# Appropriate distribution for bounded data set

I am designing a points-scored test. There is a limit on the maximum amount of points possible, as well as on the fewest amount of points possible. I have had a test group take the test and graphed their results, which form a lopsided bell curve or sorts. I am trying to figure out what kind of distribution would be best to describe this distribution, for the purpose of using it to calculate percentile scores for any future test takers. I would use normal, but intuitively, I feel that the highest percentiles would never be reached, since they would be beyond the upper bound of the data set. Would a truncated normal distribution or beta distribution be best? Any help, intuitive or direct, would be appreciated. I am mostly working in the R environment.

• Weibull looks pretty good (I simulated the density function evident in the OP's histogram with plot(density(rweibull(100000,shape=9,scale=36)))), but doesn't appear to fit the low observations. Even with a simulated sample of 10M, the lowest observation was 4.6. Mar 20, 2014 at 2:30
Of course, this is median() in the base package for R. Add na.rm=T if you have missing values.
Unless you're fitting a predictive model with distributional assumptions about the outcome that you haven't mentioned, I don't see much advantage to assuming a predefined theoretical distribution instead of just describing yours. Why not define your distribution empirically with density()? This should help you describe the probability of other scores, if you're not just interested in the most likely. Granted, this might not work ideally at the edges if it suggests nonzero probabilities out of bounds.