Note that in the outer part of the integral, the equation was used to replace an exponential factor.

[3] The studentized range is used to calculate significance levels for results obtained by data mining, where one selectively seeks extreme differences in sample data, rather than only sampling randomly.

The Studentized range distribution has applications to hypothesis testing and multiple comparisons procedures.

For example, Tukey's range test and Duncan's new multiple range test (MRT), in which the sample x1, ..., xn is a sample of means and q is the basic test-statistic, can be used as post-hoc analysis to test between which two groups means there is a significant difference (pairwise comparisons) after rejecting the null hypothesis that all groups are from the same population (i.e. all means are equal) by the standard analysis of variance.

[4] When only the equality of the two groups means is in question (i.e. whether μ1 = μ2), the studentized range distribution is similar to the Student's t distribution, differing only in that the first takes into account the number of means under consideration, and the critical value is adjusted accordingly.

In order to obtain the distribution in terms of the "studentized" range q, we will change variable from R to s and q.

is the low-end of the range, and define FX as the cumulative distribution function of fX, then the equation can be simplified: We introduce a similar integral, and notice that differentiating under the integral-sign gives which recovers the integral above,[a] so that last relation confirms because for any continuous cdf The range distribution is most often used for confidence intervals around sample averages, which are asymptotically normally distributed by the central limit theorem.

In order to create the studentized range distribution for normal data, we first switch from the generic fX and FX to the distribution functions φ and Φ for the standard normal distribution, and change the variable r to s·q, where q is a fixed factor that re-scales r by scaling factor s: Choose the scaling factor s to be the sample standard deviation, so that q becomes the number of standard deviations wide that the range is.