A recent article described how the volume of three-point shots taken in the Basketball Champions League (BCL) has shown an increasing trend – in alignment with the trend in the NBA that has been observed for several years now and has coincided with the ascent of the analytics movement in the sport. The analytical explanation of this trend is that on average, the field goal percentages for a three point shot and a two point shot are such that the expected points (i.e., %pcg * points_awarded) from a three point shot is larger than that from a two point shot in most of the court zones. Of course, in the restricted area the fielg goal percentage is very high and therefore, this is the most efficient shot. Nevertheless, the three-point shots are the next most efficient shots (and among them, corner threes are the most efficient – and the shorter distance is not the reason).

Using data for all the shots taken from the 32 teams in BCL we calculate the expected point per shot for each team and for each are of the court. The court is divided in 13 areas as shown in the figure below.

Given the field goal percentage of a team from each area of the court, we can easily compute the points per shot that the team is expected to add whenever taking a shot from that area. For example, if the field goal percentage of our team from the paint is 55%, the expected points from a shot taken from the pain are 0.55*2 = 1.1. The following figure shows the expected points per shot from each area on the court when considering all the shots taken from all the teams in BCL.

Similar to the NBA, the most efficient shots are the ones taken from the restricted area (under the basket), followed by all the three point shots. The most inneficient shots are the ones from midrange. Now, among the three point shots, the most efficient ones are the corner threes! An idea that has been thrown around for the higher efficiency of corner threes has been the shorter distance to the hoop. In particular, in the NBA, the corner threes are at a 22 feet (6.7m), while above the break the distance is 23.75 feet (7.24m). This is a fairly large difference. In contrast, the three point line in FIBA competitions varies much less, from 22.15 feet (6.24m) above the break to 21.65 feet (6.60m) at the corners. If distance was a big factor for the increased efficiency of the corner threes in the NBA, this difference in the efficiency should not be as much pronounced in FIBA competitions. This setting is exactly the setting of a **natural experiment**. The fact that in FIBA competitions the corner threes are still more efficient as compared to threes above the break, allow us to reject the hypothesis that the shorter distance at the corners is responsible for the majority of the efficiency difference. On the other hand, in both the NBA and FIBA contests, the fraction of corner threes that are assisted is much higher compared to the threes above the break. The actual fractions differ significantly between the NBA and FIBA competitions , but this is mainly an artifact of how assists are counted in the different competitions. The following table presents the fraction of shots from each three-point area in BCL.

Area |
Fraction of assisted shots |

Top of the key | 20% |

Left wing | 27% |

Right wing | 27% |

Left corner | 39% |

Right corner | 35% |

Overeall, above the break threes are assisted at a rate 25.4%, while corner threes are assisted at a rate 37.3%. A two-proportion z-test further allows us to reject a null hypothesis that the two rates are equal (p-value < 0.001). Given that assisted shots tend to be of higher quality (e.g., they tend to be open more frequent that not), the fact that corner threes are more efficient than above the break threes is not surprising. Of course, the question is why corner threes are more assisted, but this is a topic deserving its own in depth study and analysis, which we are currently performing.

Using the notion of expected points per shot we can have a quantitative way to evaluate the efficiency of a team. An efficient team will make shot choices that lead to larger expected points per shot. The above figure presents the league average expecte points per shot, which means that a team with similar efficiency as the league average, should attempt to take more shots under the basket and corner threes. However, this is not always necessarily true for all teams. Teams might not have the right personel for creating corner three opportunities. Similarly it seems inadvisible for a team to take may midrange shots, but there are several players (many of whom are future hall-of-famers – Dirk Nowitzki, Chris Paul etc.) who have made a career based on their efficiency from the midrange. Therefore, in order to estimate the expected points per shot for each team, we cannot use the league average numbers for the expected points per shot, but rather quantify these variables for each team individually. In particular, with f(z,t) being the FG% from area z for team t, n(z,t) being the number of shots team t has taken from area z and p(z) being the number of points awarded from zone z, the expected points xPTS[t] per shot for team t are:

The following figure presents the z-score for the expected points per shot for all 32 teams. As we can see the top-3 teams in terms of offensive efficiency (as captured by expected points per shot) are all from group A (having the first 3 spots in the group) !! Pinar Karsiyaka, EWE Baskets Oldenburgh and AS Monaco!

However offense is only half of the game. Defense can impact the efficiency of a team by either reducing the FG% from the different court zones, or by forcing the offense to take shots from court zones with low FG%. The following figure exhibits the z-score of the defensive expected points per shot for each team, that is the expected points of the shots allowed from the defense. Therefore the lower the better, i.e., the team allows less points per shot than an average team.

As we can see AS Monaco and Tenerife have the most efficient defense until now in the competition allowing more than two standard deviations less points compared to an average team. On the contrary EWE Basketbs Oldenburg allows more than 2 standard deviations more points compared to an average team.

Note that the above are raw numbers, i.e., they do not adjust for who a team faces. For example, scoring 1 point per shot against EWE Baskets Oldenburg is not as good as scoring 1 point per shot against AS Monaco. In order to adjust the xPTS (both offensive and defensive) for each team we solve the following optimization problem:

where **x **is a vector with the xPTS ratings for every team (offensive and defensive respectively), h is the home edge with regards to xPTS and m is the league average xPTS. Every game i will essentially provide us with two data points for the above optimization objective. Solving the above optimization problem we obtain a home edge h= 0.014 points/shot, and a league average xPTS of m = 1.038. The following table provides the results.

Note that for the defensive ratings a negative value is better (i.e., the team allows less points per shot than average). One could also get ratings for specific court zones. However, given the small number of games to begin with and the even sparser data with regards to shots in specific areas for specific games, we might not be able to provide a robust solution to the above optimization.

You can explore the (currently only offensive) efficiency of the BCL teams in the interactive app here.

Acknowledgments: I would like to thank Basketball Champions League for providing me with access to the data.

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[…] in particular corner 3s – are shoots taken after a pass more than 90% of the time! In fact, this is what it seems to mainly drive their higher efficiency, rather than the closer distance to th…. We also calculated the average distance of the closest defender for assisted and unassisted shots […]

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[…] in what resembles a natural experiment, I performed the same analysis on an International Basketball Federation competition, where the distance to the basket is almost […]

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