For clusters with ≳100 measured galaxy redshifts, we develop a new method for detecting departures from a Gaussian velocity distribution. We decompose the cluster velocity distribution into a sum of the orthogonal Gauss-Hermite functions. This method quantifies the asymmetric third-order (h3) and symmetric fourth-order (h4) terms of the distribution while minimizing the effect of interlopers in the tails.We apply the method to a sample of eight rich clusters: A426, A548, At060, Al 656, A2151, A2670, DC 2048-52, and Centaurus. Each cluster has more than 100 members with known redshifts. The clusters A548, Al 656, DC 2048-52, and Centaurus significantly depart from Gaussian. A1656 and DC 2048-52 have asymmetric velocity distributions; A548 and Centaurus have broad, symmetric distributions. The most likely interpretation is that these systems have substructure. This result is consistent with findings by other authors.