Particle tracking experiments with high speed digital microscopy yield the positions and trajectories of lipid droplets inside living cells. Angular correlation analysis shows that the lipid droplets have uncorrelated motion at short time scales (τ 10 ms, becomes persistent, indicating directed movement. The motion at all time scales is associated with the lipid droplets being tethered to and driven along the microtubule network. The point at which the angular correlation changes from anti-persistent to persistent motion corresponds to the cross over between sub-diffusive and super diffusive motion, as observed by mean square displacement analysis. Correct analysis of the angular correlations of the detector noise is found to be crucial in modelling the observed phenomena.