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The Shape of Player Performances

NBA: Cleveland Cavaliers at Philadelphia 76ers Bill Streicher-USA TODAY Sports

Joel Embiid is currently averaging about 27 points per game so far this season. In a fantasy land, that could mean that he alternates games of 54 points and 0 points, or games of 26 points and 28 points. In reality though, we know that Joel’s scoring distribution is far smoother. You can see what it looks like exactly in Figure 1, presented as two different density functions.

Figure 1: Joel Embiid points scored distribution

JJ Redick is averaging 17.6 points per game, and let’s see what his distribution looks like in Figure 2.

Figure 2: JJ Redick points scored distribution

Aside from just the raw points scored numbers (Joel obviously scores more than JJ), the shapes of the distributions differ even based on the eye test. JJ has a more defined peak that is located below his average, whereas Joel has a broader plateau type shape, with the high point above his average. It’s easy to fall into the trap of both understanding that Joel averages 27 points per game, but also thinking that scoring less than 23 points is an abject failure.

Game Score Distributions

Points are but one part of the equation when it comes to evaluating if a player has a good or bad game. I’ll use game score as kind of a generic catchall for quality. It’s an entirely box score based statistic, but we’re painting in broad strokes here, and I like the variability of the game score numbers. The biggest (and significant) miss of game score is the lack of defensive information, but I’ll address that in a little more detail later.

My idea here is to try and understand the distribution of “good games” and “bad games” both relatively and absolutely, as well as characterize the similarity of player distributions to each other. Essentially, do Player A and Player B perform well at approximately the same frequency or not. Moving forwards, all player data will be from the entirety of the 2018-19 season, including the games they might have played prior to joining the Sixers, with the exception of Jimmy Butler who will only have his Sixers games considered due to his role change. Figure 3 shows the raw game score distributions and mean scaled game score distributions for each Sixers player who sees reasonable time (also I picked an even number for aesthetically pleasing graphs). Names are above the chart for each player if you are surprised that T.J. McConnell has a game score distribution like an All-NBA player.

Figure 3: Philadelphia 76er raw and scaled game score distributions 2018-19 season

The scaled plots above show one standard deviation (unique for each player) away from the mean. Now, we’re going to take a look at what proportion of a player’s scaled game scores, and therefore relative performances, fall into the following fairly (completely) arbitrary categories.

Terrible: (<, -2)

Bad: (-2, -1)

Average Minus: (-1, 0)

Average Plus: (0, 1)

Great: (1, 2)

Amazing: (2, >)

Table 1 presents the data, with the color scheme done by column with red as low values and blue as high values. For example, in the top left cell, Ben Simmons has 3% of his games with a game score of two or more standard deviations below the mean. Important note - these are relative values. It is not saying that Ben does not have any “Amazing” games, its saying that according to the information below, Ben does not have any games with game scores two more more standard deviations above the mean.

Table 1: Occurrence of relative game score performance by player and K-means cluster

You’ll note the Cluster column on the far right. This is a k-means clustering (way to assign information to similar groups) based on four centers. Players with the same cluster have a more similar distribution of Terrible, Bad, Avg. Minus, etc. than those in another cluster. It is not a value judgement, simply a similarity measure. Cluster A is the symmetric group - they’re about equally likely to have performances +/- a certain standard deviation. Cluster B is the group where they are more likely to be slightly bad than slightly good, but more likely to be very good than very bad. Cluster C is likely to be slightly bad, but has a decent chance of being very good or excellent, but is never terrible. Jonah occupies his own group.

This isn’t a particularly advanced dive into the numbers, but I found it interesting to see how often certain players over or under performed their average performance, and how players might differ by play styles. Not saying its the best way to do this, but I found it pretty informative, and will impact how I think about game performance moving forwards

Caveat: this more or less does not consider defense. It’s not hard to imagine a game where JJ or Jimmy is not playing particularly well from a game score perspective, but Jimmy is a defensive plus and JJ is a significant minus, effecting the overall quality of their performance.

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