Bivariate Data and Analysis Part A: Scatter Plots (45 minutes)

In This Part: A Bivariate Data Question

Have you ever wondered whether tall people have longer arms than short people? We’ll explore this question by collecting data on two variables — height and arm span (measured from left fingertip to right fingertip).

Ask a Question
One way to ask this question is, “Is there a positive association between height and arm span?”

Through this question, we are seeking to establish an association between height and arm span. A positive association between two variables exists when an increase in one variable generally produces an increase in the other. For example, the association between a student’s grades and the number of hours per week that student spends studying is generally a positive association. A negative association, in contrast, exists when an increase in one variable generally produces a decrease in the other. For example, the association between the number of doctors in a country and the percentage of the population that dies before adulthood is generally a negative one.

There are many other ways to ask this same question about height and arm span. Here are two, which we will concentrate on in Part A:
• Do people with above-average arm spans tend to have above-average heights?
• Do people with below-average arm spans tend to have below-average heights?

Person # Arm Span Height 1 156 162 2 157 160 3 159 162 4 160 155 5 161 160 6 161 162 7 162 170 8 165 166 9 170 170 10 170 167 11 173 185 12 173 176

Person # Arm Span Height 13 177 173 14 177 176 15 178 178 16 184 180 17 188 188 18 188 187 19 188 182 20 188 181 21 188 192 22 194 193 23 196 184 24 200 186

Collect Appropriate Data
In Session 1, measurements (in centimeters) were given for the heights and arm spans of 24 people. Here are the collected data, sorted by increasing order of arm span:

This is bivariate data, since two measurements are given for each person.

Problem A1
The data given above are sorted by arm span. Are they also sorted by height? If not exactly, are they generally sorted by height, and, if so, in which direction? Does this suggest any type of association between height and arm span?

Problem A2
a. Measure the arm span (fingertip to fingertip) and height (without shoes) to the nearest centimeter for six people, including yourself.
b. Does the information you collected generally support or reject the observation you made in Problem A1?
c. Identify the person in the table whose arm span and height are closest to your own arm span and height.

In This Part: Building a Scatter Plot

Analyze the Data
We will now begin our analysis of the bivariate data and explore the co-variation in the arm span and height data. Here again are the collected arm spans and heights for 24 people, sorted in increasing order by arm span:

Person # Arm Span Height 1 156 162 2 157 160 3 159 162 4 160 155 5 161 160 6 161 162 7 162 170 8 165 166 9 170 170 10 170 167 11 173 185 12 173 176

Person # Arm Span Height 13 177 173 14 177 176 15 178 178 16 184 180 17 188 188 18 188 187 19 188 182 20 188 181 21 188 192 22 194 193 23 196 184 24 200 186

Bivariate data analysis employs a special “X-Y” coordinate plot of the data that allows you to visualize the simultaneous changes taking place in two variables. This type of plot is called a scatter plot. Note 1

For our data, we will assign the X and Y variables as follows:

X = Arm Span
Y = Height

To see how this works, let’s examine the 10th person in the data table. Here are the measurements for Person 10:

X = Arm Span = 170 and Y = Height = 167

Person 10 is represented by the coordinate pair (170, 167) and is represented in the scatter plot as this point:

Scatter for 1

Let’s add two more points to the scatter plot, corresponding to Persons 2 and 23:

	Person #		Arm Span		Height
	2 157 160 23 196 184

Here is the completed scatter plot for all 24 people:

Scatter for all 24

Problem A3

Judging from the scatter plot, does there appear to be a positive association between arm span and height? That is, does an increase in arm span generally lead to an increase in height?

Video Segment
In this video segment, Professor Kader introduces bivariate analysis. The participants measure their heights and arm spans and then create a scatter plot of the data. Professor Kader then asks them to analyze the association between the two variables, height and arm span.

The scatter plots illustrate the general nature of the association between arm span and height. Reading from left to right on the horizontal scale, you can observe that narrow arm spans tend to be associated with people who are shorter, and wider arm spans tend to be associated with people who are taller — that is, there appears to be an overall positive association between arm span and height.

In This Part: A Bivariate Data Question | Building a Scatter Plot | A Further Question
Quadrants

Now that we have established that there is a positive association between arm span and height, a new question emerges: How strong is the positive association between arm span and height? Here again is the data for the 24 people:

Person # Arm Span Height 1 156 162 2 157 160 3 159 162 4 160 155 5 161 160 6 161 162 7 162 170 8 165 166 9 170 170 10 170 167 11 173 185 12 173 176

Person # Arm Span Height 13 177 173 14 177 176 15 178 178 16 184 180 17 188 188 18 188 187 19 188 182 20 188 181 21 188 192 22 194 193 23 196 184 24 200 186

In order to answer this question, let’s note the mean arm span and height for these 24 adults:

Problem A4

Problem A5

Adding a vertical line to the scatter plot that intersects the arm span (X) axis at the mean, 175.5 cm, separates the points into two groups:

Problem A6

Adding a horizontal line to the scatter plot that intersects the height (Y) at the mean, 174.8 cm, also separates the points into two groups:

Problem A7

What is true about anyone whose scatter plot point appears above this line? How many such points are there?

Problem A8

Enter your own measurements or those of one of the other subjects you measured into the Interactive Activity below to plot these additional heights and arm spans against those of the people in the data set. Note that adding these measurements may affect the values of the means.

For a non-interactive version of this activity print this page, plot your additional measurements on the scatter plot in Problem A5, and calculate the new means.

In This Part: A Bivariate Data Question | Building a Scatter Plot | A Further Question
Quadrants

With bivariate data, there are four possible categories of data pairs. Accordingly, each person in the table can be placed into one of four categories:

Arm Span Height 156 162 157 160 159 162 160 155 161 160 161 162 162 170 165 166 170 170 170 167 173 185 173 176

Arm Span Height 177 173 177 176 178 178 184 180 188 188 188 187 188 182 188 181 188 192 194 193 196 184 200 186

We can represent these categories similarly on the scatter plot:

Adding both the vertical line at the mean arm span (175.5 cm) and the horizontal line at the mean height (174.8) separates the points in the scatter plot into four groups, known as quadrants:

Problem A9

Use this scatter plot to answer the following:

Problem A10