Relative fluctuations of an extensive observable ($\sqrt{\mathrm{Var}(A)}/|⟨A⟩|=O(N^{-1/2})$)

If a total observable has a nonzero extensive mean and its variance is proportional to particle number, its relative standard deviation falls as $N^{-1/2}$. Short-ranged correlations away from a critical point often justify this scaling. A vanishing mean, a shifted energy reference or anomalously large correlations requires separate treatment.