I'm working on dataset that contains a variable from 0 to 1 (0 - 100%). Distribution of the variable differs depending on the context (defined by another variable). Depending on the context the distribution of it may:
- be concentrated close to 0, e.g. [0, 0.05, 0.05, 0.06, 0.10, 0.15, 0.8]
- look similar to normal distribution, e.g. mean = 0.5, sd = 0.1, [0.31, 0.4, 0.46, 0.48, 0.5, 0.51, 0.55, 0.59, 0.72]
- be concentrated close to 1, e.g. [0.2, 0.85, 0.9, 0.94, 0.95, 0.97, 1]
Example of such variable may be test results across some tests having different difficulty. When a test is easy then most of students receive high results (90-100%), when it's hard than most results are low (0-10%). When its difficulty is well balanced, then e.g. most of results is 60-80%, but there are also significant numbers of results in ranges 45-60% and 80-95%.
I'm looking for a way standardize it, so I'll be able to compare values between contexts.
For now I've got an idea to work on percentiles of the aforementioned variable. But maybe there are smarter approaches?