What is the variance-covariance matrix?

A variance-covariance matrix is a square matrix that contains the variances and covariances associated with several variables. The diagonal elements of the matrix contain the variances of the variables and the off-diagonal elements contain the covariances between all possible pairs of variables.

For example, you create a variance-covariance matrix for three variables X, Y, and Z. In the following table, the variances are displayed in bold along the diagonal; the variance of X, Y, and Z are 2.0, 3.4, and 0.82 respectively. The covariance between X and Y is -0.86.
X 2.0 -0.86 -0.15
Y -0.86 3.4 0.48
Z -0.15 0.48 0.82

The variance-covariance matrix is symmetric because the covariance between X and Y is the same as the covariance between Y and X. Therefore, the covariance for each pair of variables is displayed twice in the matrix: the covariance between the ith and jth variables is displayed at positions (i, j) and (j, i).

Many statistical applications calculate the variance-covariance matrix for the estimators of parameters in a statistical model. It is often used to calculate standard errors of estimators or functions of estimators. For example, logistic regression creates this matrix for the estimated coefficients, letting you view the variances of coefficients and the covariances between all possible pairs of coefficients.

For most statistical analyses, if a missing value exists in any column, Minitab ignores the entire row when it calculates the correlation or covariance matrix. However, when you calculate the covariance matrix by itself, Minitab does not ignore entire rows in its calculations when there are missing values. To obtain only the covariance matrix, choose Stat > Basic Statistics > Covariance.

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