# Linear Correlation Coefficient

The **linear correlation coefficient** is the **ratio** between the **covariance** and the **product of standard deviations** of both variables.

The **linear correlation coefficient** is denoted by the letter **r**.

### Properties of the Correlation Coefficient

1. The correlation coefficient does not change the measurement scale.

That is, if the height is expressed in meters or feet, the correlation coefficient does not change.

2. The sign of the correlation coefficient is the same as the covariance.

3. The linear correlation coefficient is a real number between −1 and 1.

**−1 ≤ r ≤ 1**

4. If the **linear correlation coefficient** takes values closer to **−1**, the **correlation** is **strong and negative**, and will become stronger the closer **r** approaches −1.

5. If the **linear correlation coefficient** takes values close to **1** the **correlation** is **strong and positive**, and will become stronger the closer r approaches 1

6. If the **linear correlation coefficient** takes values close to **0**, the **correlation** is **weak**.

7. If** r = 1** or **r = −1**, there is **perfect correlation** and the line on the scatter plot is increasing or decreasing respectively.

8. If **r = 0**, there is **no linear correlation**.

#### Example

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 correlation coefficient distribution and interpret it.

x_{i} |
y_{i} |
x_{i} ·y_{i} |
x_{i}^{2} |
y_{i2 } |
---|---|---|---|---|

2 | 1 | 2 | 4 | 1 |

3 | 3 | 9 | 9 | 9 |

4 | 2 | 8 | 16 | 4 |

4 | 4 | 16 | 16 | 16 |

5 | 4 | 20 | 25 | 16 |

6 | 4 | 24 | 36 | 16 |

6 | 6 | 36 | 36 | 36 |

7 | 4 | 28 | 49 | 16 |

7 | 6 | 42 | 49 | 36 |

8 | 7 | 56 | 64 | 49 |

10 | 9 | 90 | 100 | 81 |

10 | 10 | 100 | 100 | 100 |

72 | 60 | 431 | 504 | 380 |

1º Find the arithmetic means.

2º Calculate the covariance.

3º Calculate the standard deviations.

4º Apply the formula for the linear correlation coefficient.

The correlation is positive.

As the correlation coefficient is very close to 1, the correlation is very strong.

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 |

Calculate the correlation coefficient.

Turn the double entry table into a single table.

x_{i} |
y_{i} |
f_{i} |
x_{i} · f_{i} |
x_{i}^{2 } · f_{i} |
y_{i} · f_{i} |
y_{i2 } · f_{i} |
x_{i} · y_{i }· f_{i} |
---|---|---|---|---|---|---|---|

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

0 | 2 | 1 | 0 | 0 | 2 | 4 | 0 |

0 | 3 | 2 | 0 | 0 | 6 | 18 | 0 |

2 | 1 | 1 | 2 | 4 | 1 | 1 | 2 |

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

2 | 3 | 5 | 10 | 20 | 15 | 45 | 30 |

4 | 1 | 3 | 12 | 48 | 3 | 3 | 12 |

4 | 2 | 2 | 8 | 32 | 4 | 8 | 16 |

20 | 40 | 120 | 41 | 97 | 76 |

The correlation is negative.

As the correlation coefficient is very close to 0, the correlation is very weak.