Comparing three point shot percentage by half is an interesting opportunity to test how much 3 pt shooting percentage is nested by game. So I went and (manually) entered the 2022-23 Celtic's 3 point shooting percentage for each game, split by half. First off, this is what regression to the mean looks like; x-axis is first half 3 pt shooting percentage, y axis is the change from first to second half (so positive means they shot better in the second half):

As you'd expect, when they shoot really well in the first half, their shooting in the second half is lower; and when they shoot terrible in the first half, their shooting in the second improves.

Now, if each three point shot is truly independent, there should be no relationship between shooting in the first and second half. If three point shooting is partly a function of the game, they should be positively correlated (i.e. if they shoot above average in the first half, they should be expected to shoot above average in the second). Here's what that actually looks like:

Weak, but positive correlation (r=0.14). But again, these are inherently noisy data; in each game, we're dealing with such small samples that there's going to be really different outcomes even if the "true" shooting percentage varies a lot. So I started simulated different scenarios to see what the outcome looks like if we know that shots are truly independent or nested by game.

First, here's an 82 game season simulated where each 3 pt shot has an independent probability of 0.37:

Zero correlation (r=0.01). Now, here's an 82 game season where the 3 pt shooting % for each game is randomly drawn from a normal distribution with mean=0.37 and sd=0.078 (the values for last year's Celtics team):

Stronger correlation (r=0.19), higher than observed for the C's last year. I did that a couple times, and the correlations for the independent trials and the nested trials kept changing quite a bit. So then I just simulated each type of season 1,000 times and recorded the correlation coefficient for each; red is the histogram for the truly independent outcomes; blue is nested; the vertical line is the observed correlation coefficient:

Ack. The observed correlation between first and second half 3 pt % shooting for the Celtics last year is right in the middle of the two hypotheses; it's a 90th percentile outcome for the independent shots and a 7th percentile outcome for the nested. Make of all of that what you will. Maybe 3 point shooting is a function of game but the coaches and players adjust after the half (e.g., as suggested, take their foot off the gas if they're up big, bench a player who's missing, player takes a bunch of stupid heat checks because he thinks he's on fire, etc.), plus regression to the mean.