smallrange

Smallrange is a statistical measure of dispersion that concentrates on the most compact cluster of observations in a data set. For a data set X with n observations and a chosen proportion p in (0,1], smallrange_p(X) is defined as the width of the smallest interval [a, b] that contains at least p·n data points. Equivalently, it is the minimum possible b − a such that at least p·n points lie in [a, b].

To compute, sort the data, then slide a window of length k = ceil(p·n) across the sorted array, recording the width x[i+k−1] − x[i] for each position, and take the minimum. For example, if X = [1, 2, 2, 3, 100], n = 5, and p = 0.8, then k = 4 and the windows produce widths of 2 and 98, giving smallrange_0.8 = 2.

Applications of smallrange include assessing clustering strength within a data set, comparing distributions on different scales, and providing a robust exploratory tool that highlights the tightest central cluster rather than overall spread. It can complement standard dispersion measures such as the full range or the interquartile range and is particularly useful when the analyst wants to focus on concentration of values rather than on tails or outliers.

Limitations include dependence on the chosen p, potential instability for small samples, and the possibility that different data sets yield the same smallrange_p despite differing distribution shapes. As a flexible descriptor, smallrange is most informative when used alongside other dispersion and density measures.