The response of magnetic nanoparticles (MNPs) to an oscillating magnetic field outside the linear response region is important for several applications including magnetic hyperthermia, magnetic resonance imaging and biodetection. The size and magnetic moment are two critical parameters for the performance of a colloidal MNP dispersion. We present and demonstrate the use of optomagnetic (OM) and AC susceptibility (ACS) measurements vs. frequency and magnetic field strength to obtain the size and magnetic moment distributions, including the correlation between the distributions. The correlation between the size and the magnetic moment contains information on the morphology and intrinsic structure of the particle. In OM measurements, the variation of the second harmonic light transmission through a dispersion of MNPs is measured in response to an oscillating magnetic field. We solve the Fokker-Planck equations for MNPs with a permanent magnetic moment, and develop analytical approximations to the ACS and the OM signals, which describe the change in the curve shapes with increasing field strength. We describe the influence of induced magnetic moments on the signals, by solving the Fokker-Planck equation for particles, which apart from the permanent magnetic moment may also have an induced magnetic moment and shape anisotropy. Using the results from the Fokker-Planck calculations we fit ACS and OM measurements on two multi-core particle systems. The obtained fit parameters also describe the correlations between the magnetic moment and size of the particles. From such an analysis on a commercially available polydisperse multicore particle system with an average particle size of 80 nm, we find that the MNP magnetic moment is proportional to the squareroot of the hydrodynamic size.