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

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The difference
 
The difference
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077420/r0774201.png" /></td> </tr></table>
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$$w_n=x_\mathrm{max}-x_\mathrm{min}$$
  
between the largest <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077420/r0774202.png" /> and smallest <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077420/r0774203.png" /> values in the ordered 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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077420/r0774204.png" /></td> </tr></table>
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$$(x_1,\ldots,x_n),\quad x_1\leq\ldots\leq x_n,$$
  
obtained by taking <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077420/r0774205.png" /> independent measurements of the same random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077420/r0774206.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077420/r0774207.png" /> be the distribution function of the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077420/r0774208.png" />. Then the probability distribution for the range is
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obtained by taking $n$ independent measurements of the same random variable $X$. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077420/r0774207.png" /> be the distribution function of the random variable $X$. Then the probability distribution for the range is
  
 
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077420/r0774209.png" /></td> </tr></table>

Revision as of 15:11, 1 August 2014

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

$$(x_1,\ldots,x_n),\quad x_1\leq\ldots\leq x_n,$$

obtained by taking $n$ independent measurements of the same random variable $X$. Let be the distribution function of the random variable $X$. Then the probability distribution for the range is

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=32664
This article was adapted from an original article by T.Yu. Popova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article