Covariance

Covariance

Covariance is a non-standardized measure of how much two random variables X and Y change together in a linear way. A strong positive covariance indicates that greater values in one variable correspond to greater values in the other variable. A strong negative covariance indicates that greater values in one variable correspond to smaller values in the other variable.

Requirements:

Both random variables must be at least interval scaled and bivariate normal distribution is required.

Illustration of a bivariate normal distribution

Calculation:

Suppose we have two normally distributed random variables x and y.

x_i and y_i denote the values of x and y for case i.

We then first calculate the cross-product deviation for variables x and y

then the covariance is defined as:

Example of a covariance

A psychologist is interested in his new learning program. 15 Subjects learn a list of 50 words. Learning performance is measured using a recall test. After the first test all subjects are instructed how to use the learning program and then learn a second list of 50 words. Learning performance is again measured with the recall test. In the following table the number of correct remembered words are listed for both tests.

We then first calculate the cross-product deviation for variables x and y

then the covariance is:

We get a positive covariance, so greater values in x correspond to greater values in y. The following figure illustrates this:

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References