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''quadratic variance, standard deviation, of quantities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076030/q0760301.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076030/q0760302.png" />''
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''quadratic variance, standard deviation, of quantities $x_1,\ldots,x_n$ from $a$''
  
 
The square root of the expression
 
The square root of the expression
  
<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/q/q076/q076030/q0760303.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$\frac{(x_1-a)^2+\ldots+(x_n-a)^2}{n}.$$
  
The quadratic deviation takes its smallest value when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076030/q0760304.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076030/q0760305.png" /> is the arithmetic mean of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076030/q0760306.png" />:
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The quadratic deviation takes its smallest value when $a=\bar x$, where $\bar x$ is the arithmetic mean of $x_1,\ldots,x_n$:
  
<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/q/q076/q076030/q0760307.png" /></td> </tr></table>
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$$\bar x=\frac{x_1+\ldots+x_n}{n}.$$
  
In this case the quadratic deviation serves as a measure of the variance (cf. [[Dispersion|Dispersion]]) of the quantities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076030/q0760308.png" />. Also used is the more general concept of a weighted quadratic deviation:
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In this case the quadratic deviation serves as a measure of the variance (cf. [[Dispersion|Dispersion]]) of the quantities $x_1,\ldots,x_n$. Also used is the more general concept of a weighted quadratic deviation:
  
<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/q/q076/q076030/q0760309.png" /></td> </tr></table>
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$$\sqrt\frac{p_1(x_1-a)^2+\ldots+p_n(x_n-a)^2}{p_1+\ldots+p_n},$$
  
where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076030/q07603010.png" /> are the so-called weights associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076030/q07603011.png" />. The weighted quadratic deviation attains its smallest value when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076030/q07603012.png" /> is the weighted mean:
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where the $p_1,\ldots,p_n$ are the so-called weights associated with $x_1,\ldots,x_n$. The weighted quadratic deviation attains its smallest value when $a$ is the weighted mean:
  
<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/q/q076/q076030/q07603013.png" /></td> </tr></table>
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$$\frac{p_1x_1+\ldots+p_nx_n}{p_1+\ldots+p_n}.$$
  
In probability theory, the quadratic deviation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076030/q07603014.png" /> of a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076030/q07603015.png" /> (from its mathematical expectation) refers to the square root of its variance: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076030/q07603016.png" />.
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In probability theory, the quadratic deviation $\sigma_X$ of a random variable $X$ (from its mathematical expectation) refers to the square root of its variance: $\sqrt{D(X)}$.
  
 
The quadratic deviation is taken as a measure of the quality of statistical estimators and in this case is referred to as the quadratic error.
 
The quadratic deviation is taken as a measure of the quality of statistical estimators and in this case is referred to as the quadratic error.

Revision as of 13:35, 10 April 2014

quadratic variance, standard deviation, of quantities $x_1,\ldots,x_n$ from $a$

The square root of the expression

$$\frac{(x_1-a)^2+\ldots+(x_n-a)^2}{n}.$$

The quadratic deviation takes its smallest value when $a=\bar x$, where $\bar x$ is the arithmetic mean of $x_1,\ldots,x_n$:

$$\bar x=\frac{x_1+\ldots+x_n}{n}.$$

In this case the quadratic deviation serves as a measure of the variance (cf. Dispersion) of the quantities $x_1,\ldots,x_n$. Also used is the more general concept of a weighted quadratic deviation:

$$\sqrt\frac{p_1(x_1-a)^2+\ldots+p_n(x_n-a)^2}{p_1+\ldots+p_n},$$

where the $p_1,\ldots,p_n$ are the so-called weights associated with $x_1,\ldots,x_n$. The weighted quadratic deviation attains its smallest value when $a$ is the weighted mean:

$$\frac{p_1x_1+\ldots+p_nx_n}{p_1+\ldots+p_n}.$$

In probability theory, the quadratic deviation $\sigma_X$ of a random variable $X$ (from its mathematical expectation) refers to the square root of its variance: $\sqrt{D(X)}$.

The quadratic deviation is taken as a measure of the quality of statistical estimators and in this case is referred to as the quadratic error.


Comments

The expression (*) itself is sometimes referred to as the mean-squared error or mean-square error, and its root as the root mean-square error. Similarly one has a weighted mean-square error, etc.

References

[a1] K. Rektorys (ed.) , Applicable mathematics , Iliffe (1969) pp. 1318
[a2] A.M. Mood, F.A. Graybill, "Introduction to the theory of statistics" , McGraw-Hill (1963) pp. 166, 176
How to Cite This Entry:
Quadratic deviation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quadratic_deviation&oldid=31480
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article