Scatterplots

We are often presented with questions that ask about relationships between two quantities or variables. Consider the context in Example 1.

Example
Scatterplot
Stop & Think
Analyzing Data

What is the relationship between average ACT scores and the proportion of high school seniors taking the exam? Below is a table with the average ACT scores and the proportion of seniors taking the exam by state. [from NCSSM Precal text]

Screenshot 2024-05-02 at 8.25.49 AM.png

To explore the relationship between these two variables, we can create a scatter plot of the data.

On the scatter plot shown below, average composite ACT scores are plotted on the vertical axis and the percentage of graduates taking the exam from each state are plotted on the horizontal axis. For example, the ordered pair representing New Mexico (70,19.9) implies that 70% of the graduates in 2016 living in New Mexico took the ACT and their average composite score was 19.9, while the ordered pair for California is (33,22.6), indicating that 33% of 2016 graduates took the ACT exam with an average composite score of 22.6.

Screenshot 2024-05-02 at 8.28.38 AM.png

Looking at the scatterplot should convince us that there is a relationship

between these variables. The points tend to slope downward to the right, so we observe that

the states with higher percentages of graduates taking the ACT have generally lower average

ACT scores. When one variable decreases as the other increases, we say there is a negative

association between the two variables. Notice also that the states with lower percentages of

students taking the ACT have higher average composite scores.

Can you give a plausible explanation for this?

Which display do you find easier to interpret, the table or the graph?

A scatterplot is an effective tool for analyzing data. Special characteristics of data that

may go unnoticed in a table are more obvious from a graph. If there is some relationship

between the variables, a pattern or trend is usually apparent in the scatterplot. In this case,

the scatterplot shows some spread in the data. Though the variables are related, we would

have to consider the relationship somewhat loose, or weak, in the sense that knowing a value

of one variable does not necessarily give us confidence in predicting a value for the other

variable. Notice that the data points appear to cluster with two groups with a gap between them.

In about half of the states less than 45% of the high school graduates took the ACT in 2016

with an average of around 23. In the other half more than 60% took the ACT with an average

around 21.

Screenshot 2024-09-18 at 3.13.17 PM.png

Notice that the data points appear to cluster with two groups with a gap between them.

In about half of the states less than 45% of the high school graduates took the ACT in 2016

with an average of around 23. In the other half more than 60% took the ACT with an average

around 21.

Below is a list of terms that can help us describe the relationship between variables in a scatterplot

Outlier

A single data point that goes far outside the average value of a group of data.

Negative association

When one variable decreases as the other increases.

Positive association

When one variable increases as the other increases.

Spread

Describes a large variation in the data set.

Gap

Missing data in a particular area of interest.

Strong relationship between variables

When the data exhibits a lack of spread or variation.

When you have completed the above lesson, click the next button in Canvas to progress.