Many well-known procedures in multivariate data analysis are invariant under the group, $L(p)$, of translations and nonsingular linear transformations. New maximal $L(p)$ invariant statistics are derived and are shown to have the geometrical interpretation of a scatter of points in Euclidean space. The distribution of maximal $L(p)$ invariants for the case of a single multivariate normal population is shown to follow from a result of James (1954). Finally we consider tests of the null hypothesis that $k > 1$ populations are identical and show that optimal $L(p)$ invariant tests are similar tests of randomness.