Using exact diagonalization numerical methods, as well as analytical arguments, we show that for typical electron densities in chaotic and disordered dots the peak spacing distribution is not bimodal but Gaussian. This is in agreement with experimental observations. We attribute this behavior to the tendency of an even number of electrons to gain on-site interaction energy by removing the spin degeneracy. We predict that the dot will show a nontrivial electron-number-dependent spin polarization. Experimental tests of this hypothesis based on the spin polarization measurements are proposed.