Take a striker whose true scoring rate is 0.45 goals per 90, the level of a typical starting forward. Goals are rare and random, so what we see in the stats depends on how many matches we observe. Below is the distribution of outcomes for the same player after 5 matches (450 minutes) and after 30 matches (2,700 minutes), computed from a Poisson distribution.
After 450 minutes, one in five such strikers (19%) shows 0.8 goals per 90 or more and looks like a league leader, while one in three (34%) shows 0.2 or less and looks like a player to replace. After 2,700 minutes a rate of 0.8 or more occurs in fewer than 1% of cases.
How we correct small samples
We shrink the profile rating towards the league mean in proportion to its uncertainty. Reliability grows with minutes: it reaches one half at about 220 minutes (estimated by bootstrap), 82% at 1,000 minutes and 90% at 2,000.
In practice, a player with a raw rating of 80 in a league averaging 50 gets about 70 after 450 minutes, about 75 after 1,000 and about 77 after 2,000. The ranking stops rewarding short hot streaks.
Where we set the threshold
For a full season our benchmark uses players with at least 1,000 minutes. During a season in progress we require at least 30% of the minutes available to the team. Below these thresholds we do not discard a player, but we flag the result as preliminary and state how many more minutes are needed.
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