PhysWall inverts a closed, non-linear physical law at a single measurement point — and refuses when the inverse is not unique.
On the difference between a change and noise.
A player shot 44% over his last twenty attempts. Last season he was at 34%.
The easy answer is that he improved. The correct answer is that it is still not possible to tell.
A shooter who has not changed at all — same hand, same mechanics, same everything — looks like this over twenty attempts:
15% to 55%. Forty points, with nothing about him different.
⚠ This article said 18.9% to 57.7% until an outside reviewer found it. Those figures are a Wilson interval, which answers “I observed 36% — what true rates are consistent?” The question here is the reverse: the true rate is 36%, so what can be observed. That is the binomial, and it gives 3 to 11 makes out of 20: 15% to 55%.
⚠ And the reason this article carried the old numbers after /streaks was corrected is worth stating: nothing checks the content of this page. It is linked from /puzzles, it is in the bundle, and the only test that mentions it checks that the file exists. Same claim, two pages, one corrected.
attempts 95% range width
20 15.0 - 55.0 40.0
50 24.0 - 50.0 26.0
100 27.0 - 46.0 19.0
250 30.0 - 42.0 12.0
500 31.8 - 40.2 8.4
750 32.5 - 39.5 6.9
1500 33.6 - 38.4 4.8
Only around 750 attempts does the band narrow to seven points — tight enough to say that a 36% shooter is not a 40% shooter.
Seven players in NBA history have taken 750 three-pointers in a season.
So for almost every player, almost always, a season three-point percentage is not a stable measurement of anything.
The 750 figure is from Blackport (2014) and has been known in the analytics community for a decade. What happens in practice is that analysts know it, and broadcasters, agents and front offices go on quoting samples of twenty.
If the sample is large enough and the gap survives, what causes it?
In basketball there is a geometric answer. Every shot is a ballistic arc, and given a release angle and speed, physics settles whether the ball goes in. The question becomes: what fraction of shots land inside the rim's tolerance window?
σ = the shot-to-shot scatter in release speed.
Nobody measures it. It can be derived from the shooting percentage, backwards through the same geometry:
42% shooter σ = 1.21% league, 36% σ = 1.43% 30% shooter σ = 1.74%
Half a percentage point of hand steadiness covers the whole distribution of NBA three-point shooting.
Optical tracking of 49,562 free throws puts the real scatter about three and a half times larger. A ball that touches the rim often still drops, and this model folds that forgiveness into σ instead of into the window.
In tracked units the same spread is closer to two points. Still a narrow band. Not the same number.
Two questions that a percentage cannot answer on its own: when you are allowed to say he improved, and which part of the gap is mechanical.
The distinction matters. A shooter whose open looks match the arc has a shot-selection problem rather than a mechanical one, and those need entirely different work.
Not that the mathematics is new — Wilson intervals, Blackport, and the shot-mechanics literature all got there first. And not that any of it has been measured against hardware: zero measurements of our own.
The figures here were checked against two external sources produced without knowing this exists. One of them caught three errors of presentation. They were fixed.
PhysWall was developed and architected by Gadi Zion.