Range (of variation of a sample)

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The difference $$ w_n=x_\mathrm{max}-x_\mathrm{min} $$ between the largest $x_\mathrm{max}=x_n$ and smallest $x_\mathrm{min}=x_1$ values in the ordered sample $$ (x_1,\dotsc,x_n),\quad x_1\leq\dotsb\leq x_n\,, $$ obtained by taking $n$ independent measurements of the same random variable $X$. Let $F(x) = \mathbf{P}\{X \le x\}$ be the distribution function of the random variable $X$. Then the probability distribution for the range is $$ \mathbf{P}\{w_n \le t\} = n \int_{-\infty}^\infty (F(x+t)-F(x))^{n-1} dF(x)\,,\ \ \ 0 \le t \le \infty \ . $$


[1] B.L. van der Waerden, "Mathematische Statistik" , Springer (1957)
[2] L.N. Bol'shev, N.V. Smirnov, "Tables of mathematical statistics" , Libr. math. tables , 46 , Nauka (1983) (In Russian) (Processed by L.S. Bark and E.S. Kedrova)


The range of variation of a sample is also called the sample range.


[a1] D.B. Owen, "Handbook of statistical tables" , Addison-Wesley (1962)
How to Cite This Entry:
Range (of variation of a sample). Encyclopedia of Mathematics. URL:
This article was adapted from an original article by T.Yu. Popova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article