The pattern
Pick the best performers from one period, and in the next period they usually do worse. Pick the worst, and they usually do better. Nobody has to have changed. This pattern, called regression to the mean, appears whenever results combine skill and luck and the selection is made on an extreme outcome.
The word “regression” can mislead. Nothing pulls a player back toward average like a force. The effect is a selection effect: the group chosen for an unusual result contains the players who were both good and lucky, and the luck does not repeat.
A worked example with no skill differences at all
Imagine 1000 players who are all exactly equal. Each wins a given game with probability 0.5, and each plays 10 games.
The chance that a particular player wins at least 8 of 10 is:
(C(10,8) + C(10,9) + C(10,10)) / 2^10 = (45 + 10 + 1) / 1024 = 56 / 1024, about 5.5%.
So roughly 55 of the 1000 players would post a record of 8-2 or better, purely by chance. They would look like a standout group and might attract attention, awards or rating gains.
Now have all 1000 play another 10 games. Every player still has a 0.5 chance in each game, so the 55 high performers average about 5 wins, the same as everyone else. Their group record falls back to 5-5 on average. Nothing changed except that the first sample was selected for luck.
This is an extreme case. In reality, players differ in skill, so part of an impressive record is real and part is noise.
How much comes back
A useful way to think about it: observed results are a mix of true skill and luck. The more of the variation between players that is due to skill, the more a good result carries over; the more that is due to chance, the more it fades.
If the correlation between a player’s results in one period and the next is r, then a rough expectation for the next period is:
next = average + r x (this period - average)
As a purely illustrative case, suppose a league’s average win rate is 50%, a player has gone 70% in a stretch, and the correlation between stretches is 0.3. Then the expectation is 50% + 0.3 x 20 points = 56%. Still good, but much closer to average than 70%. The size of r varies a great deal between games, stats and sample lengths, so this number is a demonstration, not a rule.
Short samples have low r because chance dominates them. Longer samples have higher r, so their extreme results are more trustworthy. This is why sample size matters so much.
A simple shrinkage estimate
One practical version of the idea is to blend the observed record with a prior belief that the player is about average. Treat the prior as worth a certain number of games.
Suppose a player is 14-6 over 20 games, and we count the average-player prior as 20 games at 50%, which means 10 wins in 20. The blended estimate is:
(14 + 10) / (20 + 20) = 24 / 40 = 0.60
The raw 70% is pulled to 60%. With 100 games at the same 70% rate (70-30), the estimate would be (70 + 10) / (100 + 20) = 80 / 120, about 0.667, much closer to the raw figure because more evidence has outweighed the prior. The choice of 20 games for the prior is a judgement call, so it is best seen as an illustration of how evidence strengthens as games accumulate.
Where it shows up
- Rating systems. A player who climbs fast on a hot streak is a candidate for a drop later, which is part of why K-factors and uncertainty terms exist in an Elo rating system.
- Best and worst lists. The top of any leaderboard built on a short window contains people who were lucky, and the bottom contains people who were unlucky.
- Coaching and changes. If a team makes a change after a terrible run, results may improve anyway. It is easy to credit the change when part of the improvement was simply a return to normal.
- Rookies and breakout stars. A strong debut is partly real and partly luck, so the next period is usually more modest.
Related ideas, and what this is not
Regression to the mean is different from the gamblers fallacy. In a sequence of independent events, a run of one result does not make the opposite result more likely next time. In the 1000 player example, each game’s chance stays at 0.5 throughout. What changes is that you chose a group because of a lucky stretch, and the group’s average goes back to the underlying value.
It is also unrelated to belief in a hot hand, the idea that streaks reflect a temporary boost. Streaks may sometimes reflect real changes in form, but a streak alone is weak evidence of it, particularly in a short window.
Practical habits
Judge players and teams on longer records where possible. Look at how many games lie behind a figure, expect extremes to soften, and avoid reading much into a change of fortunes without checking the sample first. When comparing before-and-after results of any change, remember that a team picked for being at its worst is likely to look better next time regardless.