A quantitative relation (eta = 1/2(K/alpha)(1/3)) between electric dipole polarizability alpha and hardness eta is developed. In this relation K is a constant and eta and alpha are expressed in au. With K = 0.792, using the available alpha data, the above equation is found to reproduce the experimental eta values for a number of open shell atoms and monoatomic cations of various charges very satisfactorily. This equation applies to the clusters of nonmagic numbers of sodium atoms also. However, for closed shell species, the above equation gives hardness values much tower than the experimental data. This is probably because the equation fails to reveal the "extra" amount of stability imparted to a chemical species by a closed shell configuration. Similar lower values of eta are produced by this equation for clusters of magic numbers of sodium atoms.