  Parity violating electron scattering allows model independent measurements of
neutron densities that are free from most strong interaction uncertainties. In
this paper we present statistical error estimates for a variety of experiments.
The neutron radius $R_n$ can be measured in several nuclei, as long as the
nuclear excited states are not too low in energy. We present error estimates
for $R_n$ measurements in $^{40}$Ca, $^{48}$Ca, $^{112}$Sn, $^{120}$Sn,
$^{124}$Sn, and $^{208}$Pb. In general, we find that the smaller the nucleus,
the easier the measurement. This is because smaller nuclei can be measured at
higher momentum transfers where the parity violating asymmetry $A_{pv}$ is
larger. Also in general, the more neutron rich the isotope, the easier the
measurement, because neutron rich isotopes have larger weak charges and larger
$A_{pv}$. Measuring $R_n$ in $^{48}$Ca appears very promising because it has a
higher figure of merit than $^{208}$Pb. In addition, $R_n(^{48}$Ca) may be more
easily related to two nucleon and three nucleon interactions, including very
interesting three neutron forces, than $R_n(^{208}$Pb). After measuring $R_n$,
one can constrain the surface thickness of the neutron density $a_n$ with a
second measurement at somewhat higher momentum transfers. We present
statistical error estimates for measuring $a_n$ in $^{48}$Ca, $^{120}$Sn, and
$^{208}$Pb. Again, we find that $a_n$ is easier to measure in smaller nuclei.
