Is there a standard definition of the Relative Variance? Wikipedia defines it as the square of the coefficient of variation, but some articles define it as the variance divided by the absolute value of the mean. I tend to like Wikipedia's definition best, because it is nondimensional, but I cannot find any thorough analysis of its properties.

$RV = \frac{\frac{1}{n} \sum\limits_{i=1}^n (x_i - \overline{x})^2}{\overline{x}^2} = \frac{\sigma^2}{\overline{x}^2}$

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    $\begingroup$ Perhaps the Wikipedia page has changed since this question was posted, but it does not currently define relative variance to be squared CV. I think the short answer is that no, there is no widely accepted definition of "relative variance". Presented with that term, I think most statisticians would ask: "relative to what?". I think you are free to define the term as you want, provided that you make your usage clear. $\endgroup$ – Gordon Smyth Jun 25 '17 at 6:10

At least part of the terminology problem is questionable self-consistency. For example, let us propose one plausibly self-consistent approach to relative variance. Such a system would have the property that the logarithms of the data are normally distributed, that is, that the errors of measurement are proportional type. Then, the anti-logarithms of the average logarithm of the independent or dependent variables would be good measures of location. Also, the relative measurements are then normally distributed such that transformed $\bar{x}_{new}$ and $\sigma_{new}$ become dimensionless.

In terms of the original data, for a proportional system, $$RV = \frac{1}{n} \sum\limits_{i=1}^n \left(\frac{x_i}{\bar{x}_{new}} -1 \right)^2 = \sigma^2 \; .$$ In a more general context, what relative variance means depends on the distribution types of the data, and, for some distributions, it is not as useful a concept as for other distributions.


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