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''mean deviation''
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A measure, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074980/p0749801.png" />, of dispersion for a [[Probability distribution|probability distribution]]. For a continuously-distributed symmetric random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074980/p0749802.png" /> the probable deviation is defined by
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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/p/p074/p074980/p0749803.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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''mean deviation''
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074980/p0749804.png" /> is the median of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074980/p0749805.png" /> (which in this case is identical with the [[Mathematical expectation|mathematical expectation]], if it exists). For the [[Normal distribution|normal distribution]] there exists a simple connection between the probable deviation and the [[Standard deviation|standard deviation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074980/p0749806.png" />:
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A measure,  $  B $,
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of dispersion for a [[Probability distribution|probability distribution]]. For a continuously-distributed symmetric random variable  $  X $
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the probable deviation is defined by
  
<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/p/p074/p074980/p0749807.png" /></td> </tr></table>
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$$ \tag{* }
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{\mathsf P} \{ | X- m | < B \}  = \
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{\mathsf P} \{ | X- m | > B \}  =
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\frac{1}{2}
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,
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$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074980/p0749808.png" /> is the normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074980/p0749809.png" />-distribution function. The approximate relation is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074980/p07498010.png" />.
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where $  m $
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is the median of  $  X $(
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which in this case is identical with the [[Mathematical expectation|mathematical expectation]], if it exists). For the [[Normal distribution|normal distribution]] there exists a simple connection between the probable deviation and the [[Standard deviation|standard deviation]]  $  \sigma $:
  
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$$
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\Phi \left (
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\frac{B} \sigma
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\right )  = 
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\frac{3}{4}
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,
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$$
  
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where  $  \Phi ( x) $
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is the normal  $  ( 0, \sigma ) $-
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distribution function. The approximate relation is  $  B = 0.6745 \sigma $.
  
 
====Comments====
 
====Comments====
The probably deviation is also called the mean error, [[#References|[a2]]]. The phrase  "mean deviation"  is also used to denote the first absolute moment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074980/p07498011.png" /> of the random variable around its median, [[#References|[a1]]].
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The probably deviation is also called the mean error, [[#References|[a2]]]. The phrase  "mean deviation"  is also used to denote the first absolute moment $  E ( | X - m | ) $
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of the random variable around its median, [[#References|[a1]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Cramér,  "Mathematical methods of statistics" , Princeton Univ. Press  (1966)  pp. Sect. 15.6</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  Ph.H. Dubois,  "An introduction to psychological statistics" , Harper &amp; Row  (1965)  pp. 287</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Cramér,  "Mathematical methods of statistics" , Princeton Univ. Press  (1966)  pp. Sect. 15.6</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  Ph.H. Dubois,  "An introduction to psychological statistics" , Harper &amp; Row  (1965)  pp. 287</TD></TR></table>

Latest revision as of 08:07, 6 June 2020


mean deviation

A measure, $ B $, of dispersion for a probability distribution. For a continuously-distributed symmetric random variable $ X $ the probable deviation is defined by

$$ \tag{* } {\mathsf P} \{ | X- m | < B \} = \ {\mathsf P} \{ | X- m | > B \} = \frac{1}{2} , $$

where $ m $ is the median of $ X $( which in this case is identical with the mathematical expectation, if it exists). For the normal distribution there exists a simple connection between the probable deviation and the standard deviation $ \sigma $:

$$ \Phi \left ( \frac{B} \sigma \right ) = \frac{3}{4} , $$

where $ \Phi ( x) $ is the normal $ ( 0, \sigma ) $- distribution function. The approximate relation is $ B = 0.6745 \sigma $.

Comments

The probably deviation is also called the mean error, [a2]. The phrase "mean deviation" is also used to denote the first absolute moment $ E ( | X - m | ) $ of the random variable around its median, [a1].

References

[a1] H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1966) pp. Sect. 15.6
[a2] Ph.H. Dubois, "An introduction to psychological statistics" , Harper & Row (1965) pp. 287
How to Cite This Entry:
Probable deviation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Probable_deviation&oldid=48303
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article