  We report the investigation of conductance fluctuation and shot noise in
disordered graphene systems with two kinds of disorder, Anderson type
impurities and random dopants. To avoid the brute-force calculation which is
time consuming and impractical at low doping concentration, we develop an
expansion method based on the coherent potential approximation (CPA) to
calculate the average of four Green's functions and the results are obtained by
truncating the expansion up to 6th order in terms of "single-site-T-matrix".
Since our expansion is with respect to "single-site-T-matrix" instead of
disorder strength $W$, good result can be obtained at 6th order for finite $W$.
We benchmark our results against brute-force method on disordered graphene
systems as well as the two dimensional square lattice model systems for both
Anderson disorder and the random doping. The results show that in the regime
where the disorder strength $W$ is small or the doping concentration is low,
our results agree well with the results obtained from the brute-force method.
Specifically, for the graphene system with Anderson impurities, our results for
conductance fluctuation show good agreement for $W$ up to $0.4t$, where $t$ is
the hopping energy. While for average shot noise, the results are good for $W$
up to $0.2t$. When the graphene system is doped with low concentration 1%, the
conductance fluctuation and shot noise agrees with brute-force results for
large $W$ which is comparable to the hopping energy $t$. At large doping
concentration 10%, good agreement can be reached for conductance fluctuation
and shot noise for $W$ up to $0.4t$. We have also tested our formalism on
square lattice with similar results. Our formalism can be easily combined with
linear muffin-tin orbital first-principles transport calculations for light
doping nano-scaled systems, making prediction on variability of nano-devices.
