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The Fickle Beast of Three-Point Shooting

It’s a one-eyed, one-horned, flying, purple probability.

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NBA: Milwaukee Bucks at New York Knicks Vincent Carchietta-USA TODAY Sports

One of my favorite math activities, albeit one that requires a bit of patience, involves flipping a coin 100 times. On one side of a sheet of paper, you record the actual results. On the other side of the paper, you make up the results. The final product is a combination of 200 heads and tails (hence the patience). You then pose a question of someone who didn’t observe the process: which side holds the actual results?

The answer is pretty simple: almost always, it’s the side with the longest streak of consecutive heads or tails. When we make up the results, we are reticent to record (say) eight straight heads or tails; it seems improbable. But in reality, when you flip a coin that many times, it is quite probable to obtain such a streak. (Interested readers can refer to Amos Tversky and Daniel Kahneman’s representativeness heuristic. In short, we view HTHTHTHT as a more probable streak of coin flips than HHHHHHHH, even though each has the same probability of occurring.)

The thesis of this article is that coin flips and three-pointers are essentially the same. There are key differences, namely that three-pointers are affected by extrinsic factors such as defender proximity and shot location (and, it seems, the playoffs). However, they are similar enough that I want to tap the brakes on the idea that variation in three-point shooting is immune to the same process that governs coin flips: probability. Ultimately, people are flinging a ball from a distance into a circle only slightly bigger than the ball; we should not expect every shot to go in, even if they are “feeling it.”

Accepting variation, though, does not solve two issues: first, the Bucks’ mediocre three-point shooting thus far into the regular season (although ameliorated in recent weeks), and second, the Bucks’ recent history of awful three-point shooting in the playoffs. Variation does not explain the mean, but - critically - it can tell us how far we should expect to stray from it. To that end, using our old friend Norm, I’ll look at whether key Bucks players are significantly underperforming compared to their career averages. I’ll then consider what we can expect in the playoffs.

Let’s start with a bird’s eye view of the Bucks’ three-point shooting. We’ll look at at makes, attempts, and percentage this season, and percentage prior to this season, for our key shooters (notwithstanding Khris due to low volume):

Bucks’ Three-Point Shooting

Player 3PM 3PA 3P%, 2022-23 3P%, Before 2022-23
Player 3PM 3PA 3P%, 2022-23 3P%, Before 2022-23
Bobby Portis 58 170 0.34 0.38
Brook Lopez 101 269 0.38 0.34
Grayson Allen 102 253 0.40 0.39
Jevon Carter 84 207 0.41 0.38
Joe Ingles 35 99 0.35 0.41
Jrue Holiday 102 277 0.37 0.36
Pat Connaughton 84 236 0.36 0.37
Wesley Matthews 32 91 0.33 0.38

Based on the “interocular test” (i.e., looking at it), Bobby, Jingles, and Wes appear to be shooting at a lower clip this year. The rest of the Bucks are either at (Grayson, Jrue, Pat) or above (Brook, Jevon) their clips prior to this season. All in all, that passes the sniff test, both in terms of where guys are at and in terms of a roughly even distribution of being below, at, or above historic averages.

Beyond the interocular test, though, is a significance test. Among the three Bucks mired below their averages before this season, are any of their deficits statistically significant? Notably, Jingles and Wes have the lowest amount of attempts due to coming off injury and the bench, respectively. Their smaller sample sizes might be more subject to the whims of probability.

This brings us to our old friend Norm, the normal distribution. The normal distribution represents the notion that some things are extreme, but most things are average. For example, in the population of humans, there are some short folks and some tall folks, but most are average height. One of the neat properties of the normal distribution is that you can infer the average from a sample of the population. You don’t need to measure every man on earth to realize that their average height is 5’ 10’’; a bar-full will suffice. What’s more, if you take multiple samples (perhaps from multiple bars) and sketch out a distribution of their means, you’ll see our friend Norm again: some of the sample means will be too high or too low, but most of them will be pretty accurate.

Translated to three-point percentages, we’ll treat the population as a player’s three-point attempts prior to this season. This is a little trickier than height since it is a binary variable: misses and makes. But, if we take a sample from this population, we can calculate a three-point percentage (e.g., two makes + three misses = 0.4). And, if we take multiple samples, we end up with Norm, where most of the sample means will be close to the historic player’s three-point percentage, tailing off into extremes. Armed with this distribution, we can calculate whether the extremity of a particular sample mean - in this case, the three-point percentage during the 2022-23 season - is statistically significant (i.e., that there is a less than five percent chance that we would get a result as or more extreme if we assume that the population mean is correct).

What do we find? Bupkis: none of them are at a level that we would consider statistically significant. In terms of the probability that they would be shooting this bad or worse, Wes is at 33%, Bobby is at 17%, and Jingles is at 14%. Not great, but not at the point that we should necessary doubt that this season represents a marked departure from their past performance.

Moving to the playoffs, then, we would expect the Bucks’ historic averages to hold up. But variation never sleeps. I’m curious what the worst-case scenario - if we assume a statistically significant drop-off for each player - looks like, particularly given the small sample size of the playoffs.

I’ll start with some assumptions for simplicity: each playoff series is six games, we win each series, and each Buck shoots the same number of threes per game as the regular season (all dubious, but good enough for our purposes).

I’ll then identify the worst-case scenarios at the series and playoff levels. This is kind of the opposite of what I did above. Previously, I tested the current season as a sample of three-point shots from the population of three-point shots from all of the seasons. Now, I’m taking multiple samples of all seasons, where the sample size depends on how many threes the player shoots per game. This gives me Norm, which I can use to determine the worse-case scenario: at what point we would consider the player to make a significantly lower percentage of threes than their average.

Here’s what we find:

Bucks’ Three-Point Shooting: Worst-Case Scenarios

Player 3P%, Before 2022-23 3PA (Per Game) Worst-Case Scenario (Series) Worst-Case Scenario (Playoffs)
Player 3P%, Before 2022-23 3PA (Per Game) Worst-Case Scenario (Series) Worst-Case Scenario (Playoffs)
Bobby Portis 0.38 3.6 0.18 0.28
Brook Lopez 0.34 4.9 0.17 0.26
Grayson Allen 0.39 4.9 0.21 0.31
Jevon Carter 0.38 3.7 0.18 0.28
Joe Ingles 0.41 4.3 0.23 0.32
Jrue Holiday 0.36 6.2 0.22 0.29
Pat Connaughton 0.37 5.9 0.20 0.30
Wesley Matthews 0.38 2.5 0.13 0.27

This neatly illustrates a few things about small sample sizes: they make the worst-case scenario (a) much worse during a series than the playoffs and (b) somewhat worse for players who shoot less (e.g., Wes) than shoot more (e.g., Jrue).

But it also neatly illustrates something else. The Bucks’ three-point percentages in the playoffs during the Bud era have been .34, .36, .32, and .33. As miserable as that has been, none of them technically qualify for the threshold of significantly miserable. Based on the average of the worst-case scenarios above, that would appear to be in the ballpark of .30.

That said, our average playoff three-point percentage of .33 is significantly lower than our average regular season three-point percentage of .37, even when accounting for the slight dip in playoff three-point percentage across the league.

Taken together, the Bucks do shoot significantly worse in the playoffs across the board, but none of the individual playoffs breach that barrier. Based on the former, it may behoove the Bucks to find other avenues of playoff scoring that beyond the arc (after all, it won us a championship). Based on the latter, though, we can hope that we get more help from Norm this year and end up - for the first time in the Bud era - above the mean.