Covariance/Variance Matrix

Describing the Matrix

A matrix is a two-dimensional array of numbers indicating some type of relational information among the variables. Each entry in the matrix indicates a type of relationship between the elements listed in the rows (considered the i term) and the column (considered the j term). Each entry in the matrix can be designated by a combination that becomes the i, j position in the matrix. This matrix is considered reflexive such that the same number is present for the ij terms (e.g., 2, 3) as there would be for the same term using a different order but same values (e.g., 3, 2). So the term that is the relationship of 3, 2 has the same value as 2, 3.

Table 1 Example of Variance/Covariance Matrix

The diagonal elements represent the variance generated for the particular variable and are a unique element (e.g., 1, 1 or 2, 2) such that no other element indicating the same relationship exists, unlike the off-diagonal elements, which have a mirror image. The formula for variance is as follows:

1

This formula is replicated for each variable in the analysis. In Table 1, this is done a total of four times, once for the A, B, C, and D variables, and found in the diagonal. One can examine the diagonals and see the relative size of the variance for each variable. If the same essential metric is used, then the matrices will provide the relative size of the variance of the variables when compared to each other. The variance contains the sum of squares in the numerator, an important element for use in statistics such as analysis of variance (ANOVA). The variance provided can be converted to a standard deviation by simply taking the square root of the variance. What this means is that the matrix contains a term that, with minimal statistical manipulation, can be used in a variety of applications.

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