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I have ~500 of (experimental) samples of data from a range [0,1], and I would like to see if the overall distribution is uniform or whether there is some clustering going on. I would have used Chi-Squared test as recommended here (https://math.stackexchange.com/questions/2435/is-there-a-simple-test-for-uniform-distributions), but there are two problems

  1. Each sample is small (3-10 datapoints), which is probably too small for a Chi-squared test
  2. The particular interval in which clustering may occur varies, e.g. for one samples it may be at ~0.2 and for another at ~0.5 etc
  3. I see some clustering using visualisations, but there is a small number of outliers so I can't rescale each sample based on the value of the first datapoint.

Is there a statistical test / some other approach that I could use?

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  • $\begingroup$ If you have a particular alternative model in mind, then fit the two and compare loglikelihoods. $\endgroup$ – kjetil b halvorsen Aug 4 '17 at 16:44
  • $\begingroup$ @kjetilbhalvorsen Thanks! The problem is I don't! I just expect that there'll be some clustering. Also even if I try something like Normal, do you think that I can get results with 3-10 points per sample? $\endgroup$ – GingerBadger Aug 4 '17 at 16:47
  • $\begingroup$ Is there any reason you can't combine all your samples into one big sample? The range is the same for all of them, so... $\endgroup$ – jbowman Aug 4 '17 at 16:58
  • $\begingroup$ @jbowman Unfortunately, no, as I suspect that the interval where clustering occurs is different for each sample (point 2 in the question), so I'm afraid that if I combine everything together, any clustering will be evened out. $\endgroup$ – GingerBadger Aug 4 '17 at 17:25
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    $\begingroup$ you could calculate the correlations with order statistics and the expected order statistics from a uniform sample same size, and find p-values by simulation. Then plot the p-values (which should be uniform if all he nulls are true). $\endgroup$ – kjetil b halvorsen Aug 4 '17 at 17:48
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At just 3-10 samples, you can't reliably "prove" clustering.

Unless (maybe) you get the almost exact same value every time, the test should not reject the null hypothesis, simply because of your tiny sample size. The chance of the second and third values being 0.1 close to the first is >10% each. So in >1% of cases, you will see three values within 0.1 - a naive user would call this a "cluster", but it is not statistically significant. In particular, if you have >500 data sets like this, you need to correct for multiple testing, so you don't get false discoveries.

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