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A statistic from the class of so-called Studentized statistics, obtained as a result of a special normalization of a linear combination of order statistics constructed from a normal sample.

Let  $  X _ {1} \dots X _ {n} $
be independent normally  $  N( a, \sigma  ^ {2} ) $-
distributed random variables, and let  $  X  ^ {(n)} = ( X _ {(n1)} \dots X _ {(nn)} ) $
be the vector of order statistics constructed from the observations  $  X _ {1} \dots X _ {n} $. 
Moreover, let the statistic  $  \sum _ {i=1} ^ {n} a _ {i} X _ {(ni)} $, 
which is a linear combination of the order statistics  $  X _ {(n1)} \dots X _ {(nn)} $, 
be independent of some  "chi-squared"  distribution  $  V/ \sigma  ^ {2} $
of  $  f $
degrees of freedom. Let  $  s _ {f}  ^ {2} = f ^ { - 1 } V $. 
In this case, one says that

$$ 

\frac{1}{s _ {f} }
 \sum _ { i=1 } ^ { n }  a _ {i} X _ {(ni)}
$$

is a Studentized statistic.

The Studentized range is the Studentized statistic for which  $  \sum _ {i=1} ^ {n} a _ {i} X _ {(ni)} $
is the range of the sample  $  X _ {1} \dots X _ {n} $, 
i.e. if

$$ 
\sum _ { i=1 } ^ { n }  a _ {i} X _ {(ni)}  =  X _ {(nn)} - X _ {(n1)} ;
$$

consequently, the Studentized range takes the form

$$ 

\frac{X _ {(nn)} - X _ {(n1)} }{s _ {f} }
 .
$$

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  H. David,   "Order statistics" , Wiley  (1970)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  S.S. Wilks,   "Mathematical statistics" , Wiley  (1962)</TD></TR>
</table>

====Comments====
The case

$$ 
s _ {f}  ^ {2}  =  
\frac{1}{n-1} \sum ( X _ {i} - \overline{X}\; )  ^ {2}
$$

is used for tests of normality and outlying observations, cf. [[#References|[1]]], Chapt. 8. For a table of the quantiles of the Studentized range see [[#References|[a2]]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  A.M. Mood,   F.A. Graybill,   "Introduction to the theory of statistics" , McGraw-Hill  (1963)  pp. 243</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  P.H. Müller,   P. Neumann,   R. Storm,   "Tafeln der mathematischen Statistik" , C. Hauser  (1977)  pp. 166–169</TD></TR>
</table>