# What is the intercept and slope in the regression

## Linear 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 = ∑ [(x_{i} - ∅x) × (y_{i} - ∅y)] / ∑ (x_{i} - ∅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

#### Regression line

The regression line as a linear function is then: 24 + 0.1 × height.

Generally as **formula**:

y_{i} = α + β × x_{i}

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 x_{i} or y_{i} 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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