Normalizing standard deviation


This is a simple question, I think, but one that I am too rusty now to deduce a clear answer to.

If I have a mean (m) and associated standard deviation (s), and I then normalize m (in this case by dividing by an area), is there any justification for directly normalizing s (i.e. just dividing the s by the area as well, rather than going back a step, normalizing the original data and then recalculating m and s).

Looking at the equation for s, if I replace m with m/k (where k is the normalizer) I can see no way to factor out k such that it would justify the notion of normalizing s by s/k.

Even if I am technically correct about not being able to simply normalize s as I normalize m, is this method ever used for approximate reporting of normalized s?
Am I asking this in the wrong place? This is not a course or homework question, just a general statistics question that come up during a recent review of a paper...
There are different meanings to "normalization," the most basic of which is the z-score normalization: \(z = \frac{\bar{X} - \mu}{\sigma}\). Of course when you're looking at multiple variables, things can become very complicated very quick. By dividing the mean by an area, you are getting some sort of density: average per unit area. If you are looking at single variable data, then yes, sigma may also be divided by the area to get a standard deviation of the density.


Probably A Mammal
The standard deviation is a measure of variation in your data, how it is dispersed. Unless you're interested in how your data varies on a per-area basis, I don't see why you would want to vary your standard deviation. I'm not entirely against that, though. More likely I would consider transforming your variable of interest. Suppose I'm observing some X. What you're literally asking for is to not look at X per standard deviation. You want X per standard deviation per some area. That seems odd. Instead, look at the standard deviation of X per some area. Normalize X and then look at it's z-score, for instance (i.e., center from its mean and normalize that by standard deviation). I'm doing that with some of my own data at work because how much energy somebody uses alone may not be as meaningful as the amount of energy they use per square foot. These two variables have clearly different results when it comes to their locations (mean, median, interquartile range) and variation (standard deviation, mean absolute deviation, etc.).

In short, the mean and standard deviation describe a univariate (single variable) distribution in terms of central location and dispersion of the data. You don't normalize those properties. You normalize the variable you're interested in and observe new properties (mean and std, e.g.). Make sense?