Finding Maximum Dispersion of Total Number (N)

[KFC]

For some given statistics (e.g. Fermi-Dirac or Bose-Einstein), once we know the average number of particles at state r, it is easy to calculate the dispersion by calculating

$$\overline{(\Delta n_r)^2} = -\frac{1}{\beta}\frac{\partial \bar{n}_r}{\partial \epsilon_r}$$

and the total number of particle is just the sum of all average number of particles.

My question is: how do you find maximum dispersion of TOTAL NUMBER (N)?

I know
$$\overline{(\Delta N)^2} = \overline{(N-\overline{N})^2}$$

but how do you find the maximum?

 

[KFC]

For fermi stat., the total number of particles is
$$N = \sum_r \overline{n}_r$$

so how to calculate $$\overline{N}$$ ?