Another Way to Look at Free-Throw Percentage

In a recent blog post, we linked to a New York Times article by John Branch showing that the percentage of made basketball free throws has remained steady for 50 years.

A reader named Ashley Smart (aptonym?) replied with an amplification/caveat that is well worth sharing:

I, like many of your other Freakonomics readers, was intrigued by John Branch’s article on free-throw shooting stagnation. Unlike many of your other readers, I would suppose, I instantly recalled a similar study which yielded a contrasting, if not contradictory result. You are probably not familiar with the other study, and that is quite forgivable; it was my own — impulsively undertaken, purely curiosity-driven, and very much unpublished.

Though relatively informal, my study was quantitative and easily repeatable. I simply found the average free-throw percentage of the top 20 N.B.A. free-throw shooters for each season, as listed in the N.B.A. encyclopedia, and plotted it as a function of time:


The contrast between this figure and the figure from the Times article is stark — the mean free-throw percentage in my study increased appreciably, particularly between the N.B.A.’s inception circa 1950, to the mid 1980’s. An exponential decay model (dashed line in the figure) seems to provide a reasonable fit, suggesting that the top free-throw shooters are asymptotically approaching a performance ceiling (which the model predicts is about 93.5 percent).

This result doesn’t contradict, but rather adds perspective to John Branch’s article: sure, average N.B.A. shooters have stayed nearly the same, but the best shooters have certainly gotten better. It’s not the mean; it’s the variance. There may be several potential explanations: growth of the league (more players, all else equal, increases the relative amount of poor shooters), less emphasis on free-throw shooting at certain positions, etc. Perhaps taking only the top 20 shooters has the effect of muting the statistical noise — i.e., the relatively small increase in the overall mean is real, just overshadowed by noise. No doubt your readers could produce much more convincing, more colorful explanations — I would rather leave that part up to them.


(1) The * in the figure title is because there were four seasons (’46, ’47, ’48, and ’00) for which I could only find data for the top 10 free-throw shooters.

(2) I fit the data with the model y-y_o = y_g*[1-exp(-x’/tau)]. y_o is then a baseline percentage, y_g is the gain from that percentage, so that y_max = y_o+y_g is the performance ceiling. x’ = x – x_o, where x_o is the initial time for exponential decay. Tau is then the characteristic growth time. I got (y_o = 48%, y_max = 93.5%, x_o = 1898, and tau = 47 years).


Top 20 by percentage or by volume of shots?


This suggests that the bottom tier of free-throw shooters has gotten worse over time. Not being a basketball fan, I can only assume other skills are being emphasized for certain positions. Do the top free-throw shooters represent a particular position (or two)?


Looking at the graph, you would think they would be shooting better than 0.9% :)


I said this earlier but it seems like their is a huge selection issue over time. If we assume that there is more shooting variance over time (and we assume that advance scouting has gotten better) there is a greater cost to fouling the best players. I think the fact that the "Hack a Shaq" strategy has been in vogue helps illustrate.


i wonder if (relative) # of fouls called has been constant- it may be the case that if fouls are increasing, there may be an increase in the 'hack-a-(your gangly big man here)' strategy here, where not only skill of shooter but allowable dilution of free throw percentage comes into play- come to think of it, who invented the hack-a-shaq? did it predate shaq foo?

Math Man

A brief review of 2007-08 statistics reveals a probable explanation for the difference between these two analyses. Generally speaking, players that have a higher free throw percentage attempt fewer free throws, and players with lower free throw percentages attempt more free throws.

This makes sense. Late in a close game, the losing team will foul the opposing team players, to stop the clock and have them shoot free throws, instead of letting the clock run and risking an even higher score. The strategy only works when the winning team misses the free throws. So, the losing team will foul the worst shooter in the game.

So, the analysis of the top 20 free throw percentages does show that free throws are being made more often by the best free throw shooters. The league average is pulled down by a corresponding drop in free throw percentage by the worst shooters.


Hey I agree with Chappy. Terrible free throw shooters will get fouled more, and good free throw shooters will be fouled less. But that has probably not changed over time.

I think it is very interesting that the best players in the NBA have been getting better. I do not think that the self selection factor explains it all.


I wonder if part of the explanation for these contradictory conclusions is that NBA players are becoming more specialized over time, similar to pitchers in baseball. If some players specialized in shooting (free-throw and otherwise) at the expense of other skills and some players specialized in non-shooting skills (thus hurting their FT%), then we might expect little change in the league-wide average, but a wider distribution. The wider distribution would explain Amy's findings.


As Ashley states, this could just be the increase in size of NBA. Need to look at average of top X% of freethrow % in order to test this.

MK Foster

Both studies seem to suggest that there is a lack of variables other than the shooter. I think basketball's have gotten better, ie more consistent, along with hoops and lighting.

It may be a small influence, but it does exist. Especially if you use data from the fifties.


Like in all aspects of modern sports, the athletes train more often, harder, and more scientifically. They are better than their generational predecessors.
Those who chose to concentrate on free throw shooting are better than the free throw shooters of the past.
The reason most of the league doesn't concentrate on this is that the difference in ppg between a 75% and 85% free throw team is not that great compared to the practice time it takes to get there. That's what Jamie Dixon's excuse for letting Pitt blow big games on free throws every year, anyway.


I'm going to go out on a limb and say that Ashley is both smart and a bit sassy.


This makes sense from a comparative advantage point of view. Do something very well relative to your peers and increase the odds of entering the NBA. I'd expect something similar for Punters in the NFL though I've never looked. Market need established.... here come specialists. Might find something similar with various skills 3pt shooters, blocked shots, etc.


I agree with the original article. I think coaching strategy has played a huge part. While the best free throw shooters have improved, the worst free throw players probably take an inordinate amount of foul shots. The "hack-a-Shaq" as chappy mentioned. Shaq (one of the worst free throw shooters) has MISSED more free throws then Steve Nash (one of the best free throw shooters) has EVER taken.


There's an additional statistical factor that I haven't seen anyone talk about: the growth of the league means that the top 20 players represents a much smaller percentage of the league today than it did 50 years ago. So even if every player has the same "expected" free throw percentage, the top 20 will average a much higher percentage.

As an analogy, if you flip 100 balanced coins 100 times each, the "top 10" coins will average something like 57 heads. But if you flip 1000 coins, the "top 10" will average something like 63 heads. This doesn't mean that the coins have gotten better at coming up heads, it just means that you're moving your cut-off point an extra sigma or two away from the mean.

I'd like to see the Ashley's analysis done on percentage terms, rather than absolute numbers, and see if the pattern still holds. (I suspect it will persist, but less pronounced. But that's just my personal hunch.)



This seems like bad statistical analysis. How many additional pros are there in 2008 versus 1940? With more teams, you have more players and hence a higher chance of getting a person with a good free throw percentage. I bet if we took the top 20 players in the NBA, College, High School, Europe, WNBA, and so on, the mean would be even higher.


It looks to me like there are two approximately linear trends: one for 1971 and before, one for 1972 and thereafter. There's a huge jump between those two years. Was there a rule change or something?

Also, when did granny-style free-throw shooting go out of fashion? I've heard it's actually more accurate, but it looks so stupid no one wants to do it.

Charity Stripe


You say players with a higher FT% shoot fewer FTs, and players with a lower FT% shoot fewer FTs. Not true. Great FT shooters have great FT% whether they take 100 FTs or 400 FTs.

This year, Jose Calderon made 151 of 154 FT attempts, a whopping 98%. The 2007 champ made 130 of 140. 2006 champ made 119 of 130. 2005 champ made 250 of 268. 2005 champ made 394 of 425. There is no correlation between FT% based on quantity of attempts alone.

Karl Malone and Moses Malone are #1 and #2 in career FT attempts, and they shot 74% and 76% respectively in their careers. All-time #3 (Shaq) and #4 (Wilt) shot 51% and 53% respectively. So, no clear correlation between FT quantity and FT%.

One thing is clear. If you play regularly and score a lot of points, you will get a lot of FT attempts. Whether you are good at making the FTs (Reggie Miller) or not (Shaq) is unrelated.

Joe A.

@ -a
This is where statistical analysis gets interesting- so much of it is in the interpretation.
I was thinking the exact opposite thing of what you said (and was mentioned in the original article and by Tony above). More teams means a dilution of average skill due to needing more players. The selection of NBA players isn't random- it's based on skill. If the league was smaller, the worst players would be the first to get cut- the players wouldn't get cut randomly. Now the best players might not necessarily be the best free throw shooters, so that complicates the argument a bit...


Can we put Ashley in a position of high authority in the federal government and get some reality-based process at work for all of us?