# Covariance

The **covariance** is the **arithmetic mean** of the **products of deviations** of each **variable** to their respective **means**.

**Covariance** is denoted by **cov(X,Y)**.

The covariance indicates the sign of the correlation between the variables.

If **cov(X, Y) > 0** the **correlation** is **positive**.

If **cov(X, Y) < 0** the **correlation** is **negative**.

As a disadvantage, its value depends on the chosen scale. That is, the covariance will vary if the height is expressed in meters or feet. It will also vary if money is expressed in euros or dollars.

#### Examples

The scores of 12 students in their mathematics and physics classes are:

Mathematics | 2 | 3 | 4 | 4 | 5 | 6 | 6 | 7 | 7 | 8 | 10 | 10 |

Physics | 1 | 3 | 2 | 4 | 4 | 4 | 6 | 4 | 6 | 7 | 9 | 10 |

Find the covariance of the distribution.

x_{i} |
y_{i} |
x_{i} · y_{i} |
---|---|---|

2 | 1 | 2 |

3 | 3 | 9 |

4 | 2 | 8 |

4 | 4 | 16 |

5 | 4 | 20 |

6 | 4 | 24 |

6 | 6 | 36 |

7 | 4 | 28 |

7 | 6 | 42 |

8 | 7 | 56 |

10 | 9 | 90 |

10 | 10 | 100 |

72 | 60 | 431 |

After tabulating the data, the arithmetic means can be found:

The values of the two variables X and Y are distributed according to the following table:

Y/X | 0 | 2 | 4 |
---|---|---|---|

1 | 2 | 1 | 3 |

2 | 1 | 4 | 2 |

3 | 2 | 5 | 0 |

Next, find the covariance of the distribution.

Convert the double entry table into a simple table and compute the arithmetic means.

x_{i} |
y_{i} |
f_{i} |
x_{i} · f_{i} |
y_{i} · f_{i} |
x_{i} · y_{i }· f_{i} |
---|---|---|---|---|---|

0 | 1 | 2 | 0 | 2 | 0 |

0 | 2 | 1 | 0 | 2 | 0 |

0 | 3 | 2 | 0 | 6 | 0 |

2 | 1 | 1 | 2 | 1 | 2 |

2 | 2 | 4 | 8 | 8 | 16 |

2 | 3 | 5 | 10 | 15 | 30 |

4 | 1 | 3 | 12 | 3 | 12 |

4 | 2 | 2 | 8 | 4 | 16 |

20 | 40 | 41 | 76 |