Test for difference of quantized distributions I have a bunch of ratings, which are in the set $\{1, 2, 2.5, 3, 4\}$. I want to test the hypothesis that the ratings are different between two different populations. We can model these ratings as being the result of rounding a continuous measurement to the nearest rating in the set.
I could do this by using the usual Student's t-test and treating my data as continuously distributed, but I think this is only an approximation of the correct test.
In particular, the uneven spacing of the ratings makes this harder.
Question: Is there a more precise test for this?
 A: Overall, what you're looking for here is a goodness of fit test. 
Two tests which are suitable in your case are 1) Two-Sample Chi-Squared, or 2) Two-Sample Kolmogorov-Smirnov.
Chi-Squared is a test for categorical data and therefore you'd be treating each of your ratings as its own category. This ignores the ordinal nature of your data as well as the uneven spacing between the ratings. If I'm not mistaken, disregarding this information only loses you some statistical power, but if the difference between populations is large, you'll still catch it.
Kolmogorov-Smirnov works with 1d continuous distributions, so it would retain the ordinal information in the data. I'm less familiar with K-S, so I'd check its assumptions first. 
A: Gretton, Arthur, et al. "A kernel two-sample test." The Journal of Machine Learning Research 13.1 (2012): 723-773 introduces a few two-sample test using maximum mean discrepency for seeing if two samples are drawn from different distributions. 
M. Gutman, “Asymptotically optimal classification for multiple tests
with empirically observed statistics,” IEEE Transactions on Information
Theory, vol. 35, no. 2, pp. 401–408, 1989. 
uses some maximum likelihood techniques for multinomial distributions, and 
Unnikrishnan, Jayakrishnan. "On optimal two sample homogeneity tests for finite alphabets." Information Theory Proceedings (ISIT), 2012 IEEE International Symposium on. Ieee, 2012. provides some analysis for a two-sample $\chi^2$ test. 
