probability distribution is analogous to the moment of inertia in classical mechanics of a corresponding mass distribution along a line, with respect to rotation about its center of mass. It is because of this analogy that such things as the variance are called moments of probability distributions. The covariance matrix is related to the moment of inertia tensor for multivariate distributions. The moment of inertia of a cloud of n points with a covariance matrix of is given by This difference between moment of inertia in physics and in statistics is clear for points that are gathered along a line. Suppose many points are close to the x axis and distributed along it. The covariance matrix might look like That is, there is the most variance in the x direction. Physicists would consider this to have a low moment about the x axis so the moment-of-inertia tensor is Semivariance The semivariance is calculated in the same manner as the variance but only those observations that fall below the mean are included in the calculation:It is also described as a specific measure in different fields of application. For skewed distributions, the semivariance can provide additional information that a variance does not. For inequalities associated with the semivariance, see . Generalizations For complex variables If is a scalar complex-valued random variable, with values in then its variance is where is the complex conjugate of This variance is a real scalar. For vector-valued random variables As a matrix If is a vector-valued random variable, with values in and thought of as a column vector, then a natural generalization of variance is where and is the transpose of and so is a row vector. The result is a positive semi-definite square matrix, commonly referred to as the variance-covariance matrix (or simply as the covariance matrix). If is a vector- and complex-valued random variable, with values in then the covariance matrix is where is the conjugate transpose of This matrix is also positive semi-definite and square. As a scalar Another generalization of variance for vector-valued random variables , which results in a scalar value rather than in a matrix, is the generalized variance , 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 Euclidean distance between the random variable and its mean.