Sample variance and standard deviation

Alicia Y. Tsai; Authors: Laurent EI Ghaoui; Contributors: Phan Hoang, Tony Tin, Ha To Tam, Nguyen Van Kep, Le Dinh Nam, Juliette Decugis, Nguyen Quoc Cuong.; Giuseppe C. Calafiore

Sample variance and standard deviation

The sample variance of given numbers $x_1, \cdots, x_n$ , is defined as

$\begin{align*} \sigma^2 &:= \frac{1}{n} ((x_1 -\hat{x})^2 + \cdots + (x_n - \hat{x})^2), \end{align*}$

where $\hat{x}$ is the sample average of $x_1, \cdots, x_n$ . The sample variance is a measure of the deviations of the numbers $x_i$ with respect to the average value $\hat{x}$ .

The sample standard deviation is the square root of the sample variance, $\sigma^2$ . It can be expressed in terms of the Euclidean norm of the vector $x = (x_1, \cdots, x_n)$ , as

$\begin{align*} \sigma &= \frac{1}{\sqrt{n}}\|x-\hat{x}{\bf 1}\|_2, \end{align*}$

where $||\cdot||_2$ denotes the Euclidean norm.

More generally, for any vector $p \in \mathbb{R}^n$ , with $p_i \ge 0$ for every $i$ , and $p_1 + \cdots + p_n = 1$ , we can define the corresponding weighted variance as

$\begin{align*} \sum\limits_{i=1}^n p_i(x_i - \hat{x})^2. \end{align*}$

The interpretation of $p$ is in terms of a discrete probability distribution of a random variable $X$ , which takes the value $x_i$ with probability $p_i$ , $i = 1, \cdots, n$ . The weighted variance is then simply the expected value of the squared deviation of $X$ from its mean ${\bf E}(X)$ , under the probability distribution $p$ .

See also: Sample and weighted average.

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