Now that students are more familiar with two-way tables showing relative frequency, they are ready to create their own segmented bar graphs. In this activity, students create two segmented bar graphs based on the same two-way table by considering percentages of the rows and columns separately. After creating the segmented bar graphs, they are analyzed to determine if there is an association present in the data.

Students may draw the segmented bar graph incorrectly. Most likely, they will accidentally graph frequency instead of relative frequency. They may also graph relative frequencies, but without stacking them. Both segmented bars should go from 0 to 100.

To clarify how to create and interpret segmented bar graphs, ask:

• “Why are the numbers in the top left box in the two tables different? What do they mean?” (In the first table, the number represents the percentage who also play musical instruments out of all the people who play sports. In the second table, the number represents the percentage of people who also play sports out of all the people who play musical instruments.)
• “Suppose you are in the band room with students who play instruments. To estimate what percentage of these students play sports, would you use the first or second segmented bar graph? Explain your reasoning.” (The second one, because I already know they play an instrument, so I would be looking for the percentages within that category.)
• “Is there an association between the two variables? Explain or show your reasoning.” (The answer will depend on class data, but the reasoning should include an analysis of the relative frequencies within categories. There is an association if the percentages within one category are very different from the percentages in another category.)

If there is an association, ask what the segmented bar graphs would look like if there was no association. If there is not an association, ask what the segmented bar graphs would look like if there was one. (If there is an association, the bars will look different because the percentages in each category will be very different. If there is not an association, the segmented bar graph should have bars that are very similar because the percentages will be close to one another.)

This activity provides students less structure for their work in creating segmented bar graphs to determine an association (MP4). In addition, the data in this activity is split into more than two options. Students work individually to create a segmented bar graph based on either columns or rows and then share their information with a partner who has created the other segmented bar graph. Together, partners discuss the segmented bar graphs to determine if there is an association between the variables (MP3). In particular, students should notice that there is evidence of an association if the relative frequencies within a category are very different from the relative frequencies in another category.

As students work, identify groups that use the different segmented bar graphs to explain why there is an association between the color of the eraser and flaws.

The purpose of this discussion is to identify strategies for creating segmented bar graphs and analyzing them to determine if there is an association among variables.

Ask, “What strategies did you use to create the segmented bar graphs?” (First, we created a new table of the relative frequencies. Then we approximated the heights of the segments based on the percentages from the table.)

Select previously identified groups based on whether they used rows or column relative frequencies to share their explanation for noticing an association.

After both explanations are shared, ask students, "Do you think that noticing the association was easier with one of the graphs?" (Likely the segmented bar graph based on rows is easier since there are only 2 segments and it is easier to see that the yellow and green erasers are more flawed.)

Finally, ask students, "If there was not an association between color and flaws, what might the segmented bar graph based on the rows look like? What might the segmented bar graph based on the columns look like?” (The segmented bar graph based on the rows would have each segmented bar look about the same. That is, the line dividing the 2 segments would be at about the same height in each bar. The segmented bar graph based on the columns would have segments that are all approximately equal. That is, each segment should represent about 25% of the entire bar.)