information:classification
As a scalar

Another generalization of variance for vector-valued random variables X {\displaystyle X}, which results in a scalar value rather than in a matrix, is the generalized variance det (C) {\displaystyle \det(C)}, the determinant of the covariance matrix. The generalized variance can be shown to be related to the multidimensional scatter of points around their mean.

A different generalization is obtained by considering the equation for the scalar variance, Var (X) = E {\displaystyle \operatorname {Var} (X)=\operatorname {E} \left}, and reinterpreting (X − μ) 2 {\displaystyle (X-\mu)^{2}} as the squared Euclidean distance between the random variable and its mean, or, simply as the scalar product of the vector X − μ {\displaystyle X-\mu } with itself. This results in E = tr (C), {\displaystyle \operatorname {E} \left=\operatorname {tr} (C),} which is the trace of the covariance matrix.