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Finding the mean of many commonly studied statistics on the symmetric group is often not too difficult, but the corresponding problem on conjugacy classes is more complex. In this talk, we explore some combinatorial methods that show when there are no "short" cycles in a conjugacy class, certain moments of statistics begin stabilizing. In fact, this holds in conjugacy classes of the colored permutation groups, i.e., the wreath product of a cyclic group with the symmetric group, which contains the symmetric groups and signed symmetric groups as special cases. We also discuss some connections with central limit theorems and representation theory. This talk is based on joint work with Jesse Campion Loth, Michael Levet, Sheila Sundaram, and Mei Yin.