Monday, March 26, 2012

Which NBA talent is less stable: free throws or field goals?

Last post, I wondered about why NBA players' free-throw percentage seems to fluctuate so much. If you start with the hypothesis that changes are all just luck, you'd expect only one player in 50 to be more than 2.5 SD from zero. But, from 2004-05 to 2005-06, there were 10 such players. So, clearly, it can't be just luck. Talent must be changing too.

Since then, commenter "doc" was kind enough to e-mail me similar numbers for field goal attempts. My hypothesis was that the talent change for FG should be bigger than the talent change for FT, because FG requires a bigger set of skills.

The results surprised me.

Let me take you through the math of how I did this, just so you can make sure I got it right. If you don't care, skip to the bold parts and the last section.


Start, for instance, with Allen Iverson. In 2004-05 2005-06, Iverson shot .447 in 1,822 FG attempts. The binomial SD of that is the square root of (.447 * (1-.447) / 1822), which works out to .01165.

In 2005-06 2004-05, Iverson was .424 in 1,818 attempts. That's an SD of .01159.

The SD of the *difference* between the two seasons is the square root of (.01165 squared + .01159 squared), which works out to .0164.

Iverson's actual difference between the two seasons was .023. Divide .023 by .0164, and you get 1.400. So, Iverson's Z-score for the difference is 1.4.

Repeating this for all players in Doc's sample, we wind up with 120 separate Z-scores. If the differences were all luck, we'd expect that if we took the standard deviation of those 120 numbers, we'd get exactly 1.00. Instead, we get an SD of 2.18.


So, FG shooters change year-to-year by an SD of 2.18 "luck units". Since we know 1.00 of those units are actually luck, that leaves 1.94 units of talent change (since 2.18 squared minus 1.00 squared equals 1.94 squared).

What does 1.94 units mean in real life? Well, the typical player in Doc's sample went around 45% in 930 attempts. So, the SD from luck would have been .0231 (or 2.31 percentage points). Multiply that by 1.94, and you get .045. So:

-- An NBA player's FG% talent changes from year-to-year with an SD of 4.5 percentage points.


Now, let's do the same for FT%, so we can compare.

FT shooters change year-to-year by an SD of 1.87 "luck units". That leaves 1.59 units of talent change.

The typical foul shooter went 77% in 471 attempts. So, the SD from luck would have been .0274. Multiply that by 1.59, and you get .043. So:

-- An NBA player's FT% talent changes from year-to-year with an SD of 4.3 percentage points.


They're almost exactly the same!

Part of the reason I wouldn't have expected that is that for FG attempts, what we're calling "talent" isn't really just talent. It's "everything except binomial luck." So, it also includes changes in quality of opposition, quality of teammates, role on the team, ratio of 2-point and 3-point tries, and so on -- actually, quality of shot attempts. Since we're really measuring the sum of two variances -- talent, and shot quality -- we'd expect it to be higher than just for FT attempts.

Another reason I expected FG to be higher is that there might be some selective sampling involved in the FG case: a player who has a really bad year (perhaps by luck) might not play again next year. That would remove a bunch of outliers. But, the average player in the sample actually declined the second year, by 0.7 percentage points, so it doesn't look like that's it.

On the other hand, part of the reason could be that FG percentages are lower than FT percentages. For FG, we have 4.5 percentage points out of 45. For FT, we have 4.3 percentage points out of 77. So, looking at it that way, the talent change for FG is 10% either way, but for FT, it's less than 6% either way.

What do you guys think?


P.S. I ran the same numbers for batting average in MLB. I'll save that for a future post.

UPDATE: commenter bsball points out that FG% includes both 2-point and 3-point attempts. So, that's another way shot quality is affected.

Indeed, this might be a big effect. Iverson took 338 three-point tries in 2004-05, but only 223 of them in 2005-06.

I've updated the post. Also, I corrected where I had inadvertently reversed the two seasons.

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At Monday, March 26, 2012 10:50:00 AM, Blogger Brian Burke said...

Regarding FG%, you might need to break out players by position. I noticed a similar phenomenon with regard to NFL receivers.

I noticed that receivers tended to have stable 'Yards After Catch' totals from year to year, much more stable than you'd expect.

But I realized that part of the reason was that all types of receivers were combined into one pool, when in reality there WRs, TEs, and RBs, each of which catch passes. Each position-type tends to have the same route profile, and consequently get relatively stable receiving numbers when compared to receivers among all types. A WR is a WR every year, so compared to TEs and RBs, his statistical profile appears relatively consistent.

But within-group consistency was much less stable, at least in relative terms.

For the NBA, you might want to repeat your analysis, but just look at a single position at a time. I might be out to lunch here--just a suggestion.

At Monday, March 26, 2012 10:54:00 AM, Blogger Phil Birnbaum said...

Hi, Brian,

I don't think it matters ... because we're comparing a player to himself, just one year compared to the last. So, position X can go from .600 to .650, and position Y can go from .400 to .450, and the analysis will just figure out each one separately and combine them.

But, you raise a good point, that if players switch position, we might take that as a large change in talent, when it's not. That would cause the estimate to be too high.

This is all just 2004-05 and 2005-06. Maybe someone knows if any top players changed positions between those years?

At Monday, March 26, 2012 11:53:00 AM, Anonymous bsball said...

FG% includes both 2 point shots and 3 point shots. In the 2 years you compared his shot distribution changed. If you look at 2 point shots only his percentages are:

.451 in 2004-05
.465 in 2005-06

3 point percentages were:

.308 in 2004-05
.323 in 2005-06

Each of those subsets shows less variation than his total FG%.

Also, I think your 2004-05 numbers in the example are really 2005-06, and vice versa.

At Monday, March 26, 2012 11:56:00 AM, Blogger Phil Birnbaum said...

Ah, it's FG% adjusted for 3-pointers! Didn't realize that. See how little I know about basketball? I'll update the post. Thanks!

At Monday, March 26, 2012 12:02:00 PM, Blogger Phil Birnbaum said...

Sorry, I mean it's including 3-pointers, not adjusted for 3-pointers.

At Monday, March 26, 2012 12:57:00 PM, Blogger Micah said...

Are you seeing changes in "talent" or are you seeing characteristics of the distribution? Maybe field goal and free throw percentage aren't the same standard distribution we're used to fitting data into; maybe they're flatter (and fatter-tailed) distributions?

At Tuesday, March 27, 2012 9:02:00 AM, Blogger parinella said...

There is also the usage vs efficiency as shown in "Basketball on Paper". As players take more shots per minute, they make a lower rate.

At Tuesday, March 27, 2012 11:06:00 PM, Anonymous Anonymous said...

You are surprised that it is not the change in skill component is not much higher than it is (and higher than for FT%)?

As I said in response to your other article, I would have expected that the talent change in FT% to be much greater than that for FG% because FT% requires a certain amount of practice (and that level of practice probably changes from year to year), AND I think that FT% is much more sensitive to physical and mental changes.

Golf is a very good analogy. Putting is like free throw shooting and FG% is like ball striking. PGA player gain and their lose their putting skills all the time for similar reasons. That typically don't change their ball striking skills that much, at least as far as we can tell.


At Tuesday, March 27, 2012 11:18:00 PM, Blogger Phil Birnbaum said...


I understand what you're saying about practice but ... it's the same task, putting the ball through the hoop!

You'd think trying to put the ball in the hoop while moving, in a random direction, with a defender hovering over you, would be a harder skill to maintain than doing the same thing at a standstill from exactly the same spot.

Look at it this way: suppose that instead of having to shoot from that spot, the foul shooter was allowed to run around a bit, but had to shoot within (say) 1 second of a random whistle blowing. Do you think that skill would be less variable than regular foul shooting?

At Wednesday, March 28, 2012 4:26:00 AM, Anonymous Anonymous said...

Phil, you are ignoring my other reason(s) for FT shooting having more skill variability from year to year. Sensitive to mental and physical changes like pitching and putting in golf.

And yes, I expect that FT shooting is much more sensitive to PRACTICE than FG shooting. You seem to be ignoring that point and just talking about the variability in skill.

Look at it this way:

The world record for FT made in a row is over 5,000. So you can teach yourself how to be an incredible free throw shooter if you put enough time into it. Basically an NBA player could teach himself to be a 95% FT shooter if he wanted to. Could anyone, including an NBA player teach themselves to be a 70% FG shooter in the heat of a game if they practiced enough? No.

So, if the range of FG shooting from little practice to an incredible amount of practice, for an NBA player is something like 40% to 55%, and the range of FT shooting for an NBA player with little or no practice to an incredible amount of practice is 55% to 95%, which do you think would be more sensitive to practice and which do you think would fluctuate more from year to year, IF NBA players varied their practice a lot?

Keep in mind, I am just speculating. And the bottom line is that you found that FT shooting has more variability in true talent from year to year than you expected, and FG shooting less than you expected, right? Unless I got that wrong.

So, why are you arguing against my speculation without providing an explanation of your own, unless, again, I am misreading what you are writing. I did not understand this article very well anyway...


At Wednesday, March 28, 2012 8:29:00 AM, Blogger Phil Birnbaum said...


OK, I see what you're saying. Maybe, in the heat of the game, players can hit 40% of FG from free-throw distance. Maybe, without pressure from the other team, they could hit 50%.

But they hit 75% from the free-throw line. The other 25% is practice.

OK, I see what you're saying now. Makes sense.

At Wednesday, March 28, 2012 4:41:00 PM, Anonymous Patrick (SnarkSD) said...

Interesting article, a methods question if you don't mind? I get lost here...

The SD of the *difference* between the two seasons is the square root of (.01165 squared + .01159 squared), which works out to .0164.

Iverson's actual difference between the two seasons was .023. Divide .023 by .0164, and you get 1.400. So, Iverson's Z-score for the difference is 1.4.

Can you explain what you mean by *difference*, and how did you get Iverson's 0.023? Is that just his binomal variation added together? If that's the case, isn't his z-score more attempt dependent, than actual skill?

At Wednesday, March 28, 2012 10:41:00 PM, Blogger Phil Birnbaum said...

Hi, Patrick,

Iverson was .424 the first year, and .447 the second year. The difference is .023.

His Z-score is how many standard deviations from zero his difference is, on the assumption that it's all binomial.

Yes, the Z-score is a combination of luck and skill. We start by assuming that it's all luck. But, if that were the case, then if we put Iverson's 1.4 in a list with everyone else, the SD of all those numbers should be 1.0. If it's not, that's how we know there are some skill changes involved.

At Thursday, March 29, 2012 1:31:00 AM, Anonymous Patrick said...

Thanks Phil. Just one more question. Where did you get the square root of (.01165 squared + .01159 squared) formula you used?

At Thursday, March 29, 2012 8:34:00 AM, Blogger Phil Birnbaum said...

When you have two independent variables, the variance of the difference is the sum of the variances. Therefore, the SD of the difference (which is the square root of the variance) must be the square root of the sum of the variances (each of which is the SD squared).

At Saturday, March 31, 2012 1:53:00 PM, Anonymous Patrick said...

Thanks! Appreciate it.


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