Relationships between Quantitative Datasets

Before we can evaluate a relationship between two datasets, we must first decide if we feel that one might depend on the other. In our exam example, it is appropriate to say that the score on the final depends on the score on the midterm rather than the other way around: if the midterm depended on the final, then we’d need to know the final score first, which doesn’t make sense.

Here’s another example: if we collected data on home purchases in a certain area and noted both the sale price of the house and the annual household income of the purchaser, we might expect a relationship between those two. Which depends on the other? In this case, sale price depends on income: people who have a higher income can afford a more expensive house. If it were the other way around, people could buy a new, more expensive house and then expect a raise! (This is very bad advice.)

It's worth noting that not every pair of related datasets has clear dependence. For example, consider the percent of a country’s budget devoted to the military and the percent earmarked for public health. These datasets are generally related: as one goes up, the other goes down. However, in this case, there’s not a preferred choice for dependence, as each could be seen as depending on the other. When exploring the relationship between two datasets, if one set seems to depend on the other, we’ll say that dataset contains values of the response variable (or dependent variable). The dataset that the response variable depends on contains values of what we call the explanatory variable (or independent variable). If no dependence relationship can be identified, then we can assign either dataset to either role.

Once we’ve assigned roles to our two datasets, we can take the first step in visualizing the relationship between them: creating a scatter plot.

Creating Scatter Plots

A scatter plot is a visualization of the relationship between two quantitative sets of data. The scatter plot is created by turning the datasets into ordered pairs: the first coordinate contains data values from the explanatory dataset, and the second coordinate contains the corresponding data values from the response dataset. These ordered pairs are then plotted in the [latex]xy[/latex]-plane. Let's return to our exam example to put this into practice.

Example 2

Students are exploring the relationship between scores on the midterm exam and final exam in their math course. Here are some of the scores reported by their classmates:

Name	Midterm grade	Final grade
Student 1	88	84
Student 2	71	80
Student 3	75	77
Student 4	94	95
Student 5	68	73

Create a scatter plot to visualize the data.

Step 1: Since it makes more sense to think of the final exam score as being dependent on the midterm exam score, we’ll let the final grade be the response. So let’s think of these two datasets as a set of ordered pairs, midterm first, final second:

Name	(Midterm, final)
Student 1	(88, 84)
Student 2	(71, 80)
Student 3	(75, 77)
Student 4	(94, 95)
Student 5	(68, 73)

Step 2: Next, let’s make the axes. On the horizontal axis, make sure the range of values is sufficient to cover all of the explanatory data. For our data, that’s 68 to 94. Similarly, the vertical axis should cover all of the response data (73 to 95):

Step 3: Our first point is (88, 84). So we’ll locate 88 on the horizontal axis and 84 on the vertical axis and identify the point that’s directly above the first location and horizontally level with the second:

Step 4: Repeat this process to place the other four points on the graph:

Step 5: Finally, label the axes:

Exercise 2

Create a scatter plot to visualize the following data, showing the top five NFL receivers by number of receptions for the 2019 season. Treat Yards as the response:

Player	Receptions	Yards
Stefon Diggs	127	1535
Davante Adams	115	1374
DeAndre Hopkins	115	1407
Darren Waller	107	1196
Travis Kelce	105	1416

Solution

Reading and Interpreting Scatter Plots

Scatter plots give us information about the existence and strength of a relationship between two datasets. To break that information down, there are a series of questions we might ask to help us. First: Is there a curved pattern in the data? If the answer is “yes,” then we can stop; none of the linear regression techniques from here to the end of this section are appropriate. Figure 7 and Figure 9 show several examples of scatter plots that can help us identify these curved patterns.

Notice that there’s not much vertical distance between our two starred points. This tells us that there’s not a strong relationship between these two datasets.

Positive and Negative Linear Relationships

Another way to assess whether there is a relationship between two datasets in a scatter plot is to see if the points seem to be clustered around a line (specifically, a line that’s not horizontal). The stronger the clustering around that line is, the stronger the relationship.

Once we’ve established that there’s a relationship worth exploring, it’s time to start quantifying that relationship. Two datasets have a positive linear relationship if the values of the response tend to increase, on average, as the values of the explanatory variable increase. If the values of the response decrease with increasing values of the explanatory variable, then there is a negative linear relationship between the two datasets. The strength of the relationship is determined by how closely the scatter plot follows a single straight line: the closer the points are to that line, the stronger the relationship. The scatter plots in Figure 15 to Figure 21 depict varying strengths and directions of linear relationships.

Finding the Equation of the Regression Line

So how do we find the equation of the regression line? Recall the point-slope form of the equation of a line:

The regression line has two properties that we can use to find its equation. First, it always passes through the point of means. If [latex]\bar{x}[/latex] and [latex]\bar{y}[/latex] are the means of the explanatory and response datasets, respectively, then the point of means is [latex](\bar{x}, \bar{y})[/latex]. We’ll use that as the point in the point-slope form of the equation. Second, if [latex]s_{x}[/latex] and [latex]s_{y}[/latex] are the standard deviations of the explanatory and response datasets, respectively, and if [latex]r[/latex] is the correlation coefficient, then the slope is [latex]m=r \times \frac{s_{y}}{s_{x}}[/latex]. Putting all that together with the point-slope formula gives us this:

Let's walk through an example.

Correlation coefficient and linear regression equations can easily be found with a program like Desmos. Here is a video that explains how to use the technology:

Using the Equation of the Regression Line

Once we’ve found the equation of the regression line, what do we do with it? We’ll look at two possible applications: making predictions and interpreting the slope.

We can use the equation of the regression line to predict the response value [latex]y[/latex] for a given explanatory value [latex]x[/latex]. All we have to do is plug that explanatory value into the formula and see what response value results. This is useful in two ways: First, it can be used to make a guess about an unknown data value (like one that hasn’t been observed yet). Second, it can be used to evaluate performance (meaning we can predict an outcome given a particular event). We previously created a scatter plot of final exam scores versus midterm exam scores using these data:

The equation of the regression line is [latex]y=0.687x+27.4[/latex], where [latex]y[/latex] is the final exam score and [latex]x[/latex] is the midterm exam score. If Frank scored 85 on the midterm, then our prediction for his final exam score is [latex]0.687 \times 85+27.4=85.795[/latex]. To use the regression line to evaluate performance, we use a data value we’ve already observed. For example, Allison scored 88 on the midterm. The regression line predicts that someone who scores an 88 on the midterm will get [latex]0.687 \times 88+27.4=87.856[/latex] on the final. Allison actually scored 84 on the final, meaning she underperformed expectations by almost 4 points [latex](87.856-84)[/latex].

The second application of the equation of the regression line is interpreting the slope of the line to describe the relationship between the explanatory and response datasets. For the exam data in the previous paragraph, the slope of the regression line is 0.687. Recall that the slope of a line can be computed by finding two points on the line and dividing the difference in the [latex]y[/latex]-values of those points by the difference in the [latex]x[/latex]-values. Keeping that in mind, we can interpret our slope as [latex]0.687=\frac{ \text{difference in final scores}}{\text{difference in midterm scores}}[/latex]. Multiplying both sides of that equation by the denominator of the fraction, we get [latex]0.687 \times \text{difference in midterm scores} = \text{difference in final scores}[/latex]. Thus, a one-point increase in the midterm score would result in a predicted increase in the final score of 0.687 points. A ten-point drop in the midterm score would give us a decrease in the predicted final score of 6.87 points. In general, the slope gives us the predicted change in the response that corresponds to a one-unit increase in the explanatory variable.

Correlation Does Not Imply Causation

One of the most common fallacies about statistics has to do with the relationship between two datasets. In a certain dataset, we find that the correlation coefficient between the 75th percentile math SAT score and the 75th percentile verbal SAT score is 0.92, which is really strong. The slope of the regression line that predicts the verbal score from the math score is 0.729, which we might interpret as follows: “If the 75th percentile math SAT score goes up by 10 points, we’d expect the corresponding verbal SAT score to go up by just over 7 points.”

Does the increasing math score cause the increase in the verbal score? Probably not. What’s really going on is that there’s a third variable that’s affecting them both: to raise the SAT math score by 10 points, a school will recruit students who do better on the SAT in general; these students will also naturally have higher SAT verbal scores. This third variable is sometimes called a lurking variable or a confounding variable. Unless all possible lurking variables are ruled out, we cannot conclude that one thing causes another.

Extra Practice