Editing Variance (section)

===Discrete random variable===

If the generator of random variable <math>X</math> is [[Discrete probability distribution|discrete]] with [[probability mass function]] <math>x_1 \mapsto p_1, x_2 \mapsto p_2, \ldots, x_n \mapsto p_n</math>, then

<math display="block">\operatorname{Var}(X) = \sum_{i=1}^n p_i \cdot {\left(x_i - \mu\right)}^2,</math>

where <math>\mu</math> is the expected value. That is,

<math display="block">\mu = \sum_{i=1}^n p_i x_i .</math>

(When such a discrete [[weighted variance]] is specified by weights whose sum is not&nbsp;1, then one divides by the sum of the weights.)

The variance of a collection of <math>n</math> equally likely values can be written as

<math display="block"> \operatorname{Var}(X) = \frac{1}{n} \sum_{i=1}^n (x_i - \mu)^2 </math>

where <math>\mu</math> is the average value. That is,

<math display="block">\mu = \frac{1}{n}\sum_{i=1}^n x_i .</math>

The variance of a set of <math>n</math> equally likely values can be equivalently expressed, without directly referring to the mean, in terms of squared deviations of all pairwise squared distances of points from each other:<ref>{{cite conference|author=Yuli Zhang |author2=Huaiyu Wu |author3=Lei Cheng |date=June 2012|title=Some new deformation formulas about variance and covariance|conference=Proceedings of 4th International Conference on Modelling, Identification and Control(ICMIC2012)|pages=987–992}}</ref>

<math display="block"> \operatorname{Var}(X) = \frac{1}{n^2} \sum_{i=1}^n \sum_{j=1}^n \frac{1}{2} {\left(x_i - x_j\right)}^2 = \frac{1}{n^2} \sum_i \sum_{j>i} {\left(x_i - x_j\right)}^2. </math>