On an asymptotically flat manifold, the ADM mass (or the total mass) is traditionally defined via a formula that is heavily dependent on the metric components in an admissible coordinate chart near infinity. In this talk, we discuss an equivalent way to compute the mass using the Ricci tensor of the metric. Combined with the Riemannian positive mass theorem, this geometric mass formula yields a comparison theorem for the total 2nd order mean curvature on large coordinate spheres in manifolds of nonnegative scalar curvature. We also give examples of compact manifolds with boundary on which such a comparison fails to hold. This talk is based on joint work with L.F. Tam and on joint work with L.F. Tam and N.Q. Xie.