Wiki Page, Math 13 - Perez Sandbox : Module 9 - Linear Regression (3a of 4)

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2026-01-17T21:15:56-08:00

The numbers in the equation of a line

We now know that the least squares regression line is the line that best fits the data. It gives the most accurate predictions because it has the smallest overall prediction error. More specifically, it has the smallest sum of the squared errors (SSE).

Now we focus on the numbers in the equation of the line and discuss what they tell us about the relationship between the explanatory variable and the response variable.

The general form of the equation of a line

Here are some of the equations of lines that we have used in our discussion of linear relationships:

Predicated distance = 576 - 3*Age

Predicted height = 39 + 2.7*Forearm length

Predicted monthly care insurance premium = 97 - 1.45*Years of driving experience

Notice that the form of the equations is the same. In general, each equation has the form

When we find the least-squares regression line for a data set, a and b are determined by the data. The values of a and b do not change, so we refer to them as constants.

The meaning of a and b in the equation of a line

In the equation of the line, the constant a is the prediction when x = 0. It is called initial predicted value. In a graph of the line, a is the y-intercept.

In the equation of the line, the constant b is the rate of change, called the slope. In a graph of the least-squares line, b describes how the predictions change when x increases by one unit. More specifically, b describes the average change in the response variable when the explanatory variable increases by one unit.

We can write the equation of the line to reflect the meaning of a and b:

Predicted y-value = (initial value) + (rate of change)*x

Predicted y-value = (y-intercept) + (slope)*x

The meaning of a and b in the graph of a line

The constants a and b are shown in the graph of the line below.

Comment

In algebra class you may have learned that the equation of a line is y=mx+b. In this form of the equation of a line, m is the slope and b is the y-intercept. As we noted above, a statistician will tend to use b as the slope, instead of using b to represent the y-intercept. This can be confusing! In this course, we will write the equation of the line in the form y=a+bx. The two forms of the line are equivalent, despite the confusing use of b to represent different things!

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