Ordering distributions on the circle with respect to uniformity

Orderings (denoted <<) for probability distributions on the circle are introduced for which [mu]1 << [mu]2 means roughly that [mu]1 is "more uniform" than [mu]2, or that [mu]2 is "more clumped" than [mu]1. Somewhat more precisely, [mu]1 << [mu]2 if the random variable [mu]1(As) is "less variable" than the random variable [mu]2(As) for all s, where As denotes a random arc of length s distributed uniformly on the circle. Some properties of the orderings are explored and applications are presented to random coverage problems on the circle, the distribution of spacings, and the analysis of directional data.