In a recent Dunc’d On podcast (which I highly recommend as a great listen for NBA fans), the hosts Danny and Nate briefly commented upon how the Celtics defense this season has been bolstered by the league’s second best 3-point defense, in terms of percentage of opponent makes. They also mentioned, however, that many NBA thinkers speculate 3-point percentage defense may be mostly due to luck and beyond the defense’s control. In fact, many argue that the best indicator of a good 3-point defense is the ability to limit opponent 3-point attempts, particularly the dangerous corner threes. I have previously written a little about 3-point defense for bigthreesports.com.

With the conventional wisdom in mind, it also should be noted that the Celtics under coach Brad Stevens have consistently performed well in 3-point percentage defense. Boston has been a top five ranked team in this regard during each of the five years (if we include the current season) of the Stevens tenure. Perhaps there is something more than luck going on here? I decided to investigate this more deeply.

Now for the math. Overall, Celtics opponents have attempted 8578 3-pointers and made 2898 since the 2013-14 season, good for about 33.8%. If we assume each 3-point attempt over this time period had a likelihood of going in equal to the league average 3-point percentage of the particular year it was taken, Celtics opponents would be expected to make roughly 3059 of those of those 8578 attempts, or about 35.7%. The big question is how likely would a “league average” 3-point defense see a 1.9 percentage point difference between expectation and actuality over this time period? In other words, could we reasonably argue that the Celtics just got lucky?

This question is not all that hard to answer *if *(and this is a big if) we assume that each 3-point attempt is independent. This essentially means that the result of one shot has no bearing on the result of a future attempt. Under this assumption, we can simulate a bunch (50000 to be precise) of five year periods and see how many times our theoretical “league average” 3-point defense would force as low a percentage as the Celtics.

Out of the 50000 simulations, the league average defense defended the 3-point line as well as the Celtics only *8 times (0.016% of the time).* Under this preliminary analysis, we have strong evidence that the Boston’s 3-Point defense is better than league average. Maybe there is something more than noise here?

Perhaps we just want to analyze this season’s three point defense. After all, some of the perimeter defenders on previous Celtic teams, such as Jae Crowder and Avery Bradley, have left and new faces, like Jaylen Brown and Jayson Tatum, have a more prominent role. This season (through January 5th) Celtic opponents have made 374 out of 1096 3-point attempts (34.1%). The league-wide 3-point percentage for this season is currently 36.4%. If we simulate 1096 attempts over and over again, with the independence assumption, we observe that a league average 3-point defense would achieve a percentage this low by chance alone about 6.2% of the time. As those who have experience in stats may know, this is on the edge of but is not quite a “statistically significant result” (if we use a one-tailed test at the 0.05 level). In other words we would rate this kind of performance as impressive, but maybe not impressive enough to *really *be convinced that the Celtics are a substantially better than average 3-point defense this season.

I tried out one more approach because I felt a little unsure about treating each opponent 3-point attempt as independent. After all, a player on a cold shooting night might not be expected to make his next attempt with the same probability that his season percentage would suggest. Also, my simulations assumed that *every* opponent shooter was league average, when in actuality there are of course a wide range of abilities.

For a cruder approach with less potentially dangerous assumptions, I looked at each Celtics game in the Stevens tenure and simply tracked whether or not the opponent shot 3-pointers in that game at a better or worse percentage than their season average. Assuming opponent attempts across games to be independent seems reasonable. A good 3-point defense would be expected to somewhat consistently hold opponents below their season average. A “league average” 3-point defense would be a more or less a coin flip to induce a bad shooting night in the opponent. Just to make sure this assumption was reasonable, I checked the 2016-17 season data and found teams outshot their season 3-point percentage 50.3% of the time.

The results of this approach: the Celtics have played 369 regular season games under Brad Stevens and opponents have shot *worse* than their season average on 3-pointers 198 times, or about 53.7% of the time. When we perform a simulation, we see that a league average defense would achieve this level of success through chance alone about 8.8% of the time. So by this cruder approach the Celtics 3-point defense seems a bit less impressive. If we look at the games played in just this season, Boston has held opponents below their season average 26 out of 41 times, which we would expect to be achieved by chance alone only 5.6% of the time.

So what should make of all this? For starters, I lean more towards the notion that within game 3-pointers are “independent enough” to place a bit more weight on the first analysis than the second. The game-level analysis is a bit cruder and probably undersells just how good Boston’s defense has been. Also, the Celtic’s 3-point defensive success has been very consistent over this 4.5 season period, rather than simply being the result of one great season and a few mediocre ones. That gives me more reason to believe that something real and sustainable is going on.

A further point in the Celtics favor for me is that my “priors” (previous beliefs) are more aligned with the notion that their perimeter defense is for real. Boston has had many good perimeter defenders over this period including the aforementioned Crowder and Bradley, but also Marcus Smart, Jaylen Brown, and Terry Rozier. Of course, Crowder and Bradley are gone but Jayson Tatum, another athletic wing, is also on the roster now.

Overall, the evidence is at least fairly strong (and potentially very strong if you buy the independence assumption) that the Celtics have had an above average 3-point defense in the Stevens tenure. If we only had data on the 2017-18 team, I would say there is not enough information to dissuade us from the hypothesis that chance alone was responsible for the low opponent 3-point percentage. But given the sustained poor shooting by opponents over the past 4.5 seasons, I think we can at the very least say that the Celtics defense is having at least some real success defending the 3-point line.

It would be interesting to hear an experienced NBA mind break down Boston’s defensive schemes and players to learn what might be behind their success. We should also dig more into the shot tracking data and opponent 3-point attempt rate data. I am curious to see where Boston finishes in opponent 3-point percentage at the end of this season.

*Using the league average 3-Point % of each season in which each shot was taken. **Under assumptions that shots are independent and have probability of being made equal to league average 3-Point %. ***Data from basketball-reference

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