# The Three-Point Shot Myth Continued…

I’ve been ranting a lot about the so-called “value” of the three-point shot in “modern-day” basketball. I know! But, here is yet one more entry.

The common consensus is that teams are shooting more three point shots as discussed in the articles below:

There are several more where these have come from. My issue is that on one hand these analyses seem grossly oversimplified. Second, none of the analyses have looked at a per-team trend. From my observations of these articles, they are just looking at total number of three point shots taken/made every year over the past number of seasons.

Indeed, the standard approach is to look at the league averages from the past number of years, and note that the average number of three point shots and attempts has increased (well almost) year-to-year, but this is not entirely useful.

What one should do is look at the probability that any team attempts / makes more than a given number of three point shots per game in a given season. Below, we use a kernel density method to calculate these probabilities.

Just as a reference point, looking at the past 16 seasons of NBA data per team per season (courtesy of Basketball-Reference.com), one generates the following plot:

One sees that the actual number of three point shots made has really not dramatically increased or decreased over the past number of seasons.

But, let’s break this down even further. What one really needs to do is analyze the probability that a team will attempt/make more than a certain number of three point shots per game in a given season. This is highly non-trivial. A first approach is to calculate the mean number and standard deviation of the number of three-point shots attempted and made per season for each of the previous sixteen seasons. These will generate time-dependent functions $\mu(t)$ and $\sigma(t)$.

One can in principle then solve a Fokker-Planck equation to obtain a time-dependent probability distribution $p(x,t)$ for the number of three point shots attempted and another $p(x,t)$ for the number of three-point shots made:

$p(x,t)_t = -\left[\mu(t) p(x,t)\right]_x + \left[\frac{\sigma^2(t)}{2} p(x,t)\right]_{xx}$

(where subscripts indicate partial derivatives). However, as one will quickly discover, this PDE is not separable!

My alternative approach then was to perform a non-parametric analysis using a kernel density method to fit a cumulative distribution function to each season for the past sixteen seasons.  The following set of plots was generated from this method:

One sees from this analysis, specifically, from the density analysis above, in a given season, the probability that a certain team makes more than 10 3-Point shots per game never seems to exceed 10%, so while the probability of a given team attempting more three point shots may have increased, the probability of the same team making more than say 10 3-Point shots per game has essentially stayed the same over the past number of years.

The question then remains do only “good” / “efficient” teams attempt more three point shots, in particular, does this aid in their attempt to make the playoffs or eventually be a championship-calibre team. This question has been analyzed in detail and has resulted in the following paper, which is now on the arXiv.