On the Inappropriateness of the Hodges–Lehmann Estimator for Ordinal Grading Data

The Hodges–Lehmann estimator (HL) is widely described (in statistical terms) as a nonparametric, robust, consistent and efficient estimator of central location. While this characterization is correct under standard assumptions—continuous, unimodal distributions on a meaningful metric scale—its application to ordinal grading data such as university marks is problematic. Using explicit counterexamples, we show that the HL estimator can exhibit severe jump discontinuities: minimal changes in the sample leads to large, discrete changes in the HL estimate. This behavior persists under sample size inflation by replication. This shows that the HL should not be used to represent a central location in ordinal grading data as frequently present at universities.