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Statistical Range

المؤلف:  Feller, W.

المصدر:  An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. New York: Wiley, 1968.

الجزء والصفحة:  ...

10-2-2021

1799

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

{1/C for 0<x<C; 0 for |x|>C, " src="https://mathworld.wolfram.com/images/equations/StatisticalRange/NumberedEquation2.gif" style="height:62px; width:160px" />

(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

{(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. " src="https://mathworld.wolfram.com/images/equations/StatisticalRange/NumberedEquation4.gif" style="height:126px; width:418px" />

(4)

The mean is

(5)

and the mode is

{((m-2)(m+n))/((m-1)(m+n-2)) for m-n<=2; ((n+1)(m+n-2))/(n(m+n)) for m-n>=2 " src="https://mathworld.wolfram.com/images/equations/StatisticalRange/NumberedEquation6.gif" style="height:84px; width:235px" />

(6)

(Kenney and Keeping 1962).

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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.