Score Based On Number of Games Played, and Average Score for All Games Played

Question

I have two data points

X: Number of Games Played (lowest: 1 highest: 1000)

Y: Average of all game scores[sum of all game scores / number of games played] (lowest: 4 highest: 6.5)

I don't have record of the individual game scores, I only have the average.

How can i generate a score based on these two variables, to make a good comparison mechanisms between multiple entities that have different x,y values? in a sense that when [x=1,y=6.5] is worse[has a lower score] than [x=5, y=6.1] or something similar

Answer 1

One way to correct the average is to add some "dummy" low scores (e.g. $10$), so

$$ A' = {X \cdot Y + 4 \cdot 10 \over X + 10 }. $$

People with a large number of answers see their modified percentage alter very little from their real percentage, but people with relatively few answers will see their modified percentage move considerably toward low values.

This is known as "Bayesian averaging". In effect, the people with many answers will rank higher than people with the same percentage but fewer answers.