I tried.

I consider open-mindedness and skepticism to be valuable traits. I don’t want to ignore a hypothesis because it doesn’t jive with my preconceived notions. Even if the people who hold a specified belief are the ones mocking my own thinking and my own background, I still want to understand what they are thinking and judge their theory based on the evidence. And the great, important facet of the process of science is that it is self-correcting and ultimately could revert back to a previous belief if said belief is indeed correct.

We saw this recently with peanut allergies and young children. New science has found it’s actually much better to expose young kids to peanuts, as it reduces the rates of peanut allergies later on, which has frustrated many parents who see these rules as ones that are seemingly constantly in flux. But we should act on the best evidence available, and sometimes that means rewriting guidelines.

I tried that with mid-range shots. I wanted to believe. I spent extra time on this project, tweaking the numbers and double- and triple-checking, and postulating any missing holes my methodology could have. I thought I had a smoking gun: what if we were ignoring one extremely valuable variable that does indeed make mid-range shots more efficient than they appear by FG%, and that by factoring in that additional variable we get a more palatable trade-off between those dinosaur shots and 3-pointers? Have we for all this time been forgetting to factor in free throws drawn and thus actually underrated mid-range shots? It would partially explain the survival of such a supposedly useless shot, and it would lend more justification in taking those shots, in some circumstances, with analytical evidence.

But truth — contrary to what we all want to believe — is often more boring.

Here comes the data

A while ago, I got access to some foul location data thanks to Krishna Narsu, and I thought it would answer some questions I had about mid-range shooting foul rates versus other locations. My hypothesis was that fouls were much more common on jump shots inside the line than outside, which would make mid-range shots more efficient — and remember that the players taking these shots are usually pretty good free throw shooters too.

Read More: The conundrum of the Miami Heat core

The data, unfortunately, only goes back to 2011, but that’s more than enough to work with and you can see one exploration here where I found that shooting foul rates actually correlated pretty well with good perimeter defenders. Not only did I get distance from the rim, the X- and Y-coordinates are given, which means we can create spiffy shot chart-style graphics for fouls.

You can see one below showing the location and density of shooting fouls. I’ve made the points transparent, so darker shadings indicate a higher frequency at that particular coordinate [1. If you want to make a similar chart, I’d recommend this guide for R.]. It’s fairly obvious here that the vast majority of shooting fouls occur at the rim — there’s little difference in the mid-range compared to just outside the 3-point line. In fact, there’s even a dead zone just inside the arc, and even the corners — where players have little room to maneuver and draw fouls — are much denser.

Of course, mid-range shots, especially the longer ones, are becoming increasingly rare, so let’s look at the actual rates and calculate efficiency. I did that by merging this data with play-by-play logs, and consequently the information lost a dimension because I wasn’t working with shot chart data. Nevertheless, I compiled efficiency by distance, and you can see the results in the graph below. Note that this is points per attempt, where an “and-1” free throw does not count as an “attempt” and three free throws in a row (i.e. the old fashioned 3-pointer) only counts as one shot attempt. By the way, for whatever reason — perhaps simplicity — no 3-pointers were coded as less than 24 feet, and many did not have a distance given at all, so I had to assume 24 feet [2. For reference, all the shots, and free throws, coded as 2-pointers that I saw ranged from 0 to 24 feet. And for some reason, there were a handful of 2-pointers at 24 feet and they were converted at a significantly lower percentage than 23 footers. I decided to dump all those shots into the 23 foot bin]. I did the same with layups and dunks listed without a distance; those were given distances of zero feet.

Yes, even when factoring in shooting fouls, mid-range shots are still significantly less efficient. In fact, even deep 3-pointers up to 30 feet are just as efficient, although that’s partly a case of self-selection as, typically, players proficient from that distance are the ones taking the attempts. Shooting fouls only become dominant closer to the basket, but they have the greatest effect on efficiency in zones with low field-goal percentages. Thus, they give the largest boosts to that awkward area away from the rim but too far inside to be considered mid-range. This is where players get hacked only their way to the rim, and it’s also where low-post players can draw a lot of contact.

Alternatively, it’s possible that for some players the mid-range area is actually efficient. Let’s turn to one DeMar DeRozan, a mid-range maestro who still manages to rack up the free throws and takes 3-pointers infrequently. I have a summary table below for him. His most efficient area is at the rim, but that’s no surprise. However, his most inefficient shot, with the exception of super-deep 3-pointers, is the dreaded long 2-point attempt. Even though he’s a poor shooter from outside the arc, he’s still slightly more efficient there than he is in total from from 10-to-24 feet. You may notice his efficiency is pretty good from that awkward 4-to-9 foot range, and that’s true for many players. I believe it’s because more fouls are coded in that area, partly because fouls by location are coded by where they occur and not where the ball is and partly because guys are stopped before they get to the rim and sent to the line.

Table: DeMar DeRozan, 2011-16

Zone Distances Points Attempts PPP 2PT 0 to 3 2721 1997 1.36 2PT 4 to 9 1446 1324 1.09 2PT 10 to 15 1323 1401 0.94 2PT 16 to 24 2109 2605 0.81 3PT 24 to 27 594 640 0.93 3PT 28 to 30 17 30 0.57

Player by player

Now let’s turn to a couple more interesting cases. Below are tables for James Harden, the infamous Moreyball player who eschews mid-range shots and can squeeze a couple of free throws from a stone, and Russell Westbrook, a former teammate and fast-break terror who doesn’t have the best jump shot in the world but still launches many off the dribble. Harden, indeed, is quite efficient near the rim, and beyond the arc too. However, his efficiency is above average in the mid-range areas too. He’s exceptionally potent between 10 and 15 feet, which is peculiar; he has a bizarrely high free throw rate in that range. If you wanna be efficient from the mid-range, apparently you need to derive a staggering 60% of your points from shooting fouls.

Table: James Harden, 2011-16

Zone Distances Points Attempts PPP 2PT 0 to 3 3410 2480 1.38 2PT 4 to 9 1414 1211 1.17 2PT 10 to 15 648 591 1.10 2PT 16 to 24 830 977 0.85 3PT 24 to 27 3212 2740 1.17 3PT 28 to 30 102 98 1.04

Westbrook, on the other hand, just like DeRozan, is still more efficient from beyond the arc despite being a poor outside shooter. He generates a ton of free throws, but his long 2-pointer efficiency is still below the league average for 3PT shots. I’m actually most surprised his numbers from 4-to-9 feet are so low; one would think with his free throw rate he’d generate more points.

Table: Russell Westbrook, 2011-16

Zone Distances Points Attempts PPP 2PT 0 to 3 4202 3288 1.28 2PT 4 to 9 1099 1165 0.94 2PT 10 to 15 1237 1331 0.93 2PT 16 to 24 1321 1604 0.82 3PT 24 to 27 1340 1355 0.99 3PT 28 to 30 45 57 0.79

So if volume scorers aren’t more efficient from the midrange, perhaps a low-mistake, high-accuracy point guard will be. Chris Paul is not known for his outside shooting, and his greatest skills lie in the pick-and-roll and destroying opponents who give him too much space inside the arc. However, he’s still more efficient from behind the line. And his mid-range efficiency is quite sensational, hovering around 1.0 points per attempt. In a half-court setting, that’s useful.

Table: Chris Paul 2011-16

Zone Distances Points Attempts PPP 2PT 0 to 3 1284 952 1.35 2PT 4 to 9 971 881 1.10 2PT 10 to 15 1119 1102 1.02 2PT 16 to 24 1602 1670 0.96 3PT 24 to 27 1589 1364 1.16 3PT 28 to 30 102 108 0.94

Let’s move on to a couple of power forward midrange masters. You can see LaMarcus Aldridge in the first table below. He’s actually one of the few high-volume scorers who’s just as efficient from the mid-range as he is from behind the arc, but that’s not saying much — he rarely takes 3-pointers and he’s not at all good at them, at least over this entire period. Dirk Nowitzki, however, is a good outside shooter, so his efficiency there destroys the 10-to-24 feet range, even though he’s unequivocally one of the most skilled mid-range shooters ever. The issue is that the players who are good enough to be exceptional mid-range shooters usually are just decent enough from outside the arc to make it more worthwhile there instead.

Table: LaMarcus Aldridge 2011-16

Zone Distances Points Attempts PPP 2PT 0 to 3 3403 2343 1.45 2PT 4 to 9 1558 1519 1.03 2PT 10 to 15 1328 1508 0.88 2PT 16 to 24 2401 2773 0.87 3PT 24 to 27 135 159 0.85 3PT 28 to 30 0 6 0.00

Table: Dirk Nowitzki 2011-16

Zone Distances Points Attempts PPP 2PT 0 to 3 1054 703 1.50 2PT 4 to 9 601 523 1.15 2PT 10 to 15 1854 1817 1.02 2PT 16 to 24 2343 2286 1.02 3PT 24 to 27 1698 1440 1.18 3PT 28 to 30 36 29 1.24

Finally, let’s look at the head of the snake of this 3-point revolution: Stephen Curry. He’s actually an outstanding shooter from 10-to-24 feet; it’s just that he’s awe-inspiring from further out. Somehow, he’s just as efficient from 0-to-3 feet away as he is from 28-to-30 feet, and his most efficient range is 24-to-27 feet. And remember, when this guy gets hacked at the rim, he converts his free throws at a percentage not seen since the days of Mahmoud Abdul-Rauf and Mark Price. When you can hit outside shots at 44 percent at a high volume, you can have absurd efficiency.

Table: Stephen Curry 2011-16

Zone Distances Points Attempts PPP 2PT 0 to 3 1866 1371 1.361 2PT 4 to 9 706 667 1.058 2PT 10 to 15 585 585 1.000 2PT 16 to 24 1464 1537 0.953 3PT 24 to 27 3830 2801 1.367 3PT 28 to 30 434 321 1.352

Shot efficiency charts are common now, and they’re being used in more situations and for a wider range of analyses. We really shouldn’t ignore shooting fouls, however, because it does distort the overall picture. For example, you can see an observation here that field-goal percentage stays flat when you’re away from the rim, but that’s misleading. Players pick up more fouls the closer to the rim they are, so there is actually a slight slope downwards going all the way to the 3-point line.

Sadly, these shooting fouls do not help the infamous mid-range shot. And if you doubt my methods, here’s some analysis years ago from 82games where they found similar foul rates and efficiency. Yes, there are some situations where a mi-range shot can make sense, and for some players it’s not a below average shot, and depending on the shot clock and how you’re being defended it may not be a poor decision. But the math is pretty simple: it’s just so difficult to justify a long jump shot when you can move back a few feet and receive 50 percent more points for make, not to mention the added benefit of spacing. As long as mid-range shots are worth two points and ones behind the arc are worth three, we’re going to see this disparity, even if you factor in shooting fouls.