Here we will see the step by step calculation of Linear Regression with the help of an example.

Let's suppose that we are provided with the following data.

x | y |
---|---|

1 | 3 |

2 | 1 |

3 | 5 |

4 | 2 |

5 | 6 |

6 | 9 |

7 | 10 |

8 | 4 |

9 | 8 |

Plot all these data points on the graph.

Now, our goal is to draw the regression line which passes through all these points with the least error.

### Step-1: Calculate the mean of the x and y-axis.

**Note: **The line of regression will pass through the mean of x & y always.

Mean of x-axis=5.0

Mean of y-axis=5.34

Now we will draw a point where these two values will intersect. Thus this is the point from where our line of regression will pass.

Here, the blue point is the mean value of the x and y-axis.

### Step-2: Calculate the value of m

As we know the equation of a line is y=mx+c. Here

where (x-x̄) is nothing but the distance of all the points from x=5.

( y-ȳ)s nothing but the distance of all the points from y=5.34.

x | y | x-x̄ | y-ȳ | (x-x̄)(y-ȳ) | (x-x̄)^{2} |
---|---|---|---|---|---|

1 | 3 | -4 | -2.34 | 9.34 | 16 |

2 | 1 | -3 | -4.34 | 13 | 9 |

3 | 5 | -2 | -0.34 | 0.67 | 4 |

4 | 2 | -1 | -3.34 | 3.34 | 1 |

5 | 6 | 0 | 0.67 | 0 | 0 |

6 | 9 | 1 | 3.67 | 3.67 | 1 |

7 | 10 | 2 | 4.67 | 4.67 | 4 |

8 | 4 | 3 | -1.34 | -1.34 | 9 |

9 | 8 | 4 | 2.67 | 2.67 | 16 |

Then,

Therefore,

m=0.6

### Step-3: Calculate the value of c

Equation of line,** y=mx+c.**

From the above calculations, we got

x=5, y=5.34 and m=0.6

Therefore,

5.34=5*0.6+c

c=2.34

### Step-4: Find the equation of Regression Line using Equation of a line

From all the above-calculated values, we can conclude that our regression line would intersect the y-axis at 2.94 and it will also cross the point (5,5.34).

Therefore, the equation of the Regression line would be

y=0.6x+2.34.

### Step-5: We will check whether the variables are dependent on independent variables or not for this process firstly we will predict the values of y.

For m=0.6,

c=2.34,

y=0.6x+2.34,

Then calculate the R^{2} for the given data Here R^{2} is the **goodness of fit.**

x | y | y-ȳ | y_{p} | y_{p-}ȳ | (y-ȳ)^{2} | (y_{p-}ȳ)^{2} |
---|---|---|---|---|---|---|

1 | 3 | -2.34 | 2.94 | -2.4 | 5.4 | 5.76 |

2 | 1 | -4.34 | 3.54 | -1.8 | 18.8 | 3.24 |

3 | 5 | -0.34 | 4.14 | -1.2 | 0.11 | 1.44 |

4 | 2 | -3.34 | 4.74 | -0.6 | 11.1 | 0.36 |

5 | 6 | 0.67 | 5.34 | 0 | 0.44 | 0 |

6 | 9 | 3.67 | 5.94 | 0.6 | 13.4 | 0.36 |

7 | 10 | 4.67 | 6.54 | 1.2 | 21.8 | 1.44 |

8 | 4 | -1.34 | 7.14 | 1.8 | 1.7 | 3.24 |

9 | 8 | 2.67 | 7.74 | 2.4 | 7.1 | 5.76 |

R^{2} =0.27

Here, R^{2} tends towards 0 in this case we can say that independent variables are not at all related to dependent variables. More the values of R^{2 }more will be the dependency. If we increase the value of R^{2} the error will decrease.

Here green line represents the line of regression.

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