We show that incorporating the effects of Bose-Einstein or Fermi-Dirac quantum statistics within the centroid molecular dynamics formalism leads to additional correlations in the system due to exchange effects. In the case of Bose-Einstein statistics they appear as an additional attraction between physical particles while an additional repulsion is observed for Fermi-Dirac statistics. We show that we can account for these correlations through the effective centroid Hamiltonian. Within the approach based on the phase space centroid density, this Hamiltonian depends on centroid momenta in a nonclassical way. We illustrate the above findings using a simple model of two bosons and fermions in a harmonic potential. The average of a centroid variable along centroid trajectories based on such an effective Hamiltonian can be used to study the equilibrium properties of quantum systems. Is is also shown that the dynamics of the centroid variables derived from the quantum mechanical dynamics of the corresponding physical observables does not depend on exchange effects for a harmonic system.