  We present a metric, average randomness, that predicts the compatibility of a
set of quantum states with the Haar-random distribution, by matching of
statistical moments, through a known quantum observable. We show that
Haar-randomness is connected to the Dirichlet distribution, and provide a
closed-form expression, and simple bounds of the statistical moments. We
generalize this metric to permutation- and unitary-equivalent observables,
ensuring that if the extended average randomness is compatible with a
Haar-random distribution, then the set of states is approximately Haar-random.
