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Difference between revisions of "Range (of variation of a sample)"

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between the largest $x_\mathrm{max}=x_n$ and smallest $x_\mathrm{min}=x_1$ values in the ordered sample
 
between the largest $x_\mathrm{max}=x_n$ and smallest $x_\mathrm{min}=x_1$ values in the ordered sample
 
$$
 
$$
(x_1,\ldots,x_n),\quad x_1\leq\ldots\leq x_n\,,
+
(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
 
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

Latest revision as of 12:46, 14 February 2020

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 $$ \mathbf{P}\{w_n \le t\} = n \int_{-\infty}^\infty (F(x+t)-F(x))^{n-1} dF(x)\,,\ \ \ 0 \le t \le \infty \ . $$


References

[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)


Comments

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

References

[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: http://encyclopediaofmath.org/index.php?title=Range_(of_variation_of_a_sample)&oldid=39507
This article was adapted from an original article by T.Yu. Popova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article