The term "range" has two completely different meanings in statistics.
Given order statistics , , ..., , , the range of the random sample is defined by
(1)
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(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)
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the distribution of the range is given by
(3)
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This is illustrated above for and values of from (red) to (violet).
Given two samples with sizes and and ranges and , let . Then
(4)
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The mean is
(5)
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and the mode is
(6)
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(Kenney and Keeping 1962).