Random arrays bound by a spherical volume are rigorously analyzed through a stochastic approach for uniform and Gaussian element distributions. The full mathematical derivation and process to determine the array characteristics are discussed and a key result is the form solution which describes both the sidelobe behavior and main beam characteristics. It is also shown that the proposed approach shares many conceptual similarities to the probabilistic approach applied to circular random arrays and comparisons are made. The 3-dB beamwidth, directivity, grating lobes, and sidelobe level are examined analytically and in numerical simulation for a conceptual system of isotropic wide-band radiators with a 100:1 bandwidth. \ 2014 IEEE.