Studentized range distribution

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Studentized range distribution
Parameters k > 1 — the number of groups
ν > 0degrees of freedom
Support q ∈ [0; +∞)

In probability and statistics, Studentized range distribution is a continuous probability distribution that arises when estimating the range of a normally distributed population in situations where the sample size is small and population standard deviation is unknown.

Suppose that we take a sample of size n from each of k populations with the same normal distribution N(μσ) and suppose that min is the smallest of these sample means and max is the largest of these sample means, and suppose S2 is the pooled sample variance from these samples. Then the following random variable has a Studentized range distribution.


Probability density function[edit]

Differentiating the cumulative distribution function with respect to q gives the probability density function.

Cumulative distribution function[edit]

The cumulative distribution function is given by [1]

Special cases[edit]

When the degrees of freedom approach infinity, the standard normal distribution can be used for the general equation above. If k is 2 or 3,[2] the studentized range probability distribution function can be directly evaluated, where is the standard normal probability density function.

When the degrees of freedom approaches infinity the studentized range cumulative distribution can be calculated at all k using the standard normal distribution.

How the studentized range distribution arises[edit]

For any probability distribution f, the range probability distribution is:[2]

What this means, is that we are adding up the likelihood that, given k draws from a distribution, two of them differ by r, and the remaining k-2 draws all fall between the two extreme values. If we use u substitution where and define F as the cumulative distribution function of f, then the equation can be simplified.

In order to create the studentized range distribution, we first use the standard normal distribution for f and F, and change the variable r to q.

The chi distribution is:

If we apply a change of variables we see it can also be expressed as:

Multiplying the two and integrating over S gives:


Critical values of the studentized range distribution are used in Tukey's range test.


  1. ^ Lund, R. E.; Lund, J. R. (1983). "Algorithm AS 190: Probabilities and Upper Quantiles for the Studentized Range". Journal of the Royal Statistical Society. 32 (2): 204–10. JSTOR 2347300. 
  2. ^ a b A. T. McKay (1933). "A Note on the Distribution of Range in Samples of n". Biometrika. 25 (3): 415–20. JSTOR 2332292.