Why is the coefficient of variation not valid when using data with positive and negative values? I can't seem to find a definitive answer to my question.
My data consists of several plots with measured means varying from 0.27 to 0.57. In my case, all data values are positive, but the measurement itself is based on a ratio of reflectance values that can range from -1 to +1. The plots represent values of the NDVI, a remotely derived indicator of vegetation "productivity". 
My intention was to compare the variability of values at each plot, but since each plot has a different mean, I opted for using the CV to gauge the relative dispersion of NDVI values per plot.
From what I understand, taking the CV of these plots is not kosher because each plot can have both positive and negative values. Why is it not appropriate to use the CV in such instances? What would be some viable alternatives (i.e., similar test of relative dispersion, data transformations, etc.)?
 A: Think about what CV is: Ratio of standard deviation to mean. But if the variable can have positive and negative values, the mean could be very close to 0; thus, CV no longer does what it is supposed to do: That is, give a sense of how big the sd is, compared to the mean.
EDIT: In a comment, I said that if you  could sensibly add a constant to the variable, CV wasn't good. Here is an example:
set.seed(239920)
x <- rnorm(100, 10, 2)
min(x)#To check that none are negative
(CVX <- sd(x)/mean(x))
x2 <- x + 10
(CVX2 <- sd(x2)/mean(x2))

x2 is simply x + 10. I think it's intuitively clear that they are equally variable; but CV is different.
A real life example of this would be if x was temperature in degrees C and x2 was temperature in degrees K (although there one could argue that K is the proper scale, since it has a defined 0). 
A: I think of these as different models of variation.  There are statistical models where the CV is constant.  Where those work one may report a CV.  There are models where the standard deviation is a power function of the mean.  There are models where the standard deviation is constant.  As a rule a constant-CV model is a better initial guess than a constant SD model, for ratio scale variables.  You can speculate on why that would be true, perhaps based on prevalence of multiplicative rather than additive interactions.  
Constant-CV modeling is often associated with logarithmic transformation.  (An important exception is a nonnegative response that is sometimes zero.)  There are a couple ways to look at that. First, if the CV is constant then logs are the conventional variance-stabilizing transformation.  Alternatively, if your error model is lognormal with SD constant in the log scale, then the CV is a simple transformation of that SD.  CV is about equal to log-scale SD when both are small.  
Two ways of applying stats 101 methods like a standard deviation are to the data the way you got them or (especially if those are ratio scale) to their logs.  You make the best first guess you can knowing that nature could be rather more complicated and that further study may be in order. Do take into account what folks have previously found productive with your kind of data.  
Here's a case where this stuff is important.  Chemical concentrations are sometimes summarized with CV or modeled in a log scale.  However, pH is a log concentration.
