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Difference between revisions of "Quadratic deviation"

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''quadratic variance, standard deviation, of quantities $x_1,\ldots,x_n$ from $a$''
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''quadratic variance, standard deviation, of quantities $x_1,\dots,x_n$ from $a$''
  
 
The square root of the expression
 
The square root of the expression
  
$$\frac{(x_1-a)^2+\ldots+(x_n-a)^2}{n}.$$
+
\begin{equation}\frac{(x_1-a)^2+\dots+(x_n-a)^2}{n}.\label{*}\end{equation}
  
The quadratic deviation takes its smallest value when $a=\bar x$, where $\bar x$ is the arithmetic mean of $x_1,\ldots,x_n$:
+
The quadratic deviation takes its smallest value when $a=\bar x$, where $\bar x$ is the arithmetic mean of $x_1,\dots,x_n$:
  
$$\bar x=\frac{x_1+\ldots+x_n}{n}.$$
+
$$\bar x=\frac{x_1+\dots+x_n}{n}.$$
  
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:
+
In this case the quadratic deviation serves as a measure of the variance (cf. [[Dispersion|Dispersion]]) of the quantities $x_1,\dots,x_n$. Also used is the more general concept of a weighted quadratic deviation:
  
$$\sqrt\frac{p_1(x_1-a)^2+\ldots+p_n(x_n-a)^2}{p_1+\ldots+p_n},$$
+
$$\sqrt\frac{p_1(x_1-a)^2+\dots+p_n(x_n-a)^2}{p_1+\dots+p_n},$$
  
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:
+
where the $p_1,\dots,p_n$ are the so-called weights associated with $x_1,\dots,x_n$. The weighted quadratic deviation attains its smallest value when $a$ is the weighted mean:
  
$$\frac{p_1x_1+\ldots+p_nx_n}{p_1+\ldots+p_n}.$$
+
$$\frac{p_1x_1+\dots+p_nx_n}{p_1+\dots+p_n}.$$
  
 
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: $\sqrt{D(X)}$.
 
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: $\sqrt{D(X)}$.
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====Comments====
 
====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.
+
The expression \eqref{*} 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====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Rektorys (ed.) , ''Applicable mathematics'' , Iliffe  (1969)  pp. 1318</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.M. Mood,  F.A. Graybill,  "Introduction to the theory of statistics" , McGraw-Hill  (1963)  pp. 166, 176</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Rektorys (ed.) , ''Applicable mathematics'' , Iliffe  (1969)  pp. 1318</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.M. Mood,  F.A. Graybill,  "Introduction to the theory of statistics" , McGraw-Hill  (1963)  pp. 166, 176</TD></TR></table>

Latest revision as of 14:22, 30 December 2018

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

The square root of the expression

\begin{equation}\frac{(x_1-a)^2+\dots+(x_n-a)^2}{n}.\label{*}\end{equation}

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

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

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

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

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

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

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 \eqref{*} 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