Grand-canonical single-level occupation variance ($\mathrm{Var}n=\bar{n}(1\mp \bar{n})$)

For one independent fermion level, occupations 0 and 1 form a Bernoulli distribution and give $\mathrm{Var}n=\bar{n}(1-\bar{n})$. For one ideal boson level in contact with a reservoir, its geometric distribution gives $\mathrm{Var}n=\bar{n}(1+\bar{n})$. Both approach variance equal to mean at small occupation. Pauli exclusion suppresses fluctuations of nearly full fermion states, whereas large boson occupation gives variance of order mean squared. Fixed-total-number constraints change condensate fluctuation statements.