  The empirical pairing gaps derived from four different odd-even mass
staggering formulas are compared. By performing single-$j$ shell and
multi-shell seniority model calculations as well as by using the standard HFB
approach with Skyrme force we show that the simplest three-point formula
$\Delta_C^ {(3)}(N)=\frac{1}{2}\left[B(N,Z)+B(N-2,Z)-2B(N-1,Z)\right]$ can
provide a good measure of the neutron pairing gap in even-$N$ nuclei. It
removes to a large extent the contribution from the nuclear mean field as well
as contributions from shell structure details. It is also less contaminated by
the Wigner effect for nuclei around $N=Z$. We also show that the strength of
$\Delta^ {(3)}_C(N)$ can serve as a good indication of the two-particle spatial
correlation in the nucleus of concern and that the weakening of $\Delta^
{(3)}_C(N)$ in some neutron-rich nuclei indicates that the di-neutron
correlation itself is weak in these nuclei.
