# Computing Variability Independent of mean; differences in calculation methods

This question is mostly about another post here on stats.stackexchange. In general, VIM (Variability Independent of the Mean) is usually calculated, as the name suggests, to calculate the variability in a series of numbers independent of their mean. In most of the references, VIM is calculated by: VIM = SD / mean ^ x. In this formula, x is usually calculated by a fitting a non-linear model between mean and SD. For example, in R:

nls.vim = nls(sntd ~ k*avg^p, data=df, start=c(k=1,p=1))
summary(nls.vim)


In which x = p in the above model. Consequently, to my understanding, VIM should be calculated by fitting SD and mean of each individual and the acquired p:

VIM(i)=SD(i)/mean(i)^p


However, in the above mentioned post, VIM was calculated through the following formula (SD of each individual is raised to the power of p as well and the added constant k):

VIM(i)=k⋅(SD(i)/mean(i))^p


The code and the data and relevant references are already available in the linked post. Now I have three questions:

1. Why is SD raised to the power of p as well as mean? We don't see this approach in any of the cited references.

2. In some references,(including this one, also linked in the original post), the constant k is acquired by calculating the (mean of all means)^p. However, in the mentioned post, we see that he uses the k estimate from the summary of the non-linear model. Some references don't use a k at all. what is the difference between these calculations?

3. Is the approach used in the original mentioned post statistically correct? why does it seem different compared to other references?

I would appreciate any help, as I am new to these concepts and this may well be a misunderstanding on my part, however, I can't seem to find any answers on my own.