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?