8.5: Interpreting the Slope of a Line
Learning Outcomes
- Interpret the slope of a line as the change in \(y\) when \(x\) changes by 1.
Interpreting the Slope of a Line
For every increase in the \(x\)-variable by 1, the \(y\)-variable tends to change by the amount of the slope.A common issue when we learn about the equation of a line in algebra is to state the slope as a number, but have no idea what it represents in the real world. The slope of a line is the rise over the run. If the slope is given by an integer or decimal value we can always put it over the number 1. In this case, the line rises by the slope when it runs 1. "Runs 1" means that the \(x\)-value increases by 1 unit. Therefore the slope represents how much the \(y\)-value changes when the \(x\)-value changes by 1 unit. In statistics, especially regression analysis, the \(x\)-value has real life meaning and so does the \(y\)-value.
Example \(\PageIndex{1}\)
A study was done to see the relationship between the time it takes, \(x\), to complete a college degree and the student loan debt incurred, \(y\). The equation of the regression line was found to be:
\[y=25,142\:+14,329x \nonumber\]
Interpret the slope of the regression line in the context of the study.
Solution
First, note that the slope is the coefficient in front of the \(x\). Thus, the slope is 14,329. Next, the slope is the rise over the run, so it helps to write the slope as a fraction:
\[Slope\:=\dfrac{\:rise}{run}=\dfrac{14,329}{1} \nonumber\]
The rise is the change in \(y\) and \(y\) represents student loan debt. Thus, the numerator represents an increase of $14,329 of student loan debt. The run is the change in \(x\) and \(x\) represents the time it takes to complete a college degree. Thus, the denominator represents an increase of 1 year to complete a college degree. We can put this all together and interpret the slope as telling us that:
For every additional year it takes to complete a college degree, on average the student loan debt tends to increase by $14,329.
Example \(\PageIndex{2}\)
Suppose that a research group tested the cholesterol level of a sample of 40-year-old women and then waited many years to see the relationship between a woman's HDL cholesterol level in mg/dl, \(x\), and her age of death, \(y\). The equation of the regression line was found to be:
\[y=103\:-0.3x \nonumber\]
Interpret the slope of the regression line in the context of the study.
Solution
The slope of the regression line is -0.3. The slope as a fraction is:
\[Slope\:=\dfrac{\:rise}{run}=\dfrac{-0.3}{1} \nonumber\]
The rise is the change in \(y\) and \(y\) represents age of death. Since the slope is negative, the numerator indicates a decrease in lifespan in terms of a lower age of death. Thus, the numerator represents a decrease in lifespan of 0.3 years. The run is the change in \(x\) and \(x\) represents the HDL cholesterol level. Thus, the denominator represents an HDL cholesterol level increase of 1 mg/dl. Now, put this all together and interpret the slope as telling us that:
For every additional 1 mg/dl of HDL cholesterol level, on average women are predicted to die 0.3 years younger.
Try It
The scatterplot and regression line below are from a study that collected data on the population (in hundred thousands) of cities, \(x\), and the average number of hours per week the city's residents spend outdoors, \(y\). The equation of the regression line was found to be:
\[y = 17\:-0.93x \nonumber\]
Interpret the slope of this regression line in the context of the study.
- Answer
-
For every additional 100,000 people in the population of a city, on average the number of hours per week the city's residents spend outdoors decreases by 0.93 hours.
Additional video resources:
Interpret the Meaning of the Slope of a Linear Equation - Smokers