  We develop a theory for the compressibility and quantum capacitance of
disordered monolayer and bilayer graphene including the full hyperbolic band
structure and band gap in the latter case. We include the effects of disorder
in our theory, which are of particular importance at the carrier densities near
the Dirac point. We account for this disorder statistically using two different
averaging procedures: first via averaging over the density of carriers
directly, and then via averaging in the density of states to produce an
effective density of carriers. We also compare the results of these two models
with experimental data, and to do this we introduce a model for inter-layer
screening which predicts the size of the band gap between the low-energy
conduction and valence bands for arbitary gate potentials applied to both
layers of bilayer graphene. We find that both models for disorder give
qualitatively correct results for gapless systems, but when there is a band gap
at charge neutrality, the density of states averaging is incorrect and
disagrees with the experimental data.
