# Quadratic deviation

*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:**

Standard deviation.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Standard_deviation&oldid=43775