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?

  • $\begingroup$ It really should depend on what makes sense for comparisons in your case, I don't think you'll find one general "best" way. For example you could standardize the range or interquartile range of the variable. If your standard deviations are small enough and the variables are approximately normally distributed , then maybe it would make sense to use z scores anyway. $\endgroup$ – Chris Novak Jun 6 '16 at 12:36
  • $\begingroup$ What do you mean by standarizing them and what do you want to compare? E.g. a common way to standarize a variable is to convert it to z-scores, but afterwards comparing means is pointless since they are all equal to zero... Can you describe your problem in greater detail and provide some data example illustrating it? $\endgroup$ – Tim Jun 6 '16 at 12:47
  • $\begingroup$ Added a paragraph with an example to the question. $\endgroup$ – Krzysztof Jędrzejewski Jun 9 '16 at 15:17

Without knowing much about your underlying process, I'd suggest fitting a distribution with a limited domain, e.g. Beta distribution. You can get different shapes that match your description of data: enter image description here

Then you can compare the parameters of the distribution. Otherwise, it's pointless to compare the data that "looks like normal distribution" around 50% with data that is concentrated (limited from above) at 1.

  • $\begingroup$ I don't want compare distributions but values between distributions. E.g. (as in example i've described above) I'd like to compare whether students perform better or worse in different tests (regardless of difficulty of tests). $\endgroup$ – Krzysztof Jędrzejewski Jun 9 '16 at 16:03
  • $\begingroup$ If you'r comparing samples with very different distributions, then typical ANOVA may not work as desired. $\endgroup$ – Aksakal Jun 9 '16 at 16:05

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