Based on the structural information obtained from X-STM measurements, we calculate the electron energy spectra in quantum rings (QR)s and the electron orbital magnetization as a function of the applied magnetic field. Shape anisotropy of QRs results in a mixing of electron states with different magnetic quantum numbers and tends to suppress oscillations of the electron magnetization versus magnetic field. We show that the oscillatory behavior of the average magnetization survives for ensembles of QRs with radial dispersion as large as 10-20%.