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I was reading a paper where the author mentions: "The coefficient of variation is primarily a descriptive statistic, but it is amenable to statistical inferences such as null hypothesis testing or confidence intervals. Standard procedures are often very dependent of the normality assumption and current work is exploring alternative procedures which are less dependent on this normality assumption."

The paper is from 2010. So my question is: has there been any recent "advancements" in terms of a coefficient of variation statistic that is not dependent upon the normality assumption? And, is this what one may call a non-parametric CV?

StatsScared
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2 Answers2

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The coefficient of variation is not strongly associated with the normal distribution at all. It is most obviously pertinent for distributions like the lognormal or gamma. See e.g. this thread.

Looking at ratios such as interquartile range/median is possible. In many situations that ratio might be more resistant to extreme values than the coefficient of variation. The measure seems neither common nor especially useful, but it certainly predates 2010. Tastes vary, but I see no reason to call that ratio nonparametric; it just uses different parameters.

A much better developed approach is to use the ratio of the second and first $L$-moment. The first $L$-moment is just the mean, but the second $L$-moment has more resistance than the standard deviation. Start (e.g.) here for more on $L$-moments.

Whenever the coefficient of variation seems natural, that's usually a sign that analyses should be conducted on a logarithmic scale. If CV is (approximately) constant, then SD is proportional to the mean, which goes with comparisons and changes being multiplicative rather than additive, which implies thinking logarithmically.

Note: The paper cited starts quite well, but then focuses on testing the CV when the distribution is normal. As above, if the distribution is normal, then the CV seems utterly uninteresting in practice, so the emphasis is puzzling to me. Your inclinations may differ.

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Nick Cox
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    Thanks, for the info on L-moments, very interesting. Two things, though: i) if the CV is associated w/ the log-normal distribution, wouldn't this also hold for the normal distribution since if X is normal, then exp(X) is log-normal; and ii) can you explain what you mean by "whenever the coefficient of variation seems natural"? – StatsScared Aug 28 '15 at 18:37
  • The CV is not necessarily associated with i.e. the log-normal (or any other distribution). I believe Nick's point is that the CV is interpretable for a log-normal distribution, and not for a normal. From the linked thread: "In principle and practice the coefficient of variation is only defined fully and at all useful for variables that are entirely positive." – jtobin Aug 28 '15 at 19:04
  • @StatsScared This may seem perverse but I really don't want to define or explain "natural" as if it were a technical term: it is not technical at all. It just means that an idea helps analysis because it matches something about the generating process, to a good approximation. If raspberry bush heights and redwood tree heights have roughly the same coefficient of variation despite very different SD and mean, then the CV is an idea that is of scientific use for such data. Economists too find some use for the CV of incomes, which also are often best thought of logarithmically. – Nick Cox Aug 28 '15 at 19:17
  • @jtobin can you expand on "CV is interpretable for a log-normal distribution, and not for a normal". This notion is still not clear to me. – StatsScared Sep 02 '15 at 17:58
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    @StatsScared The CV is a mean-standardized measure of dispersion; if the mean is 0 or less than 0, what useful information does the CV communicate about dispersion? A normal distribution can have a pathological CV in this sense, whereas a log-normal distribution cannot (its mean must be strictly positive). – jtobin Sep 02 '15 at 22:02
  • Loosely, normal distributions arise from additive processes; lognormal from multiplicative processes, and it is the latter where SD/mean makes most sense. – Nick Cox Sep 02 '15 at 22:07
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The coefficient of variation $\sigma / \mu$ itself has no normality assumption - just that the first two moments of the distribution are computable. I think it's a nonparametric test for the CV that you're looking for.

Googling "nonparametric coefficient of variation test" turned up a variety of papers on tests, depending on what kind of test you might be interested in. Ex:

jtobin
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