What Are Linear Models?

You use a linear model when your points lie on an approximately straight line. Each set of points has its own unique best-fit model. Because there’s an infinite amount of ways to create a collection of points that can be modeled as a linear expression, there’s an infinite number of graphs that fit the expression below. The only difference between these graphs is the values of the slope a and the constant term b.


Linear Model

A linear function (straight line) is written like this:

f(x) = ax + b

In this expression, a is the slope and b is where the graph intersects the y-axis.

When something increases or decreases by the same amount all the time, you have linear growth.

Linear regression is regression where you want to find the straight line f(x) = ax + b that best fits a set of points. You will use digital tools for this. A plot for a linear regression will look like this:

Plot of linear regression

Plot of linear regression

The Linear Correlation Coefficient

For linear regression, you use the correlation coefficient r as a measure of how well the function fits the points. The value r varies from 1 to 1, where

r = 1:

Perfectly adapted to the points, and the function increases as x increases.

r = 0:

No correlation. The variables are linearly independent.

r = 1:

Perfectly adapted to the points, and the function decreases as x increases.

This means that if we have r2 = 1, the regression matches the points perfectly, and if r2 = 0, there is no correlation between the points and the function. The larger r2 is, the less the points deviate from the line. That means you want the largest possible r2—but that’s out of your control.

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