Nobody got worse. The luck just didn't show up twice.

Score far above average once, and your next score will almost always be closer to average — not because of any curse, correction, or "regression force," but because an unusually good score is usually part skill and part luck, and luck doesn't repeat. Drag the dial to change how much of a score is durable skill, and watch how much of any extreme performance survives to the next round.

What you're seeing

Every score here is built from two ingredients: a persistent skill that stays with a person from round to round, and a one-off noise term — a good bounce, an easy set of questions, a lucky day — that hits once and is gone. Round 1 mixes skill with its own noise; round 2 mixes the same skill with a completely fresh, independent draw of noise. The skill dial sets how much of the mix is durable versus one-off — at 100% every score is pure skill and nothing ever regresses; at 0% every score is pure noise and skill plays no part at all.

Now select the amber group: everyone whose round 1 score cleared the cutoff. Some of them are there because they're genuinely skilled. Some are there because their one-off noise happened to land high that round — and for that second group, nothing carries forward, because round 2 draws an entirely new, independent noise term. The result: the amber group's round 2 average is reliably less extreme than its round 1 average, by exactly the fraction of the score that was noise. This is true whichever cutoff you choose — the lenient one or the exceptional one — because it isn't a property of the cutoff. It's a property of how much of "extreme" was ever going to repeat.

Push the skill dial to 100% and watch the amber marker snap onto the dashed line: with no noise at all, there's nothing to regress away, and the selected group's round 2 average matches its round 1 average exactly. Pull it down toward 0% and the amber marker collapses onto the centerline no matter how strict the cutoff: however impressive a purely lucky round looks, the next one owes it nothing.

The rule, exactly. Round 1 and round 2, X and Y, are each built from a shared skill draw S and an independent noise draw (A for round 1, B for round 2 — all three standard normal), mixed by the dial ρ ∈ [0,1]: X = √ρ·S + √(1−ρ)·A, Y = √ρ·S + √(1−ρ)·B By construction X and Y are each standard normal on their own, and Corr(X,Y) = ρ exactly. For any selection at all — not just a single value but any cutoff c — the conditional-expectation law for jointly Gaussian variables gives E[Y | X > c] = ρ · E[X | X > c] Verified in node (improve/verify/86-regression-to-mean.js, 96 checks):

The truncated-normal-mean machinery is itself checked against raw numeric integration before anything is allowed to trust it.
The regression law is confirmed two independent ways across eight (ρ, c) pairs: a closed-form truncated mean that never reads a Y sample, and a direct Monte Carlo average of the selected group's actual Y values.
The law is symmetric — selecting on Y predicts X equally well.
Negative control at ρ=1: Y equals X exactly for every sample, so the selected group's round 2 average matches its round 1 average to machine precision. Zero regression when there is no noise left to regress away.

This models two independent measurements of one underlying quantity — the same idea applies to a great test score, an unusually good sales quarter, or a rookie season, not only games of pure chance.