A statistics question I haven’t had any luck with: Is it possible (or useful) to generate a coefficient of covariation analogous to a coefficient of variation?

If one forms a variance/covariance matrix from a set of data, it shows by the magnitude and sign which factors covary with which others. However, because the scales of the different factors may be different, the meaning of directly comparing several covariances is confused.

I know that the usual way of normalizing a variance/covariance matrix is by dividing the entries by the product of their component standard deviations, resulting in a correlation matrix. Fine. The entries in this matrix will range from negative to positive 1.

However, I also know that you can normalize a standard deviation by dividing it by the mean, producing a coefficient of variation. So is it possible to normalize the covariance by dividing its square root by the square root of the product of the means, assigning the result the sign of the original? This seems to be a normalization analogous to the coefficient of variation, which could perhaps be called the coefficient covariation. Doing this to my data matrix produces a result very similar to the correlation matrix, except that it is not confined to the range negative to positive 1–it has values generally larger in magnitude than the correlations, and seems intuitively to be preserving more of the original information, but I’m not sure exactly how to interpret this information…

I Googled “coefficient of covariation,” and didn’t find much–only a South African paper in the *Pakistan Journal of Applied Sciences* by Seeletse in 2001, which appeared to suggest using exactly this math to produce precisely this metric. Unfortunately, two middle pages of the five-page paper are missing from the available PDF!

A friend suggested that one possible reason that this metric isn’t common is that it might not be valid to compare means of two grossly different regimes in this way–*i.e*., Teslas compared to dollars, or number of kumquats in Borneo to electoral college votes. It might no be valid to derive a unitless covariation between such variables by computing such Tesla*dollar means. This is a point that would be interesting to investigate, but I don’t think it applies to my data: I’m looking at populations of individuals in different samples, so the units should be the same. I’m more interested in eliminating strictly *size* effects, rather than unit. Also, I could see that it might be invalid to compare grossly differently-sized samples in this way–1000 individuals vs. 10, maybe–but I’m just trying to eliminate the size component of middle-of-the-road densities from their covariations to leave strictly the covariance portion without scale differences.

Any thoughts?