Bivariate Data

In the upcoming activity, you will need to identify the explanatory and response variable given a scenario and understand when linear regression analysis might be appropriate. In this page, you’ll prepare for that by looking carefully at definitions, applying the definitions in given scenarios, and seeing specific situations in which data may or may not be related linearly.

Often, we do statistical studies to find relationships between two or more variables that can help us to better predict future outcomes and perhaps make changes that will improve our lives.

In the next activity, we will be focusing on studies and relationships involving two quantitative variables. In each dataset, the two variables will be linked because both observations will be measured from the same individual or unit.

These types of linked data are called bivariate data and are often presented in scatterplots. Bivariate data are defined as pairs of data values, where each pair consists of two different measurements that come from the same individual or unit.

See the video below for a quick explanation.

Now that you have the idea of two related quantitative variables, read on to see how to determine the nature of the two variables in an existing bivariate data set or study of bivariate data. The key idea is that one of the variables will measure the outcome of the study. It will be dependent upon the other variable.

Explanatory and Response Variables

A teacher wonders if “number of absences per semester” is related to “academic performance” for students in her classes. She might look back on her class records from previous semesters and generate a dataset by observing both the final overall average grade and total number of missed classes for each student in a random sample of students. This is an example of a bivariate dataset.

When working with a bivariate dataset, there are two variables to consider:

• The explanatory variable ([latex]x[/latex]) is the variable that is thought to explain or predict the response variable of a study.
• The response variable ([latex]y[/latex]) measures the outcome of interest in the study. This variable is thought to depend in some way on the explanatory variable. It is often referred to as the “variable of interest” for the researcher. (In your previous math classes this variable may have been referred to as the dependent variable.)

In this example, the outcome the teacher is most interested in is how well her students will do in her class, so the response variable is Overall Average Grade. The other variable, Number of Absences, is the explanatory variable.

Identifying explanatory and response variables can sometimes be difficult. When trying to identify explanatory and response variables, make sure to carefully read the scenario and keep the following phrases in mind:

Explanatory is used to predict Response
(or calculate)
(or determine)

It is good practice to identify both variables and then ask, “Which one is the main outcome or focus of the study?” This variable will be the response variable, and the other variable will be the explanatory variable. When reading a pre-existing study, carefully read the context of the study to identify which variable is being used to explain (the explanatory variable) an outcome or response (the response variable).

We’ll see shortly that both variables present in a bivariate data set will need to be quantitative in order to determine a linear relationship. Note for example that both of the scenarios in the example above included only quantitative variables. To define explanatory and response variables in any bivariate data set, when the study seeks only a correlation, both variables need not be quantitative.  Keep this in mind as you answer Questions 1 and 2 below to define and identify explanatory and response variables in a given scenario.

Linear Relationships

A method we will use to make predictions about missing observations or future observations in bivariate data is called Least Squares Regression (LSR) analysis. The language might seem intimidating at first, but the ideas are quite straightforward, especially with examples to illustrate each new term. For example, LSR analysis can also be described as linear modeling, where we determine the equation of a line of best fit to make predictions based on an existing dataset. In this type of analysis, both the explanatory and response variables must be quantitative, since the linear model requires numerical values in its calculations.

Line of Best Fit

The line of best fit is simply the best line that describes the data points. For real data with natural deviations, the line cannot go through all of the points. In fact, very often, the line does not go through any of the data points.

Since no line will be perfect, the best we can do is minimize its error. In this class, we will do this by minimizing the sum total of the squared vertical errors from all data points to the line. This is why the line of best fit is also called the Least Squares Regression Line (LSRL).

The vertical error associated with each data point is called the residual of that observation. This error, illustrated by the length of the vertical line, represents how far off a prediction calculated from the line is compared to the actual, observed [latex]y[/latex] value; the larger the line, the greater the error associated with that particular observation.

Note: For data points that are above the line of best fit, the residuals are positive, and for data points that are below the line, the residuals are negative.

The equation for the line of best fit is very similar to one you may have seen in a previous math class:

[latex]\hat{y} = a+bx[/latex]

where [latex]\hat{y}[/latex] is the general predicted value of the response variable (pronounced y-hat), a is the estimated value of the y-intercept, and [latex]b[/latex] is the estimated slope.

While the actual process of finding the line of best fit might seem complicated, the concept of line of best fit is very straightforward. We can use technology to take care of long and tedious calculations.

When is LSR Analysis Appropriate?

As you answer Questions 3 and 4, keep in mind that in order to create a linear model during LSR analysis, both of the variables in the bivariate data must be quantitative.

Performing LSR Analysis

Now let’s put everything you’ve seen in this activity together to perform an LSR analysis using technology. See the example below for guidance, then answer Question 5.

Now it’s your turn to try.

You’ve seen some of these ideas before in Forming Connections [5A]. Review those notes now to answer Question 6. Make sure to have your answer for Question 6 handy as you begin the upcoming Forming Connections activity!

Summary

In this What to Know page, you learned to recognize when a linear regression analysis is appropriate, how to identify the explanatory and response variables in bivariate data, and how to calculate the line of best fit.  Let’s summarize the skills as you saw them in each question.

• In Questions 3, 4, and Question 5 Part A, you identified when a linear regression analysis might be appropriate.
• In Questions 1, 2, and Question 5 Parts B and C, you identified the explanatory and response variables in a given scenario.
• In Question 5 Parts D through F, you calculated the line of best fit and wrote it using proper notation.

If you feel comfortable with these ideas, it’s time to move on to Forming Connections in the next activity!