# 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.

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

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:

**Wiki link**

**References**

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