Luck, skill, and the long run

A striker can’t stop scoring. Eight goals in ten games, every paper calls him unplayable, the manager says he’s reached a new level. Then the goals dry up — three in the next ten — and the same papers ask what’s gone wrong. Has he lost it? Almost certainly not. He was never as hot as the hot spell, nor as cold as the cold one. Both are the same thing: a true level, plus noise, watched over too few games. The name for what just happened is regression to the mean, and it is the most misread pattern in all of sport.

The mean is the gravity

Every player and team has a true level — the rate they’d settle at over a huge number of games. Call it their mean. Real performance bounces around that mean: some games the luck runs hot, some cold. Last lesson we split a result into skill plus luck. Now watch what luck does over time.

Here’s the key move. Luck doesn’t carry over. A great run doesn’t “use up” future luck, and it doesn’t predict more of it. So after an unusually hot stretch, the most likely thing next is not more heat — it’s a stretch closer to the player’s true level, simply because the true level is where the gravity points. That drift back toward the mean is regression. It feels like cause and effect — “he peaked, then declined” — but no cause is needed. The average was always going to reassert itself.

This is why the cover jinx and the “second-season slump” are mostly illusions. A player lands the magazine cover because they just had a freak-hot run. What follows isn’t a curse; it’s the ordinary settling back down that was always coming. We notice it and invent a story.

The hot hand, measured

Take a shooter who is, truly, a 50% shot — over thousands of attempts, half go in. Will they sometimes sink six in a row? Of course. Flip a fair coin long enough and you’ll get six heads in a row too; it means the coin is hot for exactly nothing. The sixth make does not raise the seventh shot’s odds above 50%.

When researchers actually counted long sequences of shots, the streaks were about as long as plain randomness predicts — no shooter was reliably “hot” in a way you could bet on. The feeling of a hot hand is real; the predictive hot hand is mostly the mind seeing a face in the clouds. The lesson isn’t that form doesn’t exist. It’s that a short hot streak is weak evidence of a new true level — and far more likely the luck term, drawn high for a little while.

More games, more truth

If one game is a noisy sample, the fix is obvious: take more samples. The more games you watch, the more the luck averages out and the true level shows through. This is exactly why serious competitions are long.

Carry over last lesson’s two sides. City are a genuine 70 to Rovers’ 55, which works out to City winning a single game about 80% of the time — so Rovers pull off the upset one game in five. Now make them play a series instead of one match:

• One game: City win ~80%.
• Best of three: ~90%.
• Best of five: ~95%.
• Best of seven: ~97%.

The better team’s edge compounds with every extra game. A single match is a coin caught mid-flip; a seven-game series squeezes most of the luck out, so the result you get is much closer to the result that should happen. It’s the same reason a 38-game or 82-game league season is a far better test of who’s best than any one cup final. Cup upsets are thrilling precisely because the short format lets luck stay loud.

You can even see it in the numbers a season produces. A team whose true level is a 55% win rate will, over a single game, show either 100% or 0% — useless. Over 10 games their record might read anywhere from 4 wins to 7; over a full 82-game season it lands tight around 45 wins. The long sample doesn’t change who they are. It reveals it.

On the whole

Regression to the mean is one of those ideas that, once seen, you cannot unsee — and it reaches far past sport. The best-performing fund this year, the surgeon with the freak run of perfect outcomes, the school that topped the league table once: extreme results are extreme partly because luck stacked the same way, and extremes tend not to repeat. Reward the hot streak as if it’s the new normal, and you’ll be disappointed; punish the cold one as if it’s a collapse, and you’ll bail at the worst time.

The sober way to watch sport — and to judge almost any run of results, your own included — is to keep asking how big is the sample? A season tells you more than a month, a month more than a game, a game more than a moment. The mean is patient. It is always pulling everything back toward the truth, and the only thing that ever really beats the noise is more time.