Difference between revisions of "Quadratic deviation"
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− | ''quadratic variance, standard deviation, of quantities | + | {{TEX|done}} |
+ | ''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 | ||
− | + | $$\frac{(x_1-a)^2+\ldots+(x_n-a)^2}{n}.$$ | |
− | The quadratic deviation takes its smallest value when | + | 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|Dispersion]]) of the quantities | + | 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: |
− | + | $$\sqrt\frac{p_1(x_1-a)^2+\ldots+p_n(x_n-a)^2}{p_1+\ldots+p_n},$$ | |
− | where the | + | 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 | + | 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 |
Quadratic deviation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quadratic_deviation&oldid=31480