We show that in the framework of one-dimensional Bohmian Quantum Mechanics [D. Bohm, Phys. Rev. 85, 166 and 180 (1952)], for a particle subject to a potential undergoing a weak adiabatic change, the time averages of the particle's positions typically differ markedly from the ensemble averages. We apply this result to the case where the weak perturbing potential is the back-action of a measuring device (i.e. a protective measurement). It is shown that under these conditions, most trajectories never cross the position measured (as already shown for a particular example in a previous paper).
