  The trends with mass number are examined for the odd-even-staggering (OES) in
nuclear binding of neutrons and protons through the conventional measures
  $\Delta^{(3)}$. The large differences previously observed between these
trends for even and odd values of these measures is found to arise, in part,
from the slow variation of binding energies with mass and charge which provides
a background contribution. This background is estimated with the liquid-drop
model, and accounts for the greater difference found in proton removal relative
to neutron removal. The differences which persist after backgrounds are removed
can not be treated in the conventional liquid-drop model but require the
addition of a new term. Such a term is investigated, and its effect on specific
values of the OES is calcutated. The liquid-drop fitting is also applied to a
set of separation energies constrained to match the specific set of nuclei used
to determine the observed values for the odd $\Delta^{(3)}$. The resulting fit
for the pairing term is compared to the average value of even and odd measures.
The effect on this value of the new liquid-drop term is observed, and the
change in background when the new term is included is also used as an alternate
method for determining the difference between trends of the even and odd values
of the OES.
