Measuring consistency in 3 point shooting

The NBA needs a shooting variance stat, especially on 3's.

- , SSR commenter

Commenters and pundits alike often talk about how much a team can rely upon a shooter. Someone can have a good shooting percentage but still be regarded as "streaky," which is usually a pejorative. However, aside from what can be gleaned from shooting averages, there is no formal stat that calculates a player's shooting consistency. Here I describe a way to calculate an objective measure of three point shooting consistency which can be derived from game logs via basketball reference or similar sources.

Standard deviation roughly measures the degree to which a random variable, like one-game 3P%, varies from its average. If a shooter generally shoots close to his average level, he consistently performs at that level, so standard deviation provides a good starting point for a consistent shooter. Technically this means a 0% 3 point shooter is "consistent" because he is consistently terrible, but that's why you also take the average into account. Roughly speaking, standard error is an estimate of standard deviation based on the actual sample we observe. Our main estimate relies on the concept of standard error as a measure of consistency.

Weighted standard error for three point shooting

A quick and dirty way accomplish this is to go to a player's game logs, take the 3P% column, and naively calculate the standard error of the column. However, that measure of consistency has a big flaw: volume. A player who shoots 0-1 and then 1-1 is just as "inconsistent" as a player who shoots 0-10 and then 10-10. Volume is also why if you calculated the average of the 3P% line you generally wouldn't get the player's true 3P%. We can fix this by applying a frequency weight (3PA that night) to each game's shooting performance as we calculate sample variance. The formula is as follows:

Here s is the sample standard error, its square (shown) is the sample variance, 3PAi is that game's 3PA, 3P%i is that game's 3P%, 3P% is season 3P%, and total 3PA is that season's total 3PA. I prefer the standard error (s) for easier interpretation. With various minor caveats about the nature of the distribution of one-game 3P%, a player's 3P% will be within one standard error of his averages about 2/3 of the time, meaning it's entirely reasonable to expect that player to vary from his averages by around that amount. And of course about 1/3 of the time he'll shoot even better (worse) than that range.

Applying the statistic

First we'll calculate shooting standard error for two benchmark seasons: Stephen Curry's incredible second MVP season in 2015-16, and Antoine "because there are no fours" Walker's ill-fated 2003-04 campaign with the Mavs. Then, based on discussions which are somewhat current as of this post, we will compare and apply the results to this season's campaigns for Kyle Kuzma, Davis Bertans, and Bogdan Bogdanovic.

Player (season)	Efficiency (average 3P%)	Inconsistency (3P% standard error)
Stephen Curry (2015-16)	45%	14%
Antoine Walker (2003-04)	26%	20%
Kyle Kuzma (2019-20)	35%	21%
Davis Bertans (2019-20)	43%	19%
Bogdan Bogdanovic (2019-20)	38%	22%

From a very brief look, Stephen Curry's insane 2016 regular season wasn't just hyper efficient at incredible volume; that 14% standard error seems pretty low (consistent) as well. Antoine Walker's season was pretty bad, so the fact that his inconsistency was "only" 20% may just be a product of 'Toine "consistently" having 1-5 sort of nights. Was this exercise simply meant to mock a former Celtic? Who knows.

On to the current player-seasons. The difference isn't enormous, but an unsurprising result is clear: a sharpshooter like Davis Bertans is a more consistent shooter than creator-types like Bogdanovic and Kuzma. Taking the whole season's sample into account, and noting that this is a career year for Bertans, this should surprise no one.

A simpler method: Reliability

You may not like the above measure of consistency for one of three reasons:

1. You really don't want to go download game logs and modify excel sheets in weird, somewhat complicated ways every time you want to look up a stat.

2. You're more concerned with the left tail of someone's distribution (the bad shooting nights), because the right tail might turn a win into a blowout whereas the left tail might turn a close win into a close loss. (You may also have a related quibble about 3P% distribution and truncated tails, but they're relatively minor in the grand scheme of things.)

3. This method disparages your favorite player... Antoine Walker.

I can't help with 3, and I also can't help with the game log part of 1. But there is a simple way to concentrate on that left tail. Take the game log, count the number of games a player shoots below a threshold value from 3 (I'll use 30%), and divide by games played. This is no longer a measure of variance/consistency; it's more a measure of basic reliability. What is the likelihood this player's shooting at least won't hurt the team? Focusing on the same player-seasons:

Player (season)	Reliability (% games at least 30% from 3)
Stephen Curry (2015-16)	82%
Antoine Walker (2003-04)	39%
Kyle Kuzma (2019-20)	56%
Davis Bertans (2019-20)	79%
Bogdan Bogdanovic (2019-20)	53%

This basic stat incorporates both efficiency and consistency. An inefficient player only needs a small deviation from average to have a bad night, but a great shooter's "bad night" is often still pretty good. It also paints a much starker difference between shooters (stars or specialists), streaky shooters, and no-conscience chuckers.

There are many parts of basketball which are not well-measured, but shooting is pretty objective and straightforward. Hopefully this method helps future analysis more objectively cover whether players are consistent or reliable shooters.