Range (Statistics)

$\begin{displaymath} R\equiv \max(x_i)-\min(x_i). \end{displaymath}$

(1)

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

$\begin{displaymath} P(x)=\cases{ {1\over C} & for $0<x<C$\cr 0 & for $\vert x\vert<C$,\cr} \end{displaymath}$

(2)

the distribution of the range is given by

$\begin{displaymath} D(R) = N\left({R\over C}\right)^{N-1}-(N-1)\left({R\over C}\right)^N. \end{displaymath}$

(3)

Given two samples with sizes and and ranges and , let $u\equiv R_1/R_2$ . Then

$\begin{displaymath} D(u)=\cases{ {m(m-1)n(n-1)\over (m+n)(m+n-1)(m+n-2)}[(m+n)u... ...+n-2)u^{-n-1}]\hfill\cr \hfill {\rm for\ } 1\leq u<\infty.\cr} \end{displaymath}$

(4)

The Mean is

$\begin{displaymath} \mu_u = {(m-1)n\over (m+1)(n-2)}, \end{displaymath}$

(5)

and the Mode is

$\begin{displaymath} \hat u = \cases{ {(m-2)(m+n)\over (m-1)(m+n-2)} & for $m-n\leq 2$\cr {(n+1)(m+n-2)\over n(m+n)} & for $m-n\geq 2$.\cr} \end{displaymath}$

(6)

References

Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 213-214, 1962.