# Studentized range

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In statistics, the studentized range computed from a list x1, ..., xn of numbers is

$\frac{\max\{\,x_1,\dots,x_n\,\} - \min\{\,x_1,\dots,x_n\,\}}{s},$

where

$s^2 = \frac{1}{n - 1}\sum_{i=1}^n (x_i - \overline{x})^2,$

is the sample variance and

$\overline{x} = \frac{x_1 + \cdots + x_n}{n}$

is the sample mean.

Generally, studentized means adjusted by didviding by an estimate of a population standard deviation; see also studentized residual. The concept is named after William Sealey Gosset, who wrote under the pseudonym "Student".

If X1, ..., Xn are independent identically distributed random variables that are normally distributed, the the probability distribution of their studentized range is what is usually called the studentized range distribution. This probability distribution is the same regardless of the expected value and standard deviation of the normal distribution from which the sample is drawn. This probability distribution has applications to hypothesis testing and multiple comparisons.

## References and further reading Edit

• John Neter, Michael H. Kutner, Christopher J. Nachtsheim, William Wasserman, Applied Linear Statistical Models, fourth edition, McGraw-Hill, 1996, page 726.