expect the top players to decline next year. But you wouldn't necessarily expect their FT% to drop.

2. Interesting theory ... I'm not sure how you'd find evidence for it either way, though ...

At Wednesday, March 28, 2012 4:13:00 AM, Anonymous bsball said...

1. You might expect some decline from aging and some decline from injury. I wouldn't expect the aging to affect FT%, but injury might.

I repeated the study taking 2005-06 as the base year. Take the top 50 FTA from that year and compare to 2006-07. That shows 2 players with >2.5 Z scores and 2 players with < -2.5. The distribution is more balanced. SD of difference is at 1.35.

At Wednesday, March 28, 2012 9:22:00 AM, Blogger Phil Birnbaum said...


Wow, thats a big difference. Maybe the year I chose was an aberration.

Thanks for doing all that work. Be nice to have a DB to do a much larger sample, like we can for baseball ...

At Wednesday, March 28, 2012 9:53:00 AM, Anonymous bsball said...

Or maybe the year I chose was an aberration. I agree it would be nice to have a DB for larger samples.

And thanks for sharing all your work. I relly like this blog (and the Book Blog which gave me the link here). Lots of good puzzles, logical discussion and interesting comments.

At Sunday, June 17, 2012 6:34:00 AM, Anonymous wiLQ said...

"Why does free throw percentage fluctuate so much?"
Here's my theory: distribution of shots.

IMHO had players attempted 100 or 200 free throws in one session their efficiency would be pretty consistent but in the NBA games they can have 50 sessions with 2 free throw attempts or 10 sessions with 10 attempts or many other possibilities...
In each case those sessions are on different days, in different game situations/times, even in different altitudes/time zones!
So conditions for those shots change constantly which badly skews the results... not to mention that players simply can shoot many free throws on a bad day and zero on a good day or vice versa.

Do you buy that explanation?


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