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The difference
 
The difference
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$$
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w_n=x_\mathrm{max}-x_\mathrm{min}
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$$
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between the largest $x_\mathrm{max}=x_n$ and smallest $x_\mathrm{min}=x_1$ values in the ordered sample
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$$
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(x_1,\dotsc,x_n),\quad x_1\leq\dotsb\leq x_n\,,
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$$
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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
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$$
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\mathbf{P}\{w_n \le t\} = n \int_{-\infty}^\infty (F(x+t)-F(x))^{n-1} dF(x)\,,\ \ \ 0 \le t \le \infty \ .
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$$
  
<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/r0774201.png" /></td> </tr></table>
 
 
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
 
 
<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>
 
 
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
 
 
<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>
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.L. van der Waerden,  "Mathematische Statistik" , Springer  (1957)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  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)</TD></TR></table>
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<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  B.L. van der Waerden,  "Mathematische Statistik" , Springer  (1957)</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  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)</TD></TR>
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</table>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.B. Owen,  "Handbook of statistical tables" , Addison-Wesley  (1962)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  D.B. Owen,  "Handbook of statistical tables" , Addison-Wesley  (1962)</TD></TR>
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</table>
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{{TEX|done}}

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=14775
This article was adapted from an original article by T.Yu. Popova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article