English: Quantum occupancy nomograms, where for example when (ε-μ) = kT we find that f_FD = 1/(e+1) ≈ 0.269, f_MB = 1/e ≈ 0.368, and f_BE = 1/(e-1) ≈ 0.582 so that for a given value of (ε-μ)/kT, as expected, f_FD ≤ f_MB ≤ f_BE.

At top: The dashed lines denote filling fractions for Fermi-Dirac (FD) states whose energy ε is offset by integer-multiples of kT from the Fermi chemical-potential μ. In the FD case, the average number of particles per state and the fraction of states occupied are one and the same.

At bottom: The "low-density" Maxwell-Boltzmann (MB) approximation is a dashed grey curve which neglects chemical potential μ altogether, and it is also the fraction of Bose-Einstein (BE) states occupied. From bottom up the dotted curves denote contributions to the BE particle-average from those occupied energy-levels which have a population of one, two, three and four particles (or excitations), respectively.