The Slope of a Fitted Line

Student Summary

Here is a scatter plot that we have seen before. As noted earlier, we can see from the scatter plot that taller dogs tend to weigh more than shorter dogs.

Another way to say it is that weight tends to increase as height increases.

When we have a positive association between two variables, an increase in one means there tends to be an increase in the other.

Scatterplot.
A scatterplot. Horizontal, from 6 to 30, by 3’s, labeled dog height, inches. Vertical, from 0 to 112, by 16’s, labeled dog weight, pounds. 24 data points.  Trend upward and to right.

We can quantify this tendency by fitting a line to the data and finding its slope.

For example, the equation of the fitted line is w=4.27h−37w = 4.27h -37, where hh is the height of the dog and ww is the predicted weight of the dog.

The slope is 4.27, which tells us that for every 1-inch increase in dog height, the weight is predicted to increase by 4.27 pounds.

A scatterplot, horizontal, dog height in inches, 6 to 30 by 3, vertical, 0 to 112 by 16. Same scatterplot as previous, this time with a line through 9 comma 0 and 27 comma 80.

In our example of the fuel efficiency and weight of a car, the slope of the fitted line shown is -0.01.

Scatterplot, weight, kilograms, 1000 to 2500 by 250, fuel efficiency, miles per gallon, 14 to 32 by 2. Points are arranged close to the line through 1100 comma 28 down and right through 2300 comma 14.

This tells us that for every 1-kilogram increase in the weight of the car, the fuel efficiency is predicted to decrease by 0.01 mile per gallon (or, after multiplying both values by 100, every 100-kilogram increase corresponds to a predicted decrease of 1 mpg). 

When we have a negative association between two variables, an increase in one means there tends to be a decrease in the other.

Visual / Anchor Chart

Standards

Building On
8.EE.6

Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

Addressing
8.SP.1

Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

8.SP.2

Understand that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.

8.SP.3

Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.

Building Toward
8.SP.3

Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.