On the Sample Range Inequalities

We extend the classical sharp sample range inequality in terms of the sample standard deviation and provide a complete proof of the extension. In particular, we obtain a sharp inequality for the ratio of Gini’s mean difference to the sample range. By using sample ranges instead of the sample standard deviations, we also derive a novel result analogous to Benedetti’s inequality about the correlation coefficient calculated from a pair of two increasing samples of the same size. Applications of the obtained results are also discussed.