We show in this article how for single-determinant wave functions the one-electron functions derived from the diagonalization of the Fermi hole, averaged over an arbitrary domain Ω of real space, and expressed in terms of the occupied canonical orbitals, describe coarse-grained statistically independent electrons. With these domain-averaged Fermi hole (DAFH) orbitals, the full electron number distribution function (EDF) is given by a simple product of one-electron events. This useful property follows from the simultaneous orthogonality of the DAFH orbitals in Ω, Ω′=R3−Ω, and R3. We also show how the interfragment (shared electron) delocalization index, δΩ,Ω′, transforms into a sum of one-electron DAFH contributions. Description of chemical bonding in terms of DAFH orbitals provides a vivid picture relating bonding and delocalization in real space. DAFH and EDF analyses are performed on several test systems to illustrate the close relationship between both concepts. Finally, these analy