# Measuring relative variability for variables with different scales II

I'm reformulating this question to see if I might have better luck than OP did at encouraging a response.

Consider that you have two univariate datasets at different scales, and need to establish some relative measures for their variability, by which in this case we mean a measure of how much variance each dataset displays in terms of its own scale such that the respective statistics can be usefully compared.

One obvious measure is Gini coefficient of inequality. But what other equivalent/alternative methods exist?

I've been wondering about one such alternative. The Z-scores of two datasets denote their values in terms of their respective standard deviations (SD) - a useful way to normalise two datasets of different scales to make them comparable. Hence it seems to me the median of absolute deviation (MAD) of Z-scores might be considered a useful relative measure of variability. Therefore a function f(x) 1 - MAD(zscore(x)) would be almost directly equivalent to Gini coefficient, since it would yield values in the [0,1] range in an equivalent way to similar distributions. The only exception as I see it is the need for a logical safeguard to protect against cases of zero variance - since Z-scores are invalid (infinite) in such sets. Does this approach have an established name?

Apologies in advance if this is crudely formulated question - these ideas are a little beyond my grasp of mathematical notation.

• In relation to the first question, See wikipedia, or here, here, here, or here, though as Nick Cox points out in that last one, a number of the popular indices are simple transformations of the Gini. Mar 25 '19 at 3:03
• Ah some useful looking reading here - thanks Glen. Mar 25 '19 at 10:31