The chemical-binding properties of atoms belonging to the first three rows of the Periodic Table are studied within the atom-in-jellium model. The electronic structures are solved self-consistently using the density-functional theory. The extraction of the binding properties is carried out in the framework of the effective-medium theory. The emphasis is put on the systematic investigation of the trends along the 2p, 3p, and 3d series and on the effects due to different types of approximations for electron exchange and correlation. More specifically, in addition to the popular local-density approximation, the self-interaction-correction scheme and the generalized-gradient approximation are employed. The results provide insight into why the local-density approximation for solids (molecules or chemisorption systems) overestimates the cohesive (binding) energies but gives the lattice constants (bond lengths) and bulk moduli (vibration frequencies) rather well. The results obtained are also important because they give the basic parameters for the effective-medium theory, which is a versatile approximative method for calculating total energies of systems with many interacting atoms.