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Statistical Range

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The term "range" has two completely different meanings in statistics.

Given order statistics Y_1=min_(j)X_j, Y_2, ..., Y_(N-1), Y_N=max_(j)X_j, the range of the random sample is defined by

R=Y_N-Y_1
(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).

StatisticalRangeDistribution

For a continuous uniform distribution

P(x)={1/C for 0<x<C; 0 for |x|>C,
(2)

the distribution of the range is given by

P_N(R)=N(R/C)^(N-1)-(N-1)(R/C)^N.
(3)

This is illustrated above for C=1 and values of N from N=2 (red) to N=10 (violet).

Given two samples with sizes m and n and ranges R_1 and R_2, let U=R_1/R_2. 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.
(4)

The mean is

mu_u=((m-1)n)/((m+1)(n-2)),
(5)

and the mode is

u^^={((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
(6)

(Kenney and Keeping 1962).

See also

Midrange, Range, Statistical Median

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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.

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Statistical Range

Cite this as:

Weisstein, Eric W. "Statistical Range." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/StatisticalRange.html

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