In this paper, we have considered the mechanical stability of a jellium system in the presence of spin degrees of freedom and have generalized the stabilized jellium model, introduced by J. P. Perdew, H. Q. Tran, and E. D. Smith [Phys.Rev.B 42, 11627 (1990)], to a spin-polarized case. By applying this generalization to simple alkali metal clusters, we gain additional insights about the odd-even alternations, seen in their ionization potentials. In this generalization, in addition to the electronic degrees of freedom, we allow the positive jellium background to expand as the clusters' polarization increases. In fact, our self-consistent calculations of the energetics of alkali metal clusters with spherical geometries, in the context of density functional theory and local spin density approximation, show that the energy of a cluster is minimized for a configuration with maximum spin compensation. That is, for clusters with even number of electrons, the energy minimization gives rise to complete compensation (N_up = N_down), and for clusters with odd number of electrons, only one electron remains uncompensated (N_up - N_down = 1). This maximum spin compensation rule, which is brought about by the alternation in the equilibrium r_s-values for clusters with different sizes, gives rise to alternations in the ionization potentials. In other words, the odd-even alternation is partly due to the alternating expansion of the clusters' relative mean radii, R(N,zeta)/R(N,0), and partly due to shape deformations that have been considered by others.