Experimental data obtained via a Lagrangian positron emission particle tracking technique have been used to drive a new methodology for the evaluation of mixing in mechanically agitated vessels. The long-term 3D trajectories of multiphase flow components have been analyzed based on the recurrence theorem to quantify local and global mixing, using the Shannon entropy of the probability distribution of recurrence structures as a measure of mixedness. The potential of this new data-driven method has been demonstrated by considering single-phase and particle-liquid systems in a vessel agitated by different impellers over a wide range of experimental conditions. Detailed pointwise maps of local mixedness have been constructed for different flow regimes, and indices of global mixedness have also been derived. The entropy mixing maps for single-phase water show that a Rushton disk turbine achieves better global as well as local mixing than an up-pumping or down-pumping pitch-blade turbine. The results for particle-liquid flows show that, for a given impeller design, different suspensions in the just-suspended regime display the same degree of mixedness, regardless of particle size or concentration. The state of suspension mixing improves significantly as impeller speed increases above the just-suspended speed, but a plateau is reached at high speeds and it becomes thenceforth difficult to improve mixing further. This study has shown that the new methodology driven by Lagrangian flow trajectory data has potential for the design and analysis of multiphase flow systems.