# Linear Regression

The regression line is the line that best fits or represents the data on the scatter plot.

### Line of Regression of Y on X

The regression line of y on x is used to estimate the values of y from x.

The slope of the line is the quotient between the covariance and variance of the variable X.

### Line of Regression of X on Y

The regression line of x on y is used to estimate the values of x from the values of y.

The slope of the line is the quotient between the covariance and variance of the variable y.

If **r = 0** the **regression lines** are **perpendicular** to each other, and their equations are:

y =

x =

#### 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 regression lines and represent them.

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 |

1. Find the arithmetic means.

2. Calculate the covariance.

3. Calculate the variances.

4.Linear regression of y on x.

5.Linear regression of x on y.