Using the coefficient of variation to detect feature importance - but what if mean and sd are correlated? I would like to have an estimate for the importance of a feature - the rough idea would be to define importance as the ratio between the mean and standard deviation. In other words, a feature is important if it has a high mean and a low standard deviation. So far so good - there is the coefficient of variation which is very close to what I need. In my case, however, the mean of each feature is highly correlated with the standard deviation. Using the coefficient of variation would therefore not be a good measure, because it would "reward" features with low mean too much but features with a high mean too less.
Here's a figure to get an intuition for my problem (the x-axis lists each feature the y-axis represents the value, lines are representing mean with +/- sd):

Is there a way to correct for this correlation between mean and sd?
EDIT: Each feature on the figure below is a brain region/node. Mean and SD represent energy values over various state transitions for each node.
 A: Coefficient of variation has severe problems in the best of cases, due to its assumption that the origin for the measurement is zero.  So CV is not to be used for variable importance.  Use a general measure such as explained variation, i.e., $R^2$ or pseudo $R^2$ or the proportion of -2 log-likelihood $\chi^2$ explained by a part of the model.  Other measures are described here.  These approaches also easily handle the case where the variable has more than one parameter in the model (e.g., quadratic effect).
A: Let's spell out that the coefficient of variation CV is SD / mean, whether as a number or a percent.
In the simplest case with say different groups and so different means and SDs it is a feature if SD and mean are correlated and even more of a feature if they are approximately proportional, say SD $\approx$ 0.2 mean, so we can regard the CV (here 0.2) as an informative summary. However, if the relationship is even a little more complicated, say SD $= a + b$ mean, then the CV isn't going to simplify anything much. (Often progress can be made if SD is a power function of mean.)
It turns out that what a constant CV implies is that you should be thinking on logarithmic scale. Simply, if variability is multiplicative not additive, then take logarithms (or use a logarithmic link function) to get to additive variability, which typically is easier to work with.
This is all tied up with the small print alluded to by @Frank Harrell: the CV isn't of much use unless all values are positive and there is an origin that isn't arbitrary. Much more at e.g. How to interpret the coefficient of variation?
However, I can't get to close grips with the context here. Do you have a model or are you just looking at lots of variables? What does "importance" mean? Your graph isn't clear to me, but I wouldn't be surprised at taking logarithms being a good step forward.
PS Other way round, if the CV is constant, then it doesn't discriminate. Nevertheless the implication is that the geometric mean carries all the information wrapped up in mean and SD. In practice, of course, it won't be exactly constant with real data.
