Editor’s Note: This article was submitted before the 76ers’ game against the Milwaukee Bucks on 3/17. It does not include Simmons’ 0-for-2 performance from the line in that game. However, because it was only two attempts, the game would not have meaningfully affected the analysis.
While our attention was elsewhere, Ben Simmons has become a solid free-throw shooter. In his Rookie of the Year campaign Simmons hit a mere 56% of his free throws. Thankfully this is above the 52% or so at which fouling his is a positive-expected-return move by the defense, leading to endless Hack-A-Whoever. But not by much! It’s close enough to the cutoff that fouling Ben was arguably good strategy for a team that was far enough behind late in the game, as fouling can occur very fast, thus increasing the number of possessions left in the contest, sometimes by enough to more than overcome the cost of giving up 1.12 (56% * 2) points per possession. Moreover, many feared that Ben, embarrassed by his poor FT shooting, was avoiding contact so as not to be sent to the line. This not only cost the team valuable free throws, it, some said, made Ben less effective around the basket.
The good news is, Ben has been shooting more like 60% — actually a bit over — this year. Hey, every little bit helps! But the much bigger, indeed truly exciting, news is that lately Ben has taken it to a wholly new level. Over his last 25 games, Ben has hit 91 of 137 free throws, a 66.4% clip. If we look at an even more recent, though of course smaller, sample, it’s better still, 35 for 46, which is over 76%.
We are all aware of the perils of small-sample analysis. But there are statistical tools for attempting to quantify those concerns. Most important is the standard error, which gives a sense of how much the sample average will plausibly miss by.We can approximate the standard error in a situation like this with a simple formula:
s.e. = sqrt(p*(1-p)/n)
Where p is the probability of success and n is the size of the sample. So if p = 70%; i.e. Ben is around 70% in a particular sample, then p*(1-p) = .7*.3 = .21. And sqrt (p*(1-p)) = sqrt(.21) which is around .46.
But it turns out it’s easier than that! Because .46 is very close to 50%, and indeed for any probability in the range of a better-than-awful three-point shooter or a less-than-very-good FT shooter, i.e. anywhere between 25 and 75%, it’s a close enough estimate to replace the whole sqrt(p*(1-p)) part with 50%, i.e. the square root of .5*.5. So then the whole formula collapses to this: for n shots, the standard error is 50% divided by the square root of n. So for 100 shots the standard error is 5%, for 400 shots the standard error is 50/20 = 2.5%. 400 is around the number of three-pointers a reasonably-heavy sniper will take, or the number of free throws a guy like Ben will take, over a full season. So what we get out of all that math is that for a full season, such players have percentages that are likely within 2.5% of their true level of ability. And for a quarter of a season, such guys are typically within about 5% of their true level.
Of course in real life anything can happen, but what a standard error tells us is, how far from the true average the sample average will be with reasonable frequency. Being one standard error away from the mean is very common, so if Ben shot 76% over the recent sample with 7% s.e. then it’s totally plausible that he was really a 69% shooter who got a little lucky. Is it possible he was in fact a 62% shooter over the period who got very lucky, i.e. delivered results two standard errors above his true ability level? It can happen, but people only beat their true level by two standard errors (a “positive two standard error event”) around one time in 40. Now, I cheated a little because I selected the stopping point — I counted the data back through the 2/23 game against Portland where Ben was 7/8, but not through the 2/21 Miami game where he shot 3/7. Feel free to re-do the analysis with other stopping points and see how things change. One stopping point I looked at was the past 25 games; I didn’t cherry-pick that one too much, as it includes Ben’s 0-for-2 performance at the beginning of the sample (looking back 24 games instead of 25 would have been too flagrant!). That sample includes 137 shots, a pretty large sample; it gets the standard error down to around .04. Ben shot almost exactly 66.7% in that period, so one s.e. below would be around 63%. Oh, I should mention that somewhere in there was a 0-for-0 game, I just dropped that one from the sample. As Siri says, if you’re splitting cookies evenly among your friends, and you have zero cookies, and zero friends, there’s really no way to say how many cookies you can give to each friend.
How to put all this information together is as much art as science; there is no perfect algorithm for deciding how much weight to put on the early vs. the late sample. One thing we can do is look at the trend in Simmons’ FT%. The problem is, we get very different answers depending on what fraction of the season we consider. For the full campaign so far, running a simple regression of Ben’s game-by-game free throw percentages on a counter for how many games have passed, we find that Ben is improving at a rate of almost exactly 0.1% per game, which would mean he’d be 8.2% better at the end of the season than he was at the beginning if that continued into April. On the other hand, if we look at just the past few dozen games, the rate of increase is dramatically higher, suggesting that Ben has improved enormously in the past half-season, perhaps from the high 50s to the mid-70s. So there’s really no way to pin this down. But nothing is stopping me from guessing so here goes: even though Ben is at 60.5% for the year, my guess is that he is now a good enough free-throw shooter that we can expect him to hit two-thirds of his shots, and that it’s just as likely he’s a 70+% shooter as it is that he is in the low 60s.
Is this a big deal? Hell yeah! First of all, Ben is taking a lot more free throws, up more than 25% from last year, to 5.7 from 4.5. We don’t know if that’s due to greater confidence at the line, playing more in the post, or just to his general offensive improvement, but confidence at the line is likely to help at least a little and maybe a lot going forward. Since we can’t quantify that, though, let’s just assume 6 FTA per game next year based on modest further improvement, increased respect from officials, and so forth. And let’s say Ben is a 70% FT shooter next year, which is my expectation (granting that I’m a bit of an optimist, so feel free to use your own expectations in the calc below if they differ from mine by a lot). An extra 14% (70% - 56%) times 6 shots a game compared to last year is .84 extra points per game. That is huge; an extra .84 points per 36 minutes, or 1.12 points per 48, is the difference between being top 100 and top 50, or between top 50 and top 25, or between top 25 and top 15. Again, this is just from improving foul shooting, with no other effect on his game, and with no special bonus for the benefit in late-game situations of being able to step confidently to the line.
Now, the statistics may support the idea of a big improvement in Ben’s true level rather than a fluky month. But what about the videotape? If the shot looks exactly the same but has just bounced in more, it’s probably luck. My personal shot guru is Eric Sidewater. I asked him what was going on and he explained that Ben doesn’t use his lower body in shooting free throws. (New readers should understand that I am legally blind and so can no longer see the games; any errors in what follows are most likely to be a consequence of me misunderstanding what Eric told me.) The best shooters get their power from their legs by bending their knees and then pushing their whole torso upward. Ben uses little knee flex at the line and so he instead would catapult his shot toward the rim with his arms. As anyone who has tried it will attest, this is not a very accurate way to shoot a basketball. In order to avoid the small risk that this article will come in at less than 2,000 words, let me tell you something cool I learned about catapults. In olden days aristocrats built castles with thick outer walls. Inside the castle was a virtual city which housed a military encampment that included many cavalry (soldiers who fight on horseback) and dragoons (soldiers who ride into battle and then fight on foot). With such a setup it was possible to control a wide swath of territory with a population many times the number who lived in the castle. Partly this was because the men in the castle were trained soldiers while the farmers and townsfolk in the surrounding territory were not. Partly it was because the soldiers had quality weapons and armor, while the weaponry allowed to others was often limited. But mostly it was because the lord could send a group of soldiers to a particular farm or village to collect taxes, threaten destruction, or whatever, at any time. In such an engagement the troops would have not only the training and weapons but also the numerical advantage, as they arrived in a hurry and left before a countervailing force could gather, returning to the safe interior of the castle walls. Of course all that and everything below is massively oversimplified; for example, many castles had outer and inner walls; the outer walls could protect local citizens from invaders while a much smaller fraction of the population were allowed inside the inner walls.
For every strategy their is a counter strategy, and against castles there were several. Among them were:
1) Storm the castle walls; i.e. attempt to climb over them, or break through the gate, etc. Against this approach the defenders employed high walls, moats, murder holes into which boiling oil could be dropped, archers, etc. etc.
2) Siege — just assemble a gigantic force outside the castle and kill anyone who leaves, continuing the vigil until those inside run out of food or other essentials, at which point they will sue for peace. Castles would lay in stores to guarantee that a siege could not be successful unless it lasted a very long time; only an extremely wealthy and determined opponent could maintain a sufficiently large siege force for years, especially in places with bitter winters; after all the besieged were warm and dry in the castle! And any less-than-enormous force could simply be overwhelmed by an attack from within.
3) Catapult — basically, launch boulders at the castle walls until a part is knocked down, at which point the inhabitants are vulnerable to attack. In medieval times the most common type of catapult employed for this purpose was a trebuchet, which uses a counterweight to launch the ordnance. The word “catapult” is usually associated with devices that use spring tension to launch, like the catapults portrayed in old cartoons, though technically the word catapult includes both trebuchets and tension devices.
So imagine a rolling cart with a long plank, like a seesaw. On the “back” — farther from the castle — end of the plank will go the boulder or other heavy object to be launched. On the other end will be piled a much heavier set of stones. The heavy end does not pull down, though, because it’s suspended with a rope, perhaps attached to a convenient tree, or to a structure erected for this purpose either as part of the cart or next to it. At the appropriate moment, the rope is severed, the heavy end drops, and the ordance takes flight on its destructive path.
OK, here’s the cool part: a few years ago historians, physicists and other interested parties got to arguing about whether the boulder would fly farther if the trebuchet had wheels on it. Or, supposing the cart had wheels just for mobility, whether it made sense to put chocks under the wheels before firing, or whether the rock would go farther if the wheels were left to spin free. Compelling cases were made in both directions, and eventually there was nothing for it but to get empirical. They went out and built a medieval-style trebuchet and launched rocks with the wheels free and blocked. If you think you know which is better, go ahead and guess now — farther with wheels free, farther with wheels blocked, or about the same?
Turns out the ordnance flies farther — much farther — if the wheels are left free so that the cart can lurch backward during launch. Once I knew the answer it seemed obvious — conservation of momentum and all that — but believe me that before they tested it, there were many very smart people who thought that free wheels would be less effective, not more.
Anyway, catapulting is good for knocking down castle walls but not so great for accurate free-throw shooting. Many have called for Ben to start bending his knees more at the line, but this has yet to occur. If any real journalists are reading this... um, could you please ask Ben why not? Does he think that all those 85% FT shooters are succeeding in spite of the knee bends, and that no-bend is a superior approach? Do his knees hurt when he bends them? Is he too tired? Inquiring minds want to know!
So that’s the bad news, no knee flex. But the good news is, Ben and the team have found a workaround: the wrist flop. Essentially, Eric explained to me, instead of slinging the ball with his whole arm, Ben now flicks the ball toward the hoop with his wrist, the way you or I would toss a crumpled ball of paper (an “origami boulder,” in the classic formulation of Matt Groening) toward the waste-paper basket. If I tried to shoot a basketball that way it would land about 13 feet short of the hoop, but that is because I am an ordinary, perhaps less than ordinary, human. Whereas Ben is a powerful giant of a man, capable of launching a 22-ounce, 29-inch ball with the easy flinch that for me would move only a fraction of an ounce of paper.
Anyway, it’s working, so thanks and kudos to Ben and whatever members of the coaching staff worked with him in developing this new approach. Not only is his effectiveness way up, but since the approach changed very recently — around January, according to Eric — it’s likely he has additional room for improvement as he masters this new way of shooting his freebies. Again, I’m trying not to allow my optimistic nature to overcome my scientific nature, so I’ll refrain from further predictions. But it really is not beyond the realm of possibility that it’s the past 5-10 games, up in the mid-70s, that are the most accurate harbinger of things to come. And even if we aren’t that lucky, I can honestly say it will be a major surprise if Ben goes back to shooting free throws in the 50s. So, that’s one flaw in his game that has seen a significant repair. Can the corner threes be far behind?!