Is it possible to determine the average and median if we only have the upper part of a bell curve? Practical example: http://dota2.com/leaderboards shows the ~800 best solo ranked match-making scores in an Elo rating system. The lowest possible score is 1. There were 9318362 players last month, but not all of them participate in ranked match-making. From http://dota2toplist.com/statistics we can infer normal distribution; the site shows a self-reported small subset of players who are generally more knowledgeable about the game, so that curve is biased towards the larger scores.
Taking all of this into account, and assuming a participation quota of 5, 10, 15, … 100 percent of players in solo ranked match-making, can we find out what the unbiased global distribution looks like? How accurate would that model be?
 A: You can use a truncated maximum likelihood estimate assuming 800=$p\%$ top values of the population and $x_i$ are the 800 observed values:
Let $N=\lceil \frac{800}{p\%}\rceil \;\;x^-=\min\{x_i\},\;\; L(\mu,\sigma)=\left\{\Phi\left(\frac{x^--\mu}{\sigma}\right)^{N-801}\right\}\prod\limits_{i=1}^{800}\phi\left(\frac{x_i-\mu}{\sigma}\right)$, where $\Phi(\cdot)$ and $\phi(\cdot)$ represent the standard normal CDF and pdf, respectively.
Now, you can use a nonlinear optimization package (e.g., Excel Solver, Newtown-Raphson, EM, Matlab) to solve:
$\mu^*,\sigma^*=\max \limits_{\mu,\sigma} \left\{\log L(\mu,\sigma)\right\}$
The values will be your MLEs for left-censored data. Assessing the accuracy of the model will require a bootstrap procedure (I'd first try a parametric bootstrap):


*

*Simulate $N$ values from your fitted normal distribution (i.e. $\mu^*\sigma^*$)

*Take the top 800 values and re-calculate the (bootstrap) MLE's of $\mu,\sigma$ from that data

*Subtract $\mu^*\sigma^*$ from your bootstrap MLE. Record this difference as a "boostrap error".

*Do this 1,000 times or so.

*Look at the distribution of of your bootstrap errors. You can use percentile of this distribution to determine the approximate confidence of your results.

