I'm trying to estimate how many people participated in a game competition and what my percentile score is based on some public data provided by the host.
The data I have is:
- Top 50 leaderboard (Let's say the 50th player scored 940.6)
- top 5th (699.0), 20th(475.3), 40th(214.8) and 70th(107.0) percentile score
- A (not that clear) curve Showing how many player have achieved a certain score (X axis is only partially annotated.)
(1) Assuming that the final score distribution is a normal distribution, is it possible to estimate the whole distribution?
- Giving a score, is it possible to estimate the percentile score?
- Is it possible to estimate the total entries?
(2) This curve doesn't look like a trivial normal distribution. What's the best guess of this distribution? Can it be two distributions added up? If this is the case, is it possible to separate them just using these data we have?
Edit: I'll give a bit more details how this data is formed. Each player plays in a succession of 3-player match which yields a pre-adjustment zero-sum result (ex. +50, +5, -55 or +50, -15, -35). The final score of a player is generated by his or her best performance summed up in any consecutive 8 games, adding a bonus +60 to each game played (Which means that when a player didn't finish 8 games s/he still qualifies but get less bonus score).
A good simplification is that 3 players play in a poker-style game and got their earnings/lost at the end of the game summed up with an additional bonus/penalty based on their positions. Although players only carry certain chips into a game and when any one depleted their chips or had certain rounds finished the game concludes, negative chips do carry over so scores can be arbitrary large but 99% of the game won't have anything over +/-80. Players got randomly matched.
The game is actually 3 players japanese riichi mahjong. (https://en.wikipedia.org/wiki/Japanese_Mahjong)
Example 1: -50 +50 +50 +50 +50 +50 +50 +50 +50 => Total +400 (Best possible sum at Games 2-9; 1-8 only yields +300 and are not selected) +480 (Finished at least 8 games) = +880
Example 2: -50 +50 => Total +0 (Games 1-2) + 120(Finished 2 games) = +120
Players play minimum 1 game to qualify. The best guess to distribution of how many games a player played is a binomial distribution.