Ground-state atomic correlation energies, and their kinetic energy and potential energy components, are shown to be well-represented by empirical formulas of the form CN rho(0)Z(-gamma), where C and gamma are constants that are largely invariant within various sets of atoms and positive ions, Z is the atomic number, N is the number of electrons, and rho(0) is the electron density at the nucleus. Results are given for neutral atoms, singly charged positive ions, and many isoelectronic series-315 atomic species in all.