Can I calculate how good I am if I know my score and the average score?

Question

If I know the average score of all participants (46/100) and I know that there are ~1500 participants, and my score is 90/100, can I somehow calculate what percentage of people I’m in? (Eg top 10% of scores) If not, then what extra data would I need?

Answer 1

1. Think of the worst-case scenario (as a pessimist): You want to maximize the number of people who got more than or equal to 90. It is easy to realize that the best we can do in this case is to assign 90 to as many people as possible, and 0 to the rest, such that the average remains equal to 46.

   If $x$ get 90 and $1500 - x$ get $0$: $$average = \frac{90x + 0\cdot(1500-x)}{1500} = \frac{90x}{1500} = 46 \implies x = 46.1500/90 = 766.66$$.

   So, in this case around 51% of people got the score as good as yours.

2. Think of the best-case scenario (as an optimist): You want to minimize the number of people who got above or equal to 90. Again, it's easy to see that it might be the case that only you got a score of 90, and everyone else got lower than that: we can assign 1492 participants a score of 46, 1 participant gets 2, and only you got 90.

   In this case, $ average = (90 + 2 + 1492\cdot 46)/1500 = 46$ which is valid.

   So, in this case, only 0.066% (that is just you) got the score as yours.

So, without any additional data: you can be anywhere from top $0.06\%$ to $51\%$.