  We report $^{17}$O NMR measurements in single crystals of overdoped
Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ with $T_c=82$ K. We measure the full
anisotropy of the planar oxygen Knight shift, electric field gradient, and spin
lattice relaxation rate tensors, and show that the entire temperature
dependence is determined by the suppression of the density of states in the
pseudogap below $T^*\sim 94$ K. The linewidth can be explained by a combination
of magnetic and quadrupolar broadening as a result of an inhomogeneous
distribution of local hole concentrations that is consistent with scanning
tunneling microscopy measurements. This distribution is temperature
independent, does not break $C_4$ symmetry, and exhibits no change below $T^*$
or $T_c$.
