# The curious case of the 3-point line

Clay Thompson set a new NBA play-off record this weekend for most 3-point shots made in a game (that is, 11!) and it was indeed exciting.  The whole Golden State Warriors era is based on the outside threat of Stephen Curry and the likes.  However, is the three-point line diluting the game? Should it maybe be at a different distance? After all the 24 feet (22 at the corners) distance seems to be as arbitrary as a 26 feet distance. Shouldn’t the 3 point range capture a meaningful distance that essentially exhibits a discontinuity at the difficulty (or easiness – whichever you prefer) of making the shot? In other words you should be rewarded with more points only if you take a chance at hitting a shot that has significantly lower probability as compared to a different one just slightly closer to the basket.

I set to examine a few things with regards to the arbitrarity of the 3-point range by using shot data from the last regular season 2014-2015.  In particular, I began by examining the distribution of a shot’s distance from the basket.  The following figure presents the result, where as one can see there is a deep for distances slightly smaller than the 3-point range noted by the dashed line (the first dashed line corresponds to the 22 feet mark, which is the 3-point distance at the corners, while the second dashed line corresponds to the 23.9 feet mark, which is the 3-point distance at the crest).  Given the two different distances that a 3-point shot can exhibit the density of the distance by itself cannot give us a clear view on a possible bias towards taking a 3-point shot as compared to a long 2-point shot.

For that I focused on a 2-feet zone around the 3-point line (both at the corners and at the crest).  At the absence of a spatial bias (i.e., uniform distribution of the shot taken over the space of interest), half of the shot taken within this zone are expected to be taken at the 1-feet zone inside the 3-point range and half of them at the 1-feet zone outside the three-point range.  To be more accurate this is true for the corners of the half court. At the center of the half court given that the surface of the area between two arcs increases proportional to $d^2$, where $d$ is the arc radius, the outer zone should have approximately 10% more shots in the presence of no bias (i.e., 55% of all the shots taken).  To spare you the anxiety for the final outcome of the analysis, the results support a clear bias towards shooting outside the 3-point range!! Let’s now see how we reached this conclusion.  First let us visualize in the following the chart of the shots taken during the season (a random sample is depicted for visual clarity). As you can see there is some “white space” at the chart just inside the 3-point line, which can visually suggest a bias…But let’s see the numbers. Corner threes: For this case we measure the number of corner threes taken just outside the line (i.e., 22 feet), with the shots taken just inside the line, which are essentially 2-point FG attempts. From the 3028 shots taken within this area, a whooping 85% (95% confidence interval is [83.8%,86.3%]) of them corresponds to 3-point attempts! This percentage is far beyond the 50% expected in the case of a uniform distribution of the shots.

Center threes: For this case we measure the shots taken from the center of the half court just behind the 24 feet line as well as those just inside the line. There are 10,411 shots in total taken in the zone of interest.  From these 9,372, or equivalent 90% of the shots, were threes!

Compare the above results with the following control case where we focused on a 2-feet zone at distances 16 and 17 feet from the basket (i.e., the whole zone is within a 2-pt FG range).  There were in total 3,774 shots taken in the zone with 1,965, or 52%, of them taken at the 16-feet distance zone – a very slight bias towards closer shots (even if this bias was meaningful is exactly the opposite as to what we see in the 3-point case where the bias is towards the further shots!).

It becomes evident from these results that there is a clear bias for 3-point FGs when a player is near the 3-point line.  This is certainly a rational behavior, since the goal of the game is to maximize the points scored.  Or is it? In order to be rational the probability of making the 3-pt FG must be greater than 66% of the probability of making the 2-pt FG just inside the 3-point line.  Therefore, I calculated the FG percentages in the various distances around the 3-point arc line.

As we can see not only the 3PT FG percentage is higher than 66% of that for the 2PT FG at similar distances, but there is also not any statistically significant difference at the percentages at all! Hence, it is completely rational for players to take a step back and shot a 3 pointer! Is this good for the game though? What is the purpose of the 3-point line anyway?

My take is that the 3-point line should mark a location of discontinuity in the ability to score the basket.  Just inside the line you have a FG percentage of X%, while just behind the line you have a FG percentage of approximately 0.66X%. Is there such a distance/location though?  In the following figure I computed the FG percentage for all the distances.

The transition is smooth for the most part.  If you focus on the points of the plot, which are the raw data, there is some small discontinuity at the 24 feet distance, but as we saw it is not statistically significant (of course there is a large discontinuity at the 1-feet distance but I guess we all understand why I ignored it…).  A larger discontinuity appears at the transition from the 29 to 30 feet line.  At the 29 feet line the FG percentage is 28% while at the 30 feet line the percentage goes down to approximately 70% of that, namely, 20%.  The difference is certainly not statistically significant since there is a very small sample size for these distances (189 shots at distance 29 feet and 74 at 30 feet distance), and for this exact reason their is an increase in the FG percentage at 31 feet!  Nevertheless, it seems that moving the line back 6 feet might create some level of discontinuity.  Of course, moving the line back might require increasing the width of the court but this will also help the spacing between players.

One way to explore this is for the league to collect more data (e.g., through team practises, summer camps, D-league etc.).  However, since humans respond to incentives it can be the case that players might deliberately reduce their FG percentage.  Who knows? Maybe the way to create a discontinuity – even at the 24 feet distance – is to have the floor vibrate outside the 3 point line, or add some other type of distraction.  Ok now jokes aside, the line also does not have to be circular/symmetrical.  If there is a weird curve that provides the required discontinuity, why not? Shots behind the basket can potentially exhibit such a discontinuity. All in all, it might be time that the league rethinks the 3-point shot line.