# 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study, and that is quite forgivable; it was my own — impulsively undertaken, purely curiosity-driven, and very much unpublished.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

Timesarticle 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.

N.B.:

(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).

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