**The Studentized Range ( q) Distribution**

The Tukey method uses the studentized range distribution. Suppose we have *r* independent observations *y*_{1}, ..., *y _{r}* from a normal distribution with mean μ and variance σ2. Let

*w*be the range for this set; i.e., the maximum minus the minimum. Now suppose that we have an estimate

*s*2 of the variance σ2 which is based on ν degrees of freedom and is independent of the

*y*

_{i}(

*i*= 1,...,

*r*). The studentized range is defined as

Tukey's test is based on the comparison of two samples from the same population. From the first sample, the range (calculated by subtracting the smallest observation from the largest, or, where *Y*_{i} represents all of the observations) is calculated, and from the second sample, the standard deviation is calculated. The *studentized range* ratio is then calculated:

where *q* = *studentized range*, and *s* = standard deviation of the second sample.

This value of *q* is the basis of the critical value of *q*, based on three factors:

- α (the Type I error rate, or the probability of rejecting a true null hypothesis)
*n*(the number of degrees of freedom in the first sample (the one from which range was calculated))*v*(the number of degrees of freedom in the second sample (the one from which*s*was calculated))

The distribution of *q* has been tabulated and appears in many textbooks on statistics. In addition, R offers a cumulative distribution function (`ptukey`

) and a quantile function (`qtukey`

) for *q*.

Read more about this topic: Tukey's Range Test

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