Note on sample quantiles for ordinal data

Abstract

The arithmetic mean computed over ordinal data has to be interpreted with care. Instead it is often recommended to compute the sample median, or more generally sample quantiles. The virtue of raw quantiles is that they are not affected by an arbitrary rescaling of the data. Unfortunately, they are not very informative when an ordinal variable falls into only few categories. The informational content is larger for so-called grouped quantiles interpolating around one possible value. In this note we show, however, that they turn out to be not invariant with respect to strictly monotonous transformations as long as the possible outcomes are not equidistant. This motivates to suggest the new centered interpolated quantiles. They are designed to be invariant with respect to transformations that preserve the ranking of the possible values.