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''sample dispersion''
 
''sample dispersion''
  
A scalar characteristic of the disperson, or spread, of a sample (consisting of real numbers) relative to a fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083210/s0832101.png" /> (called the centre of dispersion). It is numerically equal to the sum of the squares of the deviations of the values from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083210/s0832102.png" />. For real-valued random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083210/s0832103.png" />, the variable
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A scalar characteristic of the disperson, or spread, of a sample (consisting of real numbers) relative to a fixed point $  x $(
 +
called the centre of dispersion). It is numerically equal to the sum of the squares of the deviations of the values from $  x $.  
 +
For real-valued random variables $  X _ {1} \dots X _ {n} $,  
 +
the variable
  
<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/s/s083/s083210/s0832104.png" /></td> </tr></table>
+
$$
 +
S _ {n} ( x)  = \
 +
\sum _ {i = 1 } ^ { n }
 +
( X _ {i} - x)  ^ {2}
 +
$$
  
is the sample variance about <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083210/s0832105.png" />. The variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083210/s0832106.png" /> are often assumed to be independent and identically distributed in discussions about <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083210/s0832107.png" />. Since, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083210/s0832108.png" />,
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is the sample variance about $  x $.  
 +
The variables $  X _ {1} \dots X _ {n} $
 +
are often assumed to be independent and identically distributed in discussions about $  S _ {n} ( x) $.  
 +
Since, for any $  x $,
  
<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/s/s083/s083210/s0832109.png" /></td> </tr></table>
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$$
 +
S _ {n} ( x)  = S _ {n} ( \overline{X}\; ) + n ( \overline{X}\; - x)  ^ {2}  \geq  \
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S _ {n} ( \overline{X}\; )  \equiv  S _ {n} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083210/s08321010.png" />, the sample variance about <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083210/s08321011.png" /> will be minimal when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083210/s08321012.png" />. A small value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083210/s08321013.png" /> indicates a concentration of the sample elements about <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083210/s08321014.png" /> and, conversely, a large value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083210/s08321015.png" /> indicates a large scattering of the sample elements. The concept of a sample variance extends to that of a sample covariance matrix for multivariate samples.
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where $  \overline{X}\; = ( X _ {1} + \dots + X _ {n} )/n $,  
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the sample variance about $  x $
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will be minimal when $  x= \overline{X}\; $.  
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A small value of $  S _ {n} $
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indicates a concentration of the sample elements about $  \overline{X}\; $
 +
and, conversely, a large value of $  S _ {n} $
 +
indicates a large scattering of the sample elements. The concept of a sample variance extends to that of a sample covariance matrix for multivariate samples.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.S. Wilks,  "Mathematical statistics" , Wiley  (1962)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.S. Wilks,  "Mathematical statistics" , Wiley  (1962)</TD></TR></table>

Latest revision as of 08:12, 6 June 2020


sample dispersion

A scalar characteristic of the disperson, or spread, of a sample (consisting of real numbers) relative to a fixed point $ x $( called the centre of dispersion). It is numerically equal to the sum of the squares of the deviations of the values from $ x $. For real-valued random variables $ X _ {1} \dots X _ {n} $, the variable

$$ S _ {n} ( x) = \ \sum _ {i = 1 } ^ { n } ( X _ {i} - x) ^ {2} $$

is the sample variance about $ x $. The variables $ X _ {1} \dots X _ {n} $ are often assumed to be independent and identically distributed in discussions about $ S _ {n} ( x) $. Since, for any $ x $,

$$ S _ {n} ( x) = S _ {n} ( \overline{X}\; ) + n ( \overline{X}\; - x) ^ {2} \geq \ S _ {n} ( \overline{X}\; ) \equiv S _ {n} , $$

where $ \overline{X}\; = ( X _ {1} + \dots + X _ {n} )/n $, the sample variance about $ x $ will be minimal when $ x= \overline{X}\; $. A small value of $ S _ {n} $ indicates a concentration of the sample elements about $ \overline{X}\; $ and, conversely, a large value of $ S _ {n} $ indicates a large scattering of the sample elements. The concept of a sample variance extends to that of a sample covariance matrix for multivariate samples.

References

[1] S.S. Wilks, "Mathematical statistics" , Wiley (1962)
How to Cite This Entry:
Sample variance. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sample_variance&oldid=48610
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article