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Say I work out the mahalanobis distance 'D' to measure the separation between two objects (which aren't normally distributed).

Say I now want to use 'D' against some critical values to decide if it's an outlier or not.

I've read that using Chi-Square Distribution is one way, using N-1 degree of freedom and converting the distance to Chi-square p values. However, it states that because isn't normally distributed some conversion is recommended.

In cases where the predictor variables are not normally distributed, the >conversion to Chi-square p-values serves to recode the Mahalanobis >distances to a 0-1 scale. Mahalanobis distances themselves have no upper >limit, so this rescaling may be convenient for some analyses.

In my case, where I have one distance 'D' and I can't re-scale it, is using the above still advisable?

If if it advisable, could one briefly explain the conversion where e.g., N=2 (used in degree of freedom N-1?) to get some critical value and what it represents?

If it isn't, what alternative method could I use to generate some critical values?

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You need to have some assumed distribution, normal or not, in which to compare your 'D' to. If you only have one D and no other information, then there's no way to tell whether it's an outlier or not. If you have information about the population distances then your first step would probably be to plot the distances in a boxplot, then overlay your one D and see where it lies in terms of a percentile. Then you could use that information to deem whether it's an outlier in the practical and contextual sense.

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To my mind, you have a distance $D$ and a number of degrees of freedom $N$, and you just have to compute the value of the chi-squared distribution with these two parameters.

For instance in MATLAB you can use:

p = chi2pdf(D,N);
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    $\begingroup$ This does not address the problem raised in the post: when the underlying values are not normally distributed, it doesn't appear that the chi-squared distribution is relevant. $\endgroup$ – whuber Sep 23 '15 at 16:49

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