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I'm measuring distances of various samples from a reference point. The distance is defined as a non-negative number, where $d=0$ means that the test case is identical to the reference.

My general question is: Given a set of "typical" distances, what is the proper way to tell whether a given $d_1$ "too large", compared to the "typical"?

In my particular case the distance distribution is shown on the following graph

enter image description here

I failed to transform these data to anything symmetrical, so I can't use normal approximation. Any suggestions?

chl
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Boris Gorelik
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2 Answers2

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My first instinct is to say that it would be silly to make such a determination absent any knowledge of the topic. "Too large" for what, or for whom? But perhaps what you're looking for is really a test for outliers in the distribution--not that you're likely to find any in the one you've shown. Check out Dixon's Test for Outliers (sometimes called the Q-Test). I'm not thrilled with what Wikipedia provides, so you might want to check around further than that. Sorry I don't have a good web reference; I use the guidelines in the book 100 Statistical Tests by Gopal Kanji.

rolando2
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  • From what I read (e.g. http://www.chem.uoa.gr/applets/AppletQtest/Appl_Qtest2.html), Q-Test assumes normal distribution of the data. In my case this assumption is most probably wrong (see the asymmetric shape of the histogram). – Boris Gorelik May 15 '11 at 08:04
  • @bgbg - You're right, for p-values to be exactly correct, you need a normal distribution. With your case and its "outliers," the distribution is slightly-to-moderately skewed. I think you could make a convincing argument that if p is nowhere near your alpha and is, say, .5, then it would not fall below your alpha under a normal distribution either. I was trying to hint earlier that you really don't need to run a test, since the lack of outliers is so apparent. – rolando2 May 15 '11 at 19:39
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Can you not use the empirical distribution's 95% (or whichever you prefer) confidence limit? If your sample size is big enough, this ought to be a reasonable approximation.

Nick Sabbe
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  • I thought about such an approach, but this would mean that we assume a-priory that 5% of the existing observations are "atypical", "faulty" etc. Which might not be the case – Boris Gorelik May 15 '11 at 07:57
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    Well, if you're not willing to make other assumptions (like normality), it's all you really have... How else would you define "atypical"? – Nick Sabbe May 15 '11 at 15:36