  We present a statistical analysis of the X-ray flux distribution of Sgr A*
from the Chandra X-ray Observatory's 3 Ms Sgr A* X-ray Visionary Project (XVP)
in 2012. Our analysis indicates that the observed X-ray flux distribution can
be decomposed into a steady quiescent component, represented by a Poisson
process with rate $Q=(5.24\pm0.08)\times10^{-3}$ cts s$^{-1},$ and a variable
component, represented by a power law process ($dN/dF\propto F^{-\xi},$
$\xi=1.92_{-0.02}^{+0.03}$). This slope matches our recently-reported
distribution of flare luminosities. The variability may also be described by a
log-normal process with a median unabsorbed 2-8 keV flux of
$1.8^{+0.9}_{-0.6}\times10^{-14}$ erg s$^{-1}$ cm$^{-2}$ and a shape parameter
$\sigma=2.4\pm0.2,$ but the power law provides a superior description of the
data. In this decomposition of the flux distribution, all of the intrinsic
X-ray variability of Sgr A* (spanning at least three orders of magnitude in
flux) can be attributed to flaring activity, likely in the inner accretion
flow. We confirm that at the faint end, the variable component contributes ~10%
of the apparent quiescent flux, as previously indicated by our statistical
analysis of X-ray flares in these Chandra observations. Our flux distribution
provides a new and important observational constraint on theoretical models of
Sgr A*, and we use simple radiation models to explore the extent to which a
statistical comparison of the X-ray and infrared can provide insights into the
physics of the X-ray emission mechanism.
