# Calculating NBA shooting consistency

What would be the proper way to evaluate/determine an NBA player's 3-point shooting consistency? For instance, I have a player that shoots 37% from 3-point range and takes 200 attempts all year.

I was considering taking the rolling average 3-point% of an arbitrary number of shots (say 20). Then using those averages to determine the standard deviation from the 37% mean. Using a rolling sample size of 20 shots only allows for a precision of 5% in shooting percentage, but I'm concerned that using too many shots won't reveal the inconsistencies in performance.

Is there a better approach to determine consistency?

• What are you going to use this measurement for? e.g. do you want to compare players to each other? Do you just want to see who is the most consistent? Or do you have some more specific question such as that consistency is higher when your team is ahead or something like that? Commented May 17, 2014 at 17:21
• I'm using the measurement to determine how consistent each 3-point shooter on a team is (of players that have some minimum number of attempts). I'll want to compare consistency between players, understanding that they'll have an unequal number of attempts in the season.
– Will
Commented May 17, 2014 at 19:18
• I think your basic idea is good. But why a rolling average? Why not "first 10 shots", "11th-20th" etc? You could try different numbers of shots. You should also probably limit this to players with at least a certain minimum number of shots in the season Commented May 17, 2014 at 19:23
• My thinking is that by taking the data in chunks instead of a rolling average, I might miss out on periods of inconsistent shooting. An extreme example being if a player makes shots 1-5, misses shots 6-15, and makes shots 16-20. Using 10-shot groupings results in two 50% shooting groups, but a 10-shot rolling average would reveal the 0% shooting slump.
– Will
Commented May 17, 2014 at 19:54
• Do an analysis of runs. Also, you need to be clear about what it is you mean when you say "consistent"--I'm going to interpret it as meaning that the probability of making a shot is constant for every single shot (i.e. it is completely independent of any and all previous outcomes). Agree? Nevertheless, do an analysis of runs... Commented Jul 22, 2014 at 23:20

I would fit a hierarchical model with a hierarchal level across games and look at the variance across games. If $$y_g$$ is the number of makes out of $$N_g$$ attempts for game $$g$$

$$y_g \sim Binomial(\theta_g, N_g)$$ $$\theta_g \sim Beta(\alpha, \beta)$$

with some prior put on $$\alpha$$ and $$\beta$$.

The posterior variance of $$\theta_g$$ represents the variance in percentage across games. You can get posterior sample $$s$$ of the variance of $$\theta_g$$ using the following,

$$\frac{\alpha^{(s)}\beta^{(s)}}{(\alpha^{(s)} + \beta^{(s)})^2(\alpha^{(s)} + \beta^{(s)} + 1)}$$

where the $$(s)$$ subscript represents posterior sample $$s$$ of the respective parameter.

• +1 I was literally about to write the exact same answer after reading the original post. Commented Apr 11 at 17:40

As another user stated in the comments above, a runs test is the way to analyze your shooting data. It tests the hypothesis that the elements of the sequence are mutually independent. If the hypothesis is rejected, then you could say that the player's 3-point shooting is inconsistent.

I would like to also point you towards this article since it's directly related to your analysis.

• It's good to include the links but even better to explain what they contain. Commented Feb 18, 2017 at 18:45

I think a runs test is a good idea. To me, by analysing the data in "chunks," your intention is to create a proxy for or control for "hot hands" in player consistency. There's a huge literature on this phenomenon out there. One of the best papers was discussed by Gelman on his blog back in July 2015. The title of his post was, "Hey-guess what? There really is a hot hand!" (http://andrewgelman.com/2015/07/09/hey-guess-what-there-really-is-a-hot-hand/). The paper Gelman reports on is a rebuttal to much of the previous literature insofar as it details the errors made by previous analyses of the hot hands phenomenon. The earlier work focused on overall as opposed to conditional probabilities. This paper posits a new sequential probability model (see the link for a reference to the paper).

One good metric of consistency which should control for differences in, e.g., number of shots taken, is the coefficient of variation. The CV is a dimensionless, scale invariant measure of variability and is calculated by dividing the std deviation by the mean. The problem it attempts to solve is that std deviations are expressed in the scale of the unit under measure, i.e., it is not scale invariant. This means that metrics with high average values will also tend to have higher std deviations than metrics with low average values. So, for instance, due to differences in their average values, measures of the variability in diastolic and systolic blood pressure are not directly comparable. By taking the CV, their variability becomes comparable. The same thing holds for many other metrics such as stock prices, online metrics such as the number of impressions or hits to a web page, and so on.

Thus, the CV can be calculated for many metrics and scale types, excluding categorical information and measures with negative values.