Editing Variance (section)

====As a scalar====
Another generalization of variance for vector-valued random variables <math>X</math>, which results in a scalar value rather than in a matrix, is the [[generalized variance]] <math>\det(C)</math>, the [[determinant]] of the covariance matrix. The generalized variance can be shown to be related to the multidimensional scatter of points around their mean.<ref>{{cite book |last1=Kocherlakota |first1=S. |title=Encyclopedia of Statistical Sciences |last2=Kocherlakota |first2=K. |chapter=Generalized Variance |publisher=Wiley Online Library |doi=10.1002/0471667196.ess0869 |year=2004 |isbn=0-471-66719-6 }}</ref>

A different generalization is obtained by considering the equation for the scalar variance, <math> \operatorname{Var}(X) = \operatorname{E}\left[(X - \mu)^2 \right] </math>, and reinterpreting <math>(X - \mu)^2</math> as the squared [[Euclidean distance]] between the random variable and its mean, or, simply as the scalar product of the vector <math>X - \mu</math> with itself. This results in <math>\operatorname{E}\left[(X - \mu)^{\mathsf{T}}(X - \mu)\right] = \operatorname{tr}(C),</math> which is the [[Trace (linear algebra)|trace]] of the covariance matrix.