We present a statistical analysis of the X-ray flux distribution of Sgr A* from the Chandra X-ray Observatorys 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 $Q=(5.24pm0.08)times10^{-3}$ cts s$^{-1},$ and a variable component, represented by a power law process ($dN/dFpropto F^{-xi},$ $xi=1.92_{-0.02}^{+0.03}$). 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 $sigma=2.4pm0.2,$ 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.
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