Abstract In this work, we analyze the Levy mutation operations based on the Levy probability distribution in the evolutionary programming via the mean square displacement and the distinctness. The Levy probability distribution is characterized by an infinite second moment and has been widely studied in conjunction with the fractals. The Levy mutation operators not only generate small varied offspring, but are more likely to generate large varied offspring than the conventional mutation operators. Based on this fact, we prove mathematically, via the mean square displacement and the distinctness, that the Levy mutation operations can explore and exploit a search space more effectively. As a result, one can get better performance with the Levy mutation than the conventional Gaussian mutation for the multi-valued functional optimization problems