The term "range" has two completely different meanings in statistics.
Given order statistics 
, 
, ..., 
, 
, the range of the random sample is defined by
 |
(1)
|
(Hogg and Craig 1995, p. 152).
For small samples, the range is a good estimator of the population standard deviation (Kenney and Keeping 1962, pp. 213-214).
For a continuous uniform distribution
 |
(2)
|
the distribution of the range is given by
 |
(3)
|
This is illustrated above for 
and values of 
from 
(red) to 
(violet).
Given two samples with sizes 
and 
and ranges 
and 
, let 
. Then
![D(u)={(m(m-1)n(n-1))/((m+n)(m+n-1)(m+n-2))[(m+n)u^(m-2)-(m+n-2)u^(m-1)]; for 0<=u<=1; (m(m-1)n(n-1))/((m+n)(m+n-1)(m+n-2))[(m+n)u^(-n)-(m+n-2)u^(-n-1)]; for 1<=u<infty.](/images/equations/StatisticalRange/NumberedEquation4.svg) |
(4)
|
The mean is
 |
(5)
|
and the mode is
 |
(6)
|
(Kenney and Keeping 1962).
