Variance-Covariance Matrix

This lesson explains how to use matrix methods to generate a variance-covariance matrix from a matrix of raw data.

Variance

Variance is a measure of the variability or spread in a set of data. Mathematically, it is the average squared deviation from the mean score. We use the following formula to compute variance.

Var(X) = Σ ( Xi - X )2 / N = Σ xi2 / N

where

N is the number of scores in a set of scores

X is the mean of the N scores.

Xi is the ith raw score in the set of scores

xi is the ith deviation score in the set of scores

Var(X) is the variance of all the scores in the set

Covariance

Covariance is a measure of the extent to which corresponding elements from two sets of ordered data move in the same direction. We use the following formula to compute covariance.

Cov(X, Y) = Σ ( Xi - X ) ( Yi - Y ) / N = Σ xiyi / N

where

N is the number of scores in each set of data

X is the mean of the N scores in the first data
set

Xi is the ithe raw score in the first set of scores

xi is the ith deviation score in the first set of scores

Y is the mean of the N scores in the second data
set

Yi is the ithe raw score in the second set of scores

yi is the ith deviation score in the second set of scores

Cov(X, Y) is the covariance of corresponding scores in the two sets of data

Variance-Covariance Matrix

Variance and covariance are often displayed together in a variance-covariance matrix, (aka, a covariance matrix). The variances
appear along the diagonal and covariances appear in the off-diagonal elements, as shown below.

V =	Σ x12 / N Σ x1 x2 / N . . . Σ x1 xc / N Σ x2 x1 / N Σ x22 / N . . . Σ x2 xc / N . . . . . . . . . . . . Σ xc x1 / N Σ xc x2 / N . . . Σ xc2 / N

where

V is a c x c variance-covariance matrix

N is the number of scores in each of the c data sets

xi is a deviation score from the ith data set

Σ xi2 / N is the variance of elements from the ith data set

Σ xi xj / N is the covariance for elements from the ith and jth data sets