Broken symmetry is characteristic of arrays of quantum dots and can be observed in the failure of selection rules of optical spectroscopy or in the dielectric properties. Here we discuss scanning tunneling spectroscopy, where electrons are detached or attached. In the lowest order of description (sometimes known as Koopmans theorem), the orbitals of a system are regarded as given and, one adds or removes electrons from these orbitals. if one has a half-full band of states whose energies have a reflection symmetry about the center, the density of states should be symmetric about the energy of the highest occupied state. Features that are special to arrays of nanodots and lead to the breaking of the expected symmetry are identified. Computations of the density of states of an array of Ag nanodots that are in accord with the available experimental observations are also provided. For a disordered array, the response of the STM probe can be qualitatively different at different lattice points and we interpret this in terms of a change in the nature of the ground electronic state of the array when it is more disordered.