Part One introduced the two of my teammates during my time playing basketball: A fantastic mid-range scorer (Derek Wolhowe, PPP of ~1.12 on his signature mid-range attempt) and a fantastic three-point shooter (Isaiah Zierden PPP of 1.38 on his three-pointers). The question that I’ll seek to answer in installment is how often do players of this caliber come about.

It’s been established that an elite three-point shooter in Isaiah Zierden can have a PPP of 1.38 on three-point attempts. But, because this was a Power Conference-bound prospect’s mark against high school competition, is this number within a realm of possibility for most players? To answer shortly, it’s rare, but it is actually a possible mark for the best of the best three-point shooters: In 2018-19, of the 855 Division I players that Synergy tracked as having at least 100 three-point attempts in the half-court, 17 of them shot at or above 1.38 PPP. When expanding this to Divisions II and III, even more shooters meet the mark. 25 of 548 DII players reach that bar, while 20 of 498 DIII players reach it. It’s rare, but those 62 players show that it’s absolutely an attainable goal, though a very lofty one.

By the same criteria, taking Derek’s The Wolhowe (his signature, mid-post face-up move from Part One), how many players in college last year scored at a 1.12 PPP on all their mid-range shots, or even just the ones from 17 feet and in? That’s an easy answer: None.

And that was across all levels of the NCAA, NAIA, and two-year schools. Not one college player scored at that rate, and that was only on jumpers 17 feet and in. In fact, among players who had the requisite 100 attempts, one of Reid’s clients and contributor to the site Ryan Bruggeman has the highest PPP on those attempts at 1.00.

I could continue with more examples of efficiency thresholds reached and not reached by shots from the mid-range or three, but instead, I’ll just supply a chart with some raw college data.

The minimum amount of possessions to qualify for the player pool was 100 total attempts from three, and, since there were only 15 players to take 100 mid-range shots from 17 feet and in across all levels of the NCAA, 60 total attempts from mid-range. All efficiency values are represented in the form of Points Per Play:

For those unfamiliar with the percentile system, it’s a measure of relative value. For players in the 99th percentile, it means they scored higher than 99% of the data population. For players in the 70th percentile, it means they scored higher than 70% of the data population, and so on.

The discrepancy in shot efficiencies are staggering, which is not a new development, with these being the most notable findings:

1,901 players across the NCAA attempted at least 100 threes in the half-court last season. A player was 126 times more likely to be allowed to attempt 100 threes in the half-court as they were to be allowed to take 100 mid-range shots from 17 feet and in.

The average three-point shooter was about as efficient as a 99th percentile mid-range shooter.

There’s the argument to be made that in reaching the top of the mid-range dataset, a player has much higher autonomy over his shot-selection than the average three-point shooter, as covered briefly in Part One. I agree strongly, and this is the great advantage of the mid-range shot when it can be made by a player at a high rate, but clearly the odds are stacked against any single player from reaching those levels of efficiency. And even at that point, players who are able to reach those levels of mid-range efficiency are likely to be even more efficient from three. Another common argument for the mid-range is that with a higher field goal percentage, mid-range jumpers are a less volatile shot on small samples on a game-by-game basis, compared to threes.

To illustrate, let’s just simulate just 5 shots attempts from both the mid-range and three, using the benchmarks of 90th percentile Division players as the input. Division Three represents a happy medium between players who work very hard on their skill sets, and the physical measurements and athleticism that can compare somewhat to the average high school player. For the mid-range shooter, Player M, the corresponding PPP of 1.03 is equivalent to a 51.5% shot from two. For the three-point shooter, Player T, a PPP of 1.31 is equivalent to a 43.7% shot from three.

First, here are the expected distributions of how many points we would expect to see scored by both player on just these 5 dissimilar shot attempts:

To walk you through the graph, the idea that a lower-reward, but higher-percentage shot from mid-range will outpace the impact of the more volatile stock of three-pointers quickly falls apart, even in a 5 shot sample size. The three-pointer distributions trails off stronger towards the right side of the graph, representing more points scored, than the upward bound the mid-range shot even allows. Actually, there is one final representation of the graph that will make this even easier to identify, the graph below shows the percentage chance of scoring at least X amount of points over the 5 shot sample for both shots:

This graph shows there is minimal team upside to a developing 90th percentile mid-range jumper if it comes at the cost of instead developing a 90th percentile three-point shot, especially in an era of basketball where even many high school guards are able to knock down threes at a high rate off the dribble. If there is no risk being averted in mid-range shots, and no dramatic upside like there are with three-pointers, why draw up sets and practice plans that prioritize, or even give equal considerations, to mid-ranges and three-pointers?

Now, obviously no coach can wave a wand and magically turn mid-range possessions into a three-pointer when many mid-ranges come off the dribble and most threes come off the catch, but, what a coach or skills trainer can do is value the three-pointer more in practice and scheme. That process will over time turn mid-ranges into threes, if the coach or skill trainer stays the course and provides the correct feedback when necessary, either to reinforce or curb their players’ shot selection and practice habits. The 0.28 PPP differential between the two shots sounds small, but using average Division III team offensive benchmarks, that small advantage gained over just 5 possessions a game would lead to an expected benefit of one more win in an average, 26-game season.

Finally, the most common reasoning I hear for working on mid-range shots for the average player is the argument that sometimes a game situation mandates that raw field goal percentage is more important than PPP. This is referring to the highest-leverage situations in games, either down a point or tied in the final seconds.

First and foremost, this is a rare situation. If 99.5% of offensive possessions don’t meet this situational criterion, why spend value gym time practicing for it. This is especially true when considering that spending that time to become even slightly more proficient at higher-volume skills would make being down less likely. Simply, the easiest way to ensure that a team enters into these high-leverage, late-game possessions less often in the actual games is to over-prioritize them in practice plans, reducing higher-volume skill sets to a lesser role via opportunity cost.

Second, even in these situations, using a win-probability calculator shows that equal efficiencies are about a wash until the final seconds, regardless of the point-value of made baskets. And remember, a 90th percentile mid-range shooter is equal to a 34% shooter from three in expected PPP of attempts. Using Mike Beuoy’s fantastic win probability calculator on his website Inpredictable, we can “simulate” these high-leverage situations quite easily for the NBA level, though I’m comparing apples to oranges slightly with numbers taken from NCAA percentiles in an NBA win probability calculator (this was a choice made only for consistency’s sake, inputting NBA numbers reaches the same general outcome with a similar selection criteria to determine the 90th percentile PPP’s).

Let’s take this situation: Down 1 point, 10 seconds to go, with possession of the ball. Mike’s win probability calculator gives us a 34.4% chance of winning this game. We’ll have to make some assumptions to keep the calculations clean, and just use the same 90th percentile shot benchmarks as before. First, let’s assume every miss is rebounded by the opponent. Second, let’s assume it takes 2 seconds for the shot to occur. When running these numbers, the team has an expected win percentage of 42.5% when taking the mid-range shot, and 44.7% when taking the three. The difference? Though the team is more likely to take the lead by a point with the mid-range shot, they only end up winning an expected 67.9% of the time the shot is made. With a made three, that number jumps up to 85.2%.

I often see this exact situation relayed, but with the percentages as 45% from mid-range, and 30% from three, both equal to a 0.90 PPP shot. The most important thing to remember is that in this hypothetical, the premise is taking a decidedly above-average mid-range attempt, and comparing it to a bottom-of-the-barrel three-pointer. On pretty much every team I’ve encountered that is routinely playing in competitive, late-game situations, there are many, many plays that can be drawn up to get a 30% (or higher) three-pointer, but often only one player capable of manufacturing a 45% mid-range shot. It’s a false equivalence on every level, except the equal PPP’s.

Like all of the thought-experiments described in this series, there are all sorts of ways that it could be improved (as well as further complicated), if this was to be used as a more air-tight simulation of game scenarios. However, the lesson is the same: Efficiency is almost always more important than just getting a bucket, even in the most high-leverage situations. It is also worth noting in a tie game, the result is flipped, with the mid-range shot having an expected win percentage of about 3% higher than the three.

All of this cycles back to an analytical philosophy outlined in Dean Oliver’s *Basketball On Paper*. Variance is an inescapable aspect of basketball, a team will always encounter some amount either favorable or unfavorable runs of luck at multiple times over the course of the entire game. Going back to Part One, there is very little that can be done to trump a decently-sized, single-game PPP advantage in a single game, but when facing an underdog scenario, often the best strategy for a team to employ is a higher-variance one, even if it has a lower-expected, but higher-upside PPP than a less-risky one. The inverse is also true, as teams who enter a game as the favorite should consider adopting less risk-averse strategies.

Now, care needs to be taken to not adopt these strategies too hastily, especially when looking to reduce variance, as a team that finds itself obsessed with minimizing variance over an entire game will quickly find its overall efficiency plummeting. But, these late-game scenarios are perfect to show the relationship between when to invite additional variance in, and when to seek to further minimize variance’s effect. It’s common sense at a certain point, and best embodied in situation that coaches constantly toss at me: In the final minute and up 3, pushing the lead to 5 by embracing a higher field goal percentage shot is probably wiser than risking the lower-percentage shot to try to push the lead to 6. In this specific instance, the expected win-probability of 90th percentile, NBA shot attempts would have the mid-range shot as a higher expected winning percentage option by 0.5%, at that exact time of the game, for these shooting percentages.

So, the data supports taking a high-percentage mid-range attempt over a high-percentage three, in this niche instance.

The secret is to not let a niche instance completely dominate a team’s or player’s offensive principles.

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