Statistical Range

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

(4)

The mean is

(5)

and the mode is

(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.