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Date: 15-2-2016
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Date: 27-4-2021
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Date: 17-4-2021
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The term "range" has two completely different meanings in statistics.
Given order statistics ,
, ...,
,
, the range of the random sample is defined by
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(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
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(2) |
the distribution of the range is given by
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(3) |
This is illustrated above for and values of
from
(red) to
(violet).
Given two samples with sizes and
and ranges
and
, let
. Then
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(4) |
The mean is
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(5) |
and the mode is
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(6) |
(Kenney and Keeping 1962).
REFERENCES:
Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. New York: Wiley, 1968.
Hogg, R. V. and Craig, A. T. Introduction to Mathematical Statistics, 5th ed. New York: Macmillan, p. 152, 1995.
Kenney, J. F. and Keeping, E. S. "The Range." §6.2 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 75-76, 213-214, 1962.
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