rel="nofollow" href="#ub071c1fc-45c3-5727-8b9b-04a19625492c">13, we will learn statistical tools that quantify the strength of the relationship between two scale variables displayed in a scatterplot. For now, understand that there are three types of relationships between scale variables that a scatterplot can reveal. First, there is a linear relationship; that is, the relationship can be displayed with a straight line. The relationship between role overload and burnout in Wendt’s (2013) research is an example of a linear relationship because the general pattern of data points flows from the lower left to the upper right (what is called a “positive” linear relationship, which we touched on quickly in Chapter 1 and will discuss in detail in Chapter 12). Another example of a linear relationship would be the relationship between sleep quality and depression (Davidson, Babson, Bonn-Miller, Soutter, & Vannoy, 2013). You can see an example of this linear relationship in Figure 3.8. Here, the data points flow from the upper left corner to the lower right corner (what is called a “negative” linear relationship).
Figure 3.7 Using a Scatterplot to Display a Positive Linear Relationship Between Role Overload and Burnout in Wendt’s (2013) Research
Linear relationship: relationship between two variables that is displayed with a straight line.
If one type of relationship is called linear, it will come as no surprise that another type of relationship between scale variables is called a nonlinear relationship. That is, the relationship is displayed by a curve rather than by a straight line. Let’s take an example that you are likely familiar with. Have you ever been so excited or nervous about a test or some sort of performance (e.g., sports or music) that you couldn’t concentrate on doing what you needed to do to succeed at the task? Conversely, have you ever been so unmotivated in a situation that you simply did not care that your performance would be bad? According to Yerkes–Dodson’s Law (1908), our performance on a task is at its best not when we are highly motivated or lacking motivation but at some optimal (midlevel) point of arousal. This nonlinear relationship is displayed in Figure 3.9. As you can see, too much motivation can lead to a decrement in performance because at those overly motivated levels, it becomes difficult to focus on the task itself.
Figure 3.8 Using a Scatterplot to Display a Negative Linear Relationship Between Sleep Quality and Depression
The final type of relationship that a scatterplot can reveal is no relationship between the two variables. That is, the dots on the scatterplot look like they were randomly thrown onto it with no linear or nonlinear relationship. I know of no research suggesting any sort of relationship between shoe size and intelligence. You can see such this relationship in Figure 3.10.
Nonlinear relationship: relationship between two variables that is displayed by a curved line.
Figure 3.9 Yerkes–Dodson’s Law: A Nonlinear Relationship
Figure 3.10 Scatterplot That Displays What “No Relationship” Between Shoe Size and Intelligence Would Look Like
Line Graphs
Similar to scatterplots, line graphs are used to depict how two scale variables are related. Typically in psychology, many line graphs are used to predict the variable on the y-axis from the variable on the x-axis. In fact, this type of line graph starts with a scatterplot such as the one in Figure 3.7. However, we use the scatterplot to draw a line of best fit through the data points, as observed in Figure 3.11. We will learn more about the line of best fit in Chapter 13. For now, if you understand that the line of best fit allows us to predict the variable on the y-axis from a value of the variable on the x-axis, you’re in great shape.
Line graph: graph used to depict the relationship between two scales variables.
One type of line graph we encounter in psychological research is called a time plot, or sometimes a time series plot. An example of a time plot appears in Figure 3.12. This figure displays the value of one of the major U.S. stock market gauges, the Dow Jones Industrial Average (DJIA), from 1989 through 2015. On a time plot, time is plotted on the x-axis, and the value of interest is plotted on the y-axis. As you can see, the DJIA has generally increased in value since 1989, with some notable exceptions along the way (e.g., 2000 and 2008).
Figure 3.11 “Line of Best Fit” That Allows Us to Predict Burnout (on the y-axis) from Role Overload (on the x-axis)
Figure 3.12 Time Plot of The Dow Jones Industrial Average Since 1989
Time plot: type of line graph that plots the value of a variable on the y-axis as it changes over time, which is displayed on the x-axis.
How might a time plot be useful in making decisions? Let’s discuss an example. There is a local pizza place in the small town in which my college is located. They asked me to help predict how much pizza people order so that they can have enough fresh ingredients on hand to meet customers’ demands. To start this process, we plotted how many pizzas the restaurant sold each day for two months. These data are displayed in Figure 3.13. As we can see in this time plot, we know a lot of pizza is ordered on Sundays, with not many ordered on Mondays, Tuesdays, or Wednesdays, but then sales progressively pick up again Thursday, Friday, and Saturday. Armed with this time plot, the restaurant is now in a better position to predict when it will need to have fresh ingredients on hand to meet customer orders.2
Let’s consider another use of line graphs. In their research, Caroline Campbell and Katherine White (2015) had undergraduate students complete a mood scale (Mayer & Gaschke, 1988) before and after engaging in moderately strenuous exercise. Half of the students listened to music while exercising, whereas the other half of the students did not listen to music while exercising. Their results are presented in a line graph in Figure 3.14. Along the x-axis is time, which was preexercise and postexercise. Along the y-axis is the average mood rating. There are two lines, one for the group of students who listened to music and one for the group of students who did not listen to music (the control group). As you can see in this line graph, both groups of students were in a better mood after exercising than before exercising. However, the mood of the group of students who listened to music while exercising improved even more than did the mood of the control group, who did not listen to music while exercising.
Figure 3.13 How Can This Time Plot Help a Restaurant Manage Its Pizza Business?
Figure 3.14 Results of Campbell and White’s (2015) Experiment
