Sort $X$, then scale the first differences by $\bar X$: what, if anything, is this used for? 
Sort data vector $X$, take first differences of the sorted data, and divide by $\bar X$.

I came across this transformation in someone's notes, without any citation. It would be applied to non-negative data.
It seems odd to me but I can't put my finger on why. What, if anything, would it be useful for? What are the properties of the result?
 A: *

*Gaps in sorted data are called "gaps", "gap statistics" or "spacings" so if you divide by $\bar{X}$ you'd have a scaled gap statistic or a normalized gap statistic (or scaled spacing, etc). 
An unusually large value might indicate a "gap" in the distribution of values, suggesting clusters on that variable. 
(Using gaps to spot clusters is used with k-means clustering for example, but more normally applied to more than one dimension). Note that gaps divided by mean will be (as with coefficient of variation) unit-free. 
The gaps themselves are in the original data units, so for this kind of application, you'd want to scale(/standardize/normalize) in some way to remove the effect of the unit.
If the original values are approximately exponential, the mean would be an obvious thing to scale by -- you want to divide the gaps by the scale parameter of that exponential to standardize them, which parameter has $\bar{X}$ as its MLE; this would allow one to derive a statistic which would denote a surprisingly large gap for random exponentials. 
Even when it's not exponential (but is still positive), dividing by the mean will still remove the units from the distribution of gaps, at least making them comparable across different scales.  
Indeed, if the original values are uniform, the gaps have the same distribution (though the mean is then a less obvious thing to divide by in that case). The fact that the gaps have the same distribution would perhaps make it a more sensible starting point for a "looking for big gaps" type calculation.

*Somewhat related to 1: It might be related to some kind of goodness of fit statistic.
