The theory of the internally studentized range distribution revisited

Abstract

The present paper intends to revisit the distribution of the ratio of therange to the sample standard deviation, known as the distribution of the internallystudentized range, in the normal case. This distribution has its importance recognizedin several areas, as quality control and inference, for testing the lack of homogeneity ofthe data or kurtosis. An alternative distribution to the one presented by David et al.(1954), based on the distribution of the maximum, is proposed. We exhibit a detailedproof for the distribution of the internally studentized range in the normal case andsample size 3. We also provide a new result: the distribution for the uniform case withsample of size 3.