  The distance standard deviation, which arises in distance correlation
analysis of multivariate data, is studied as a measure of spread. The
asymptotic distribution of the empirical distance standard deviation is derived
under the assumption of finite second moments. Applications are provided to
hypothesis testing on a data set from materials science and to multivariate
statistical quality control. The distance standard deviation is compared to
classical scale measures for inference on the spread of heavy-tailed
distributions. Inequalities for the distance variance are derived, proving that
the distance standard deviation is bounded above by the classical standard
deviation and by Gini's mean difference. New expressions for the distance
standard deviation are obtained in terms of Gini's mean difference and the
moments of spacings of order statistics. It is also shown that the distance
standard deviation satisfies the axiomatic properties of a measure of spread.
