# The Distance Standard Deviation

## TL;DR

This paper introduces and analyzes the distance standard deviation as a new measure of spread for multivariate data, deriving its properties, asymptotic distribution, and applications in hypothesis testing and quality control.

## Contribution

It provides the first detailed study of the distance standard deviation, including its asymptotic distribution, inequalities, and connections to classical measures, expanding tools for multivariate analysis.

## Key findings

- Derived the asymptotic distribution under finite second moments.
- Compared the distance standard deviation with classical measures for heavy-tailed distributions.
- Established inequalities relating it to standard deviation and Gini's mean difference.

## Abstract

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.