The median of a random interval Beatriz Sinova 1 , Mar´ ıa Rosa Casals 1 , Ana Colubi 1 , and Mar´ ıa ´ Angeles Gil 1 Abstract In dealing with real-valued random variables, the median of the distribution is the ‘central tendency’ summary measure associated with its ‘middle position’. When available random elements are interval-valued, the lack of a universal ranking of values makes it impossible to formalize the extension of the concept of median as a middle-position summary measure. Nevertheless, the use of a generalized L 1 Hausdorff-type metric for interval data enables to formalize the median of a random interval as the central- tendency interval(s) minimizing the mean distance with respect to the ran- dom set values, by following the alternate equivalent way to introduce the median in the real-valued case. The expression for the median(s) is obtained, and main properties are analyzed. A short discussion is made on the main different features in contrast to the real-valued case. 1 Introduction Interval data in connection with random experiments usually come either from the observation/measurement of an intrinsically interval-valued random attribute (say fluctuations, ranges, etc.), from an uncertain measurement or from a grouping of real-valued data in accordance with a given list of intervals (like often happens with age or income groups). The statistical analysis of interval data, and especially the inferential de- velopments, requires an appropriate formalization within the probabilistic setting. In this respect, compact convex random sets represent a suitable tool to handle the random mechanisms producing interval data. In the literature, one can find several statistical studies devoted to interval data, most of them being based on descriptive techniques and approaches. University of Oviedo, 33071 Oviedo, Spain sinovabeatriz.uo@uniovi.es · rmcasals@uniovi.es · colubi@uniovi.es · magil@uniovi.es 1