Math Core

When both variables are quantitative, a two-way table no longer works. Instead you plot each individual as a point. A scatterplot is the most important graph in this unit: before you compute a correlation or fit a line, you always look at the picture first.

Explanatory and response variables

Many studies are designed to see whether one variable helps explain or predict another.

Definition

Explanatory and response variables

A response variable measures an outcome of a study. An explanatory variable may help explain or predict changes in the response variable.

On a scatterplot, the explanatory variable goes on the horizontal ( $x$ ) axis and the response variable goes on the vertical ( $y$ ) axis.

Sometimes the roles are obvious: rainfall helps explain crop yield, not the other way around. Sometimes there is no clear choice, such as the math and reading scores of the same students. Then either variable can go on either axis, and you are simply exploring the relationship. Calling a variable "explanatory" does not mean it causes changes in the response. It only describes the role it plays in the analysis.

Making a scatterplot

To make a scatterplot, label both axes with the variable names and units, choose scales that cover the data (the axes don't need to start at $0$ ), and plot one point for each individual. Here are the arm spans and heights of $10$ students, in centimeters.

Arm span (cm)	$150$	$155$	$158$	$162$	$165$	$170$	$172$	$176$	$180$	$185$
Height (cm)	$152$	$154$	$160$	$161$	$167$	$168$	$174$	$175$	$182$	$183$

x-axis: arm span (cm). y-axis: height (cm).Open in grapher →

Describing a scatterplot

AP Statistics expects a four-part description, always in context.

Direction, form, strength and unusual features

Direction: In a positive association, larger values of $x$ tend to go with larger values of $y$ . In a negative association, larger values of $x$ tend to go with smaller values of $y$ .
Form: Is the overall pattern linear or nonlinear (curved)?
Strength: How closely do the points follow the form? Strong, moderate or weak.
Unusual features: Are there outliers (points that fall outside the overall pattern) or distinct clusters?

For the arm span data: there is a strong, positive, linear association between arm span and height for these students, with no obvious outliers. Students with longer arm spans tend to be taller.

Notice the word tend. In this small sample, every longer arm span happens to go with a greater height, but that won't be true in a larger group: a new student with a $170$ cm span could easily be shorter than one with a $166$ cm span. Associations describe overall trends, not guarantees for every pair of individuals.

Worked example: A negative association

A café records the daily high temperature (°F) and the number of hot cocoas sold on $9$ days. Describe the relationship.

x-axis: daily high temperature (°F). y-axis: hot cocoas sold.Open in grapher →

Temperature is the explanatory variable and cocoa sales is the response. There is a strong, negative, linear association between daily high temperature and hot cocoas sold, with no outliers. On warmer days, the café tends to sell fewer hot cocoas.

Worked example: A curved pattern

An engineer measures the fuel efficiency of a test car at several steady speeds. Describe the relationship.

x-axis: speed (mph). y-axis: fuel efficiency (miles per gallon).Open in grapher →

The form is nonlinear: fuel efficiency rises as speed increases up to about $50$ mph and then falls. The points follow the curve closely, so the relationship is strong. There are no outliers. Direction doesn't apply to the whole plot: the association is positive on the left and negative on the right, so a single word like "positive" would be misleading.

Worked example: Clusters and an outlier

A park ranger records how long each eruption of a geyser lasts and the waiting time until the next eruption. Describe the plot.

x-axis: eruption length (minutes). y-axis: wait until next eruption (minutes).Open in grapher →

There is a positive association: longer eruptions tend to be followed by longer waits. The most striking feature is two clusters: short eruptions (about $2$ minutes) followed by waits of $50$ to $60$ minutes, and long eruptions (about $4$ to $4.7$ minutes) followed by waits of $78$ to $90$ minutes. The point at $(3.0, 95)$ is a possible outlier: a medium-length eruption followed by the longest wait of all, far from both clusters. Within each cluster the association is weak.

Common mistake

Don't describe a scatterplot with only "it's positive." A complete answer names direction, form, strength and any unusual features, and uses the variable names in context. "Strong, positive, linear" alone earns less credit than "a strong, positive, linear association between arm span and height."

Why form matters so much

The rest of this unit is about linear models: the correlation coefficient and the least-squares regression line. Both are only appropriate when the form is roughly linear. A curved scatterplot, like the fuel efficiency data, needs a different approach, which you'll see in the last lesson of the unit. So the scatterplot is not a formality. It decides which tools you are allowed to use.

Tip

Judge strength by how tightly the points hug the pattern, not by how steep the pattern is. A nearly flat cloud of points can show a very strong relationship, and a steep one can be weak. Changing the scale of an axis changes how steep a plot looks, but not how strong the association is.

Practice

Practice 1

A researcher wants to know whether the number of hours a student works at a part-time job each week helps predict the student's GPA. Which choice correctly assigns the variables?

Practice 2

Which description best fits this scatterplot?

x-axis: age of a laptop (years). y-axis: battery life (hours).Open in grapher →

Practice 3

In the geyser example, which statement is true?

Practice 4

For $50$ countries, a scatterplot shows internet access (percent of the population) on the $x$ -axis and life expectancy (years) on the $y$ -axis. The points rise steeply at first and then level off near $80$ years. Which description is best?

Practice 5

A scatterplot of shoe size ( $x$ ) and height ( $y$ ) for $30$ adults shows a strong positive linear pattern. The researcher now converts every height from inches to centimeters and redraws the plot with a new $y$ -axis scale. What happens to the strength of the association?

Practice 6

In the arm span data, which point, if added, would be an outlier?

Practice 7

A student says: "The scatterplot of ice cream sales versus drowning deaths by month is strong and positive, so ice cream is the explanatory variable and it causes drownings." What is the main flaw?