Schottky signals are used for non-invasive beam diagnostics as they contain information on various beam and machine parameters. The instantaneous Schottky signal is, however, only a single realisation of a random process, implicitly depending on the discrete distribution of synchrotron and betatron amplitudes and phases among the particles. To estimate the expected value of the Schottky power spectrum, and reveal the inner structure of the Bessel satellites described by the theory, the averaging of instantaneous Schottky spectra is required. This study describes this procedure quantitatively by analysing the statistical properties of the Schottky signals, including the expected value and variance of Schottky power spectra. Furthermore, we investigate how these quantities evolve with the number of particles in the bunch, the observed harmonic of the revolution frequency, the distribution of synchrotron oscillation amplitudes, and the bunch profile. The theoretical findings are compared against macro-particle simulations as well as Monte Carlo computations.