The least squares regression line
Posted in Maths, Statistics 2The least squares regression line is the line which produces the smallest value of the sum of the squares of the residuals. A residual is the vertical distance from a point on a scatter diagram to the line of best fit. Therefore the least squares regression line can be seen as the best line of best fit.
The equation of the least squares regression line of y on x is:
Where b is:
As you can see there are 3 different ways to calculate the value for b, based on what information you are given in the question
Example
Calculate the least squared regression line of y on x from the following data:
| x | 20 | 30 | 40 | 50 | 60 | 70 |
|---|---|---|---|---|---|---|
| y | 2.49 | 2.41 | 2.38 | 2.14 | 1.97 | 2.03 |
Firstly draw up a table of the values you need and fill it out. In this example we'll use the third equation for b so we need all the values of x2 and xiyi:
| x | y | x2 | xiyi | |
|---|---|---|---|---|
| 20 | 2.49 | 400 | 49.9 | |
| 30 | 2.41 | 900 | 72.3 | |
| 40 | 2.38 | 1600 | 95.2 | |
| 50 | 2.14 | 2500 | 107 | |
| 60 | 1.97 | 3600 | 118.2 | |
| 70 | 2.03 | 4900 | 142.1 | |
| Sum: | 270 | 13.42 | 13900 | 584.7 |
Next step is to calculate the means of the x and y values:
Now the value of b can be calculated:
Hence the equation is therefore:
Cleaning it up: