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.

中文翻译：

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距离标准差

在多元数据的距离相关分析中出现的距离标准偏差被研究作为传播的度量。经验距离标准差的渐近分布是在有限二阶矩的假设下推导出来的。提供了对来自材料科学的数据集的假设检验和多元统计质量控制的应用。将距离标准偏差与经典尺度度量进行比较，以推断重尾分布的传播。推导出距离方差的不等式，证明距离标准差在经典标准差和基尼平均差的范围内。根据 Gini' 获得距离标准偏差的新表达式 s 平均差和阶次统计的间距矩。还表明距离标准偏差满足传播度量的公理性质。