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Is There an Optimal Way to Divide Shots?

NBA: Boston Celtics at Philadelphia 76ers Bill Streicher-USA TODAY Sports

So far in the 2018-19 season, the Oklahoma City Thunder lead the NBA in field goals attempted per game at 93.6, and the Memphis Grizzlies are at the bottom, with 83.5 per game. The rather finite number of shots per game sometimes seem to be forgotten when there are claims that Player X needs to “get more involved on offense” (see the Bucks game thread for examples). I realize this is a bit of a strawman, but hey, I need some inspiration for these pieces. Aside from the numerous ways that a player can contribute on offense besides hoisting shots, this lead me to wonder what the general shot distribution looks like across the NBA and for the Sixers in particular.

Shot Equivalents

One of the downsides of relying on field goal attempts is that it does not take into account free throws, so we’ll be using shot attempts instead, which is simply FGA + (FTA/2), with the FTA/2 rounded down. Using data from the 2018-19 season, I calculated the average shot equivalents for each player and game, creating both a rank and proportion of overall, which can be seen in Figure 1.

Figure 1: Shot equivalents and shot equivalent percentage by team rank

For example, we can see that the average player who takes the most shot equivalents in a game (across the entire NBA season so far) averages about 20 raw shot equivalents, and about 22% of all the shot equivalents in a given game - reminder that the rank does not correspond to a single player. The overall distribution generally tracks with the idea that offensively there is a single focal point for most teams for each game. Now, let’s look at the Miami Heat, a fairly egalitarian offense devoid of a true superstar, and the Sixers in comparison to the NBA as a whole.

Figure 2: Shot equivalents and shot equivalent percentage by team rank - Miami Heat, Philadelphia 76ers, and total NBA

Now, these differences are relatively small for the Sixers compared to the NBA writ large, but they do fit the narrative that the Sixers have a top heavy lineup in terms of usage (before you scream at me to just use usage percentage, we’ll get to that in a bit). The Heat are the opposite, with a lower than average number of shot equivalents for the top few players, and higher than average for the bottom few. You can see the same data below in Table 1.

Table 1: Shot equivalent percentage by team rank and team

You’ll note that Mr. Harden is rather likely to be the main culprit in the Houston Rockets having a raw increase of 4.7% greater than the second highest #1 ranked team slot. However, just getting up shots isn’t in and of itself necessarily valuable, as you will know if you have ever played pick up with the person who yells “Kobe!” after 12 successive missed 18 foot fadeaways (this person without exception wears a Jordan t-shirt and Duke/UNC shorts). That brings us to our next section...

Points Per Shot Equivalents

Using the same methodology as before with the shot equivalent rankings and such, we now look at points per shot equivalent for each of the 1-10 ranking positions. Since we used the Heat for the previous example, we’ll continue to use them for Figure 3 as well. Confidence intervals were removed for this example due to some squirrely values manipulating the scale.

Figure 3: Points per shot equivalent and percent of shot equivalents taken by rank

Again, as you might expect, the Sixers are a good team with a highly rated offense - they exceed the points per shot equivalent for the NBA more or less across the board. The incredible average efficiency of the average 8th spot is something that is either some bizarre artifact, or the Sixers need to somehow make their 8th most shooty player the centerpiece of the offense. Table 2 breaks it out by team. In a huge surprise, GSW leads the way with this efficiency stat amongst their top two slots.

Table 2: Points per shot equivalent by shot equivalent taken rank by team

Overall Usage

Traditionally, usage is determined exclusively by the time that player spends on the court. I’m going to here keep the first part the same, but modify it to be the entire course of the game. For example, if Player A has 20% usage when they’re on the court and plays for 50% of the possessions in a game, their overall usage is 10%. This should generally track with shot equivalents, since part of the possession ending events are covered there. We’ll stick with the Sixers, Heat, and NBA overall for Figure 4, and you’ll see the overall info in Table 3.

Figure 4: Overall usage and points per shot equivalent
Table 3: Overall usage by shot equivalent taken rank and team


Now, let’s get into the “so what”. Do we find so far this season that there is a common shot equivalent/overall usage/efficiency profile amongst winning teams vs. non-winning teams? There are a multitude of ways to think about this question, and the way I’m going to get at it is by no means the best or only method. The primary metric I’m going to use is Euclidean distance, as it’s a nice balance of easily understood, easy to use, and fits well for data where magnitude of the numbers actually matters. In Figure 5 below, squares that are red indicate low dissimilarity/high similarity, and blue squares the opposite. The diagonal represents comparisons to self, hence the zero dissimilarity. Teams are also ordered along the y-axis in terms of similarity from the top down.

Figure 5a: Dissimilarity matrix for top ten overall usage averages

The way you can interpret this is that the Houston Rockets are breaking my graphs and I resent them deeply for that. Figure 5b is the exact same information and methodology with the Rockets excised, because they (Harden) are such a monstrous outlier.

Figure 5b: Dissimilarity matrix for top ten overall usage averages (Houston Rockets removed)

That looks a lot better. This is intentionally an eyeballing based technique, but you can see that there are loosely three or four clusters by looking for large blocks of red squares along the diagonal. Orlando to Jazz, Warriors to Thunder, and Pelicans to Timberwolves are the broad strokes groups. There are six playoff teams (as of March 21) in the upper group, two in the middle, and eight in the bottom.

Next, Figure 6 uses the same methodology with points per shot equivalent data for the top ten slots by shot equivalents taken. Houston remains in this one, as the efficiency numbers are not as bonkers as the usage numbers.

Figure 6: Dissimilarity matrix of PPEQ for top ten shot equivalent takers by team

This one appears a little more noisy, with some groups in the Thunder to Wizards, Heat to Bulls, LeBrons to Raptors, and then it gets fuzzy. If you can make sense of those groups in any way, be my guest. However, when we break it out to just the top-5, there is a smidge of clarity. Emphasis on the smidge.

Figure 7: Dissimilarity matrix of PPEQ for top five shot equivalent takers by team

High (Avg. Win % of 59.5): Bucks, Raptors, Sixers, Rockets, Blazers, Celtics, Clippers, Wizards

Middle (Avg. Win % of 48.2): Nuggets, Pacers, Jazz, Nets, Thunder, Timberwolves, Lakers, Hawks, Mavericks, Suns

Low (Avg. Win % of 39.4): Pistons, Heat, Kings, Magic, Hornets, Bulls, Cavs, Knicks

Unknown (Win % 55.9): Warriors, Spurs, Grizzlies

Unfortunately, this doesn’t do much in terms of new knowledge. The high group has consistently higher PPEQ across the board than the mid, which has higher PPEQ across the board compared to the low group. Efficiency = wins is not a novel conclusion since about 2011 or so when coaches and GMs collectively figured out that three is more than two.

In Table 4, I counted up the number of different players who occupied the 1-5 overall usage ranking slots since the trade deadline, and sorted by descending ortg. For example, you can see that Portland has had three different players have the highest overall usage in their 19 games under consideration. The information for the Sixers specifically will be presented at the end of the article due to length.

Table 4: Count of players with specific overall usage rankings by team

If we can assume relatively safely that Joel will be the most often #1 ranked overall usage player for the Sixers on a per game basis, that leaves Ben, Jimmy, Tobias, and JJ to fill the remaining four spots in the top five. I believe the primary advantage that the Sixers have is that you could put those in any order in a given game and there wouldn’t be much of an outcry. In fact, take a look at this table below, where I average the 2-5 spots in overall usage by team, post trade deadline. OBPM and DBPM provided via

Table 5: Average OBPM and DBPM by team for players with overall usage ranks from 2-5

The Sixers are 0.6 raw OBPM points ahead of the 2nd place team (LA Clippers) by this metric, a massive difference.


I didn’t really know where this one would take me when I started, and there certainly isn’t a gigantic takeaway that nicely shows a good/bad way to break up shots. However, I think there is something to be taken from this in conjunction with some of my other pieces. I’m beginning to believe that your best and highest usage players need to be great, and that bench offensive production is overrated. What the Sixers currently have with Joel, Ben, Jimmy, Tobias, and JJ is averaging slightly over 100 points per game. It seems possible that down the line, trying to get a bench player who can also score 20 per night will not be as valuable as a guy who can just play defense and not screw up when the some of the starters are on the bench.

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