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Range (Statistics)

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
{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
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 $m$ and $n$ and ranges $R_1$ and $R_2$, let $u\equiv R_1/R_2$. Then

{m(m-1)n(n-1)\over (m+n)(m+n-1)(m+n-2)}[(m+n)u...
\hfill {\rm for\ } 1\leq u<\infty.\cr}
\end{displaymath} (4)

The Mean is
\mu_u = {(m-1)n\over (m+1)(n-2)},
\end{displaymath} (5)

and the Mode is
\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)


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

© 1996-9 Eric W. Weisstein