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

 

bivariate normal distribution

Calculation:

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

xi and yi denote the values of x and y for case i.

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

 

cross-product deviation

 

cross-product deviation 2

 

then the covariance is defined as:

 

covariance

 


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.

 

  x y x*y
  2 1 2
  1 2 2
  9 6 54
  5 4 20
  3 2 6
Σ 20 15 84

 

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

 

cross-product deviation

 

then the covariance is:

 

covariance

 

covariance

 

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

 

scatterplot covariance

 


Wiki link


References

Bortz, J. (2005). Statistik für Human- und Sozialwissenschaftler (6th Edition). Heidelberg: Springer Medizin Verlag.




 

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