## Houston Rockets’ Record-Breaking Shooting Display a Statistical Anomaly

When the Houston Rockets and Golden State Warriors faced off in a battle of two playoff hopefuls back on Feb. 5, fans were expecting to see some distance shooting between the two.
After all, Golden State and Houston ranked in the top 10 in three-point shooting goal percentage (first and eighth, respectively), as well as the top 11 in three-point shots attempted (11th and first, respectively).
However, no fans -- not even some of the most optimistic Rockets fans -- could have expected to see what unfolded at the Toyota Center. The Rockets shot 40 three-pointers and made 23 of them, tying the previous NBA record set by the Orlando Magic in 2009.
Maybe it wasn’t all too surprising the Rockets shot 40 three-pointers, as they attempt the most per game in the NBA. But to make 23 of them? That seems to be out of this world hot from distance. So I set out to discover the probability that the Rockets would make 23 out of 40 from beyond the arc.
What I found was that the Rockets three point barrage last Tuesday night had very little chance of happening on that day.
Prior to last Tuesday’s game, the Rockets, as a team, were shooting 35.9% from distance on the season. Using this field goal percentage, the Rockets had a .27% chance of making 23 out of 40 three pointers. That’s not a 27% chance or even a 2.7%, but less than a half percent chance of making 23 out of 40.
Also, Carlos Delfino, a 39.6% three-point shooter and one of the Rockets' better three-point shooters, was hurt and did not play. So I limited my scope and looked at the nine Rockets players who attempted a three pointer (side note: all nine made at least one too) and redid my calculations using only the players who attempted a shot.
Those nine players (James Harden, Jeremy Lin, James Anderson, Patrick Patterson, Chandler Parsons, Patrick Beverley, Toney Douglas, Donatas Motiejunas, and Marcus Morris) had combined to shoot 35.3% from three-point range on the year up to this point. Using the nine players’ combined three-point percentage; the Rockets had an even smaller chance of making 23 out of 40, as those players had a .22% chance of accomplishing what they did.
To break this down a little further, under both circumstances the Rockets could have been expected to make approximately 14 out of 40 threes. As we know, percentages are not accurate indicators of what will happen in the future, but of what has happened in the past. So after some calculations, under both the above circumstances, the Rockets' standard deviation is approximately three made three-point baskets.
Statistically speaking we can expect a majority of our outcomes to fall within two standard deviations of the mean (our average). This means, that if the Rockets took 40 threes, we could expect them to make, on average, 14. But, if the Rockets shot 40 three-pointers a game for a whole season we could expect to see a majority of the results fall within plus or minus two standard deviations of our average; meaning the majority of our results would be in the range of 9–20 three pointers made per game when attempting 40 threes (note: our exact range is 8.1 to 20.1, but because our low range number is greater than 8 we cannot use 8 in our assumption).
This means the Rockets would only have a 2% chance of hitting our ‘high’ range number of 20 made threes. The Rockets' 23 makes were three standard deviations above the average. As proven above, that has a 99.8% chance of NOT happening.
When I was watching the game I thought the Rockets' three-point shooting was an abnormality. But what if the Rockets had a better chance of making the threes they took? The flaw with the above statistics is they do not take into consideration the location of the made/taken three pointers. For instance, a corner three (taken from below the break on the three-point line) is closer than a three-pointer above the break, and typically easier for players to hit.
The nine players mentioned above hit 12 of 18 corner threes, a remarkable 66.7%. Prior to the game, those nine players averaged a combined 39.3% on corner threes. Thus we could expect them to make approximately seven out of 18 corner three attempts. Given the standard deviation is approximately two made baskets, we could expect that if the Rockets shot 18 corner threes, they would make in between three and 11 corner threes a majority of the time. After running the probability, the Rockets had a 1.3% chance of hitting 12 out of 18 corner threes. Compared to the previous observations, it was more realistic for the Rockets to hit 12 out of 18 corner threes, but not much more.
In addition to their corner three exhibition, the nine Rocket players hit 11 out of 22 from above the break of the three-point line. On the season, those nine players combined to make 33.5% of threes from above the break. In an effort to not bore you with statistical breakdowns, we could expect the majority of results from 22 Rocket above the break three point attempts to fall within 3 and 12 makes.
You may notice that our range includes what the Rockets actually made, 11. I can assume there is a much higher probability for those nine players to hit 11 out of 22. To be exact they have a 4.7% chance of hitting 11 out of 22 threes from above the break.
In summary, the Rockets’ three-point shooting was a statistical anomaly. But have they done this before?
The answer is yes and no.
Twice this season the Rockets' three-point shooting has fallen outside of two standard deviations above the mean. The day after Thanksgiving the Rockets hit 14 of 25 from deep and on New Year’s Eve they hit 16 of 29 from distance. These numbers fall slightly outside of two standard deviations above the average. BUT if we decided to round our numbers (which we could because you can’t make a percentage point of a basket, it’s either you make it or you don’t) they would fall within two standard deviations of the mean. With that said, the Rockets had a 2% and 1.6% chance, respectively, of shooting that percentage in those games. Obviously the biggest difference between these games and the game against the Warriors was the amount of three-pointers taken. The Rockets have only attempted 40 threes one other time in a game this year and only made 11 of them, which falls at one standard deviation below our average.
The key point to remember is over the course of a season fans should expect to see shooting displays that are outside of two standard deviations of the mean, both above and below our range. However, all the above probabilities are the chances the Rockets had of shooting that way, on that night.
If you also step away from the statistical aspect and just observe the game, the Warriors' defense was nothing short of atrocious. Lin was getting into the lane at will and distributing to open players squared up around the arc like vultures circling their prey.
This should come as no surprise because the Warriors give up a league leading amount of three point attempts per game and allow a top ten three point field goal percentage against.
To truly understand how bad the defense was I re-watched the made three pointers and broke them into our two location categories (above the break and corner threes) as well as assigning my own metrics for the type of defense faced on the shot: open, closed out and contested shots.
Out of the 12 corner makes, three were completely wide open, six were being closed out on and three were contested where the defense was solid and a hand was up.
Out of the 11 above the break makes, four were completely wide open, six were closed out on and only one was contested.
As bad as that looks, I was probably pretty generous to the Warriors' defense with my discretionary picks between open shots and those being closed out on. If Warriors coach Mark Jackson had been announcing the game he would have used his famous catchphrase, “Hand down, man down” so much that he would have been sick of it.
Despite the horrendous defense, the shooting display the Rockets put on from distance should still be considered a statistical anomaly. There is a lot to be said about the hot hand theory and many casual fans (or those not statistically inclined) could have just assumed the whole team was ebbing and flowing out of “ON FIRE” mode in an NBA Hangtime game. But the truth is the Rockets have a very, very, very miniscule chance of replicating that performance.
This Tuesday night, the Warriors and Rockets will face off again, this time at Oracle Arena in Oakland. I am going to go out on a limb and say the defense will be a lot better from the Warriors. But I’m not going out on a limb when saying I do not expect the Rockets to replicate last Tuesday’s shooting performance. When the Magic hit 23 out of 37 three pointers in 2009 they had a .0004% chance of accomplishing that. The Magic didn’t strike twice back in 2009, and I don’t expect the Rockets to in 2013.

@EricSports here. Very good article. Two things I would tell you to edit is you have to adjust the predicted shooting percentage to who actually shot how many 3’s. Not sure how that would effect the #’s because I know Lin shot a high amount of them, but is needed. Also I would factor in Golden State’s 3 Point Percentage allowed this season into the #’s. I’d imagine it would be much much easier to make 23 threes against the Warriors than the Bulls. Hopefully I don’t seem like I’m bashing the article. I enjoyed it a lot and have bookmarked your website to read going forward.

@ericsports Really appreciate the feedback, I just added a little bit about the Warriors defense. The statistics on defensive shot locations were just not available for me to analyze. Also, I felt the sample sizes of the individual player shots were to small to make any significant observations, but as an aggregate it’s much easier to analyze. Definitely not 100% accurate but the best I could do… Thanks for the follow and I definitely appreciate the feedback! Follow me @domlucq. Great to hear you’ll be reading our blog. Should be great all year!

The problem with saying “what are the chances that the Rockets would do this” looks at the one night in particular. The probability of them or another team doing this on any given game during a season may be low, but not as low as you’ve made it look. Factor in the possibility of a 21-21 night at the FT line, a team getting 80% of the rebounds in a game, shooting 20-21 in a quarter, and any other number of possibility weird records and the fact that something happened on a random night isn’t really that remarkable.

If the goal was to discover what the probability that the Rockets would make 23 out of 40, all that is needed is a simple binomial based on the Rockets e3P% against the Warriors. Story stops there. Everything else that happens is just explaining why we saw a rare event. A team played horrible defense, a team made higher than expected shots, and took a lot of corner threes. The prob of the event happening doesn’t change.

If you wanted to determine the likelihood of them tying the record with 23 threes, you might want think a little more about the distribution you use.

“As we know, percentages are not accurate indicators of what will happen in the future, but of what has happened in the past.” Might want to be careful here, especially given the methods used in this paper.

I think Ric Bucher, live at the Warriors game, just mentioned you as his “friend” who does analytics. Amazing stuff Dom!