Why do I have to use variance and not mean value to calculate noise? Why, when I have to calculate some types of noise (for example shot noise and others which are in the denominator of SNR ratio) or attenuation from turbulence, I have to use variance of the of the density probability and not the mean value? In all of the formulas they use  variance (the second moment) and this is not clear for me.
Typically the first moment (mean) is zero and therefore contains (relatively) no information. The most obvious thing then, to distinguish different levels of noise, is to look at the variance.

And if the variance happens to be zero, then surely there is no noise at all.