= Studentized Range Distribution =

The '''Studentized range distribution''' is a continuous probability distribution function parameterized to a number of groups and degrees of freedom.

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== Description ==

Consider ''m'' groups, all of equal size ''n'', from each of which a [[Analysis/NormalDistribution|normally distributed]] statistic is calculated. For example, sample means denoted ''y̅,,i,,'' for each group ''i''.

The general application is that these sample statistics are collected from the sample population, i.e. there is a common population mean and variance. These become important crutches for the formulation of a distribution.

The range between the maximum sample statistic and the minimum sample statistic is itself a statistic ''q'' that follows the Studentized range distribution. Continuing to use the example of sample means, and implicitly using a null hypothesis that the difference between population means is 0, the ''q'' statistic is calculated as:

{{attachment:q.svg}}

where ''y̅,,{max},,'' refers to the maximum of all sample means and ''s,,p,,'' is a [[Statistics/PooledVariance|pooled standard deviation]]. Recall that when samples are of equal size, that formula simplifies to:

{{attachment:pooled.svg}}

Conceptually, this distribution is a development upon [[Analysis/StudentsTDistribution|Student's t]]. It is intuitive to expect that range, or '''Studentized range''' (i.e., range normalized by dividing over the standard deviation), will grow with the number of groups. Like a repeated trial, there are simply more opportunities for a rare event. Note however that the Studentized range distribution with ''m=2'' is '''not''' equal to Student's t. Continuing to use the example of sample means, the ''t'' statistic for a [[Statistics/StudentsTTest#Two_Sample_Test|two sample test]] under these assumptions (i.e., ''n = n,,1,, = n,,2,,'') would be calculated as:

{{attachment:t.svg}}

This should demonstrate that the two are related, but the Studentized range distribution is larger by a factor of ''√2''.



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