A quantitative measure of the delocalization of electrons from the region inside a sphere of radius R centered on a nucleus to the region exterior to this sphere is the atom delocalization index A(R). A(R) is defined by the double integral of the point-point sharing index over two distinct volumes: over the interior of a sphere of radius R, and zeta' over the remaining space. This index, as a function of R, has a remarkable shell-like structure with the values of the maxima being close to the number of electrons traditionally assigned to a particular shell times 0.25. The origin of these almost "magic" numbers and the reasons for deviations of these numbers from the simple rule is discussed in terms of simple models of the electronic structure of some light atoms. Results from numerical computations of electronic structure are presented for the atoms Li, Be, Al, Ne, Ar, Kr, Xe, Rn, Zn, and Au. The distinct shell-like structure of A(R) persists even to the heaviest of these. For atoms with zero total orbital angular momentum, the delocalization index can be split into a sum of noninterfering terms of different single particle angular momenta. These contributions also have distinctive shell-like structures. The maxima of the different angular momentum contributions do not always coincide. A similar decomposition also holds for spin "up" and "down" contributions. The shell-like structure allows for the identification of the spatial region in which the valence electrons lie. The angular momentum contributions can be used to identify the nature of the valence regions. The invariance of the core electron regions and the changes in the valence regions upon bond formation are illustrated by the calculation of the delocalization index for the heavy atoms in CH4, NH3, H2O, and SiH4. The identification of the spatial location of the valence regions should aid choosing the location of fixed points when analyzing the behavior of electrons using sharing amplitudes.