What is the intercept and slope in the regression
Example linear regression
For 3 men (Anton, Bernd and Claus) the height in cm and the shoe size are recorded. Is there a linear relationship? - and how can this be expressed in a formula? In this case, the shoe size (variable y) from the height (variable x) can be derived or forecast.
First, the arithmetic mean values (averages) for body height and shoe size are calculated:
- Height: (170 cm + 180 cm + 190 cm) / 3 = 180 cm.
- Shoe size: (41 + 42 + 43) / 3 = 126/3 = 42.
Then (in the columns of the table)
- the respective deviations from the mean value are calculated for the body size
- the height deviation is squared
- the respective deviations from the mean value are calculated for the shoe size and
- the deviations from the mean are multiplied in each case.
Calculate the slope
Now the sum of the multiplied deviations is divided by the sum of the squared deviations in body height: 20/200 = 0.1.
As formula: β = ∑ [(xi - ∅x) × (yi - ∅y)] / ∑ (xi - ∅x)2
The slope of the regression line determined in this way corresponds to the quotient of the covariance (20/3) and the variance of the body size (200/3).
Calculate the intercept
As a final step, the value just calculated times the average height is subtracted from the average of the shoe size: 42 - 0.10 × 180 = 24.
As formula: α = ∅y - β × ∅x
The regression line as a linear function is then: 24 + 0.1 × height.
Generally as formula:
yi = α + β × xi
Here, α (24) is the point of intersection with the y-axis (the shoe sizes in the example therefore start at 24, with the theoretical body size 0), β (0.1) is the slope of the regression line and xi or yi are the respective body and shoe sizes.
Here, α and β are also used as Regression coefficients denotes or α as Regression constant; ß indicates by how many units y increases when x increases by one unit (in the example: if the body height increases by 1 cm, the shoe size increases by 0.1 shoe sizes).
For Anton: 24 + 0.1 × 170 = 41.
For Bernd: 24 + 0.1 × 180 = 42.
For Claus: 24 + 0.1 × 190 = 43.
The regression formula can now be used to estimate or predict the shoe size (for other men) based on a body size.
The scatter diagram for the 3 measurement data including the regression line:
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