What’s in My Sandwich?

In this activity, we’re going to extend the skills you obtained learning to read and interpret scatterplots in the previous page into an understanding of how the correlation coefficient describes the strength and direction of the linear relationship between two quantitative variables. Let’s begin by picking up here where the previous page left off.

In the What to Know page for this activity, you examined the relationship between the energy content and carbon footprint of 24 readily-made sandwiches. This relationship is illustrated in the following scatterplot:

Previously, you identified the the bivariate data represented in the scatterplot and learned how to identify and describe any apparent trend in the data. You also made note of a particular value, called the Pearson Correlation Coefficient. If you don’t have that value available, you can obtain from a classmate or redisplay the graph in the data analysis tool at https://dcmathpathways.shinyapps.io/Association_Quantitative/ by choosing the Dataset Carbon Footprint.

Question 1 below asks you to consider the scatterplot of energy content and carbon footprint. Read the question individually first before discussing it with a classmate. Consider all the information available to you in the scatterplot. Then you’ll work in pairs to answer it, using what you learned in the What to Know page to write a thorough description of the relationship between the variables.

Continue to work in pairs or move into groups of four for the remainder of this activity. In Question 2, you’ll consider the possible implications of the correlation coefficient. Rather than memorizing a definition, let’s build up understanding by identifying connections between the graph and the correlation coefficient.  How do you think the value of the measure you obtained from the tool connects to any trend that might be apparent in the scatterplot?

 Correlation Coefficient

Let’s look at a different situation now as we attempt to understand more fully what the correlation coefficient tells us about the relationship between the variables.

The following table displays four variables collected on 18 animals. Refer to the data dictionary below the table for details. In a moment, you’ll look at three scatterplots that each compare two of the four variables from the table. Some of the plots will indicate a negative trend, some positive, and in some the data will be more tightly or loosely associated. You’ll be given three possible [latex]r[/latex] values and asked to match each to the scatterplot that is most likely associated with it.

Below is the data dictionary for the four variables collected on these animals:

Gestation Period (days): length of pregnancy
Heart Rate (beats/minute): average resting heart rate
Weight (pounds): average weight of an adult
Longevity (years): average lifespan

Continue to work in groups (or pairs) to answer the remaining questions about this data.

Strength of Relationship

Now that you are hopefully feeling more comfortable with the sign of the correlation coefficient and its association to a positive or negative trend, let’s confirm what we have begun to understand about the strength of the relationship.

Recall in the scatterplot showing carbon footprint and energy content that the [latex]r[/latex]-value was [latex]0.621[/latex] and the points in the graph appeared to be only moderately associated. Now imagine what the graph would have looked like if [latex]r[/latex] had been closer to [latex]1[/latex]. Do you think the points would have been placed more closely together along a more clearly defined line?

The following table contains general guidelines for describing the strength of a linear relationship based on the value of the associated correlation coefficient.