Lesson 7

In earlier lessons, students learned that the mean absolute deviation (MAD) is a measure of variability. In this warm-up, they study two distributions that appear very different but turn out to have the same MAD. Students notice that the MAD may not fully tell us about the variability of a data set. The work here motivates the need to have a different way to quantify variability, which is the focus of this lesson. While students may notice and wonder many things about these images, highlight ideas related to the variability of the data sets.

Arrange students in groups of 2. Display the dot plots for all to see. Ask students to identify at least one thing they notice and at least one thing they wonder about the dot plots, and to give a signal when they have both. Give students 1 minute of quiet think time, and then 1 minute to discuss their observation and question with their partner. Follow with a whole-class discussion.

Invite students to share what they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After each response, ask the class if they agree or disagree and to explain alternative ways of thinking, referring back to the dot plots each time. Discuss:

• “Do you think the ages of the people at the first party are alike or different? What about the ages of the people at the second party?”
• “The MAD for both data sets is approximately 10.5 years. What does a MAD of 10.5 years tell us in this context?”
• “Is the MAD a useful description of variability in the first data set? What about in the second data set?”

Two key ideas to uncover here are:

• The MAD is a way to summarize variation from the mean, but the single number does not always tell us how the data are distributed.
• The same MAD could result from very different distributions.

If the key ideas above are not uncovered during discussion, be sure to highlight them.

This activity introduces students to the five-number summary and the process of identifying the five numbers. Students learn how to partition the data into four sets: using the median to decompose the data into upper and lower halves, and then finding the middle of each half to further decompose it into quarters. They learn that each value that decomposes the data into four parts is called a quartile, and the three quartiles are the first quartile (Q1), second quartile (Q2, or the median), and third quartile (Q3).  Together with the minimum and maximum values of the data set, the quartiles provide a five-number summary that can be used to describe a data set without listing or showing each data value. 

Students reason abstractly and quantitatively (MP2) as they identify and interpret the quartiles in the context of the situation given.

To allow more time for the synthesis, find the median together with the whole group. Then, assign half of the groups to find the 25th percentile and the other half to find the 75th percentile. After 2 minutes of group work time, ask a group from each half to share their results with the class.

During the synthesis, define and compute the range (\(42 - 7 = 35\)) and interquartile range (\(29 - 10.5 = 18.5\)) of the data.

Explain to students that they previously summarized variability by finding the MAD, which involves calculating the distance of each data point from the mean and then finding the average of those distances. Explain that we will now explore another way to describe variability and summarize the distribution of data. Instead of measuring how far away data points are from the mean, we will decompose a data set into four equal parts and use the markers that partition the data into quarters to summarize the spread of data.

Remind students that when there is an even number of values, the median is the average of the middle two values.

Arrange students in groups of 2. Give groups 8–10 minutes to complete the activity. Follow with a whole-class discussion.

Ask a student to display the data set they have decomposed and labeled, or display the following image for all to see.

Focus the conversation on students' interpretation of the five numbers. Discuss:

• “In this context, what do the minimum and maximum values tell us?” (The ages of the youngest and oldest partygoers.)
• “Why are Q1 called 25th percentile, Q2 50th percentile, and Q3 75th percentile?” (Each quartile tells us how many quarters of the ordered data values are accounted for up to that point. The first quartile tells us that one quarter, or 25 percent, of data values are less than or equal to that value. The second quartile tells us that two quarters, or 50 percent, of data values are less than or equal to that value, and so on.)
• “In this context, what does Q1 (10.5) tell us?” (That a quarter of the partygoers are 10.5 years old or younger.)
• “What does Q3 (29) tell us?” (That three quarters of the partygoers are 29 years old or younger.)
• “How do the five numbers help us to see the distribution of the data?” (It divides the values in the data into sections containing one fourth of the values each. This gives us an idea about the distribution of the data by looking at how varied each section is.)

Previously, students learned to identify the median, quartiles, and five-number summary of data sets. They also calculated the range and interquartile range of distributions. In this activity, students rely on those experiences to make sense of box plots. They explore this new representation of data kinesthetically: by creating a human box plot to represent class data on the lengths of student names, which they collected in the Finding the Middle activity in an earlier lesson. 

Before the lesson, use thin painter’s tape to make a number line on the ground. If the floor is tiled with equal-sized tiles, consider using the tiles for the intervals of the number line. Otherwise, mark off equal intervals on the tape. The number line should cover at least the distance between least data value (the fewest number of letters in a student's name) to the greatest (the most number of letters).

Provide each student with a copy of the data on the lengths of students’ names from the Finding the Middle activity. If any students were absent then, add their names and numbers of letters to the data set.

Give students 4–5 minutes to find the quartiles and write the five-number summary of the data. Then, invite several students to share their findings and come to an agreement on the five numbers. Record and display the summary for all to see.

​Explain to students that the five-number summary can be used to make another visual representation of a data set called a box plot. Tell students that they will create a human box plot in a similar fashion as when they were finding the median. 

• Return to students the index cards from the lesson on finding the median. If any students were absent when the cards were made, give them each an index card and ask them to record on the card their full name and the number of letters in their name. If any student who made a card is absent, have another student with the same number of letters in their name hold the card of the absent student.

• Ask students to stand up, holding their index card in front of them, and place themselves on the point on the number line that corresponds to their number. (Consider asking students to do so without speaking at all.) Students who have the same number of letters should stand one in front of the other.

• Hold up the index cards that have been labeled with “minimum.” Ask students who should claim the card, then hand the card to the appropriate student. Do the same for the other labels of a five-number summary. If any of the quartiles falls between two students' numbers, write that number of the index card and have both students hold that card together.

Now that the five numbers are identified and each associated with one or more students, use wide painter's tape to construct a box plot. 

• Form a rectangle on the ground by affixing the tape around the group of students between Q1 and Q3. If a quartile is between two people, put the tape down between them. If a quartile has the value of a student's number, put the tape down at that value and have the student stand on it. 
• Put a tape segment at Q2, from the top side of the rectangle to the bottom side, to subdivide the rectangle into two smaller rectangles. If Q2 is a student's number, have the student stand on the tape.
• For the left whisker, affix the end of tape to the Q1 end of the rectangle; extend it to where the student holding the “minimum” card is standing. Do the same for the right whisker, from Q3 to the maximum.
• Tape the five-number summary cards and students’ cards that correspond to them in the right locations.

This image shows an example of a completed human box plot.

Explain to students that they have made a human box plot. Consider taking a picture of the box plot for reference and discussion later.

Tell students that a box plot is a representation of a data set that shows the five-number summary. Discuss:

• “Where can the median be seen in the box plot?” (It is the line inside the box.) What about the first and third quartiles? (The left and right sides of the box.)
• “Where can the IQR be seen in the box plot?” (It is the length of the box.) 
• “The two segments of tape on the two ends are called ‘whiskers.’ What do they represent?” (The lower one-fourth of the data and the upper one-fourth of data.)
• “How many people are part of the box, between Q1 and Q3? Approximately what fraction of the data set is that number?” (About half. Note that the number of people that are part of the box may not be exactly one half of the total number of people, depending on whether the number of data points is odd or even, and depending on how the values are distributed.)
• “Why might it be helpful to summarize a data set with a box plot?” (It could help us see how close together or spread out the values are, and where they are concentrated.)

Explain to students that we will draw and analyze box plots in upcoming activities and further explore why they might be useful.

In the last activity, students constructed a box plot based on the five-number summary of their name length data. In this activity, they learn to draw a box plot and they explore the connections between a dot plot and a box plot of the same data set.

Tell students that they will now draw a box plot to represent another set of data. For their background information, explain that scientists believe people blink their eyes to keep the surface of the eye moist and also to give the brain a brief rest. On average, people blink between 15 and 20 times a minute; some blink less and others blink much more.

Arrange students in groups of 2. Give 4–5 minutes of quiet work time for the first set of questions and a minute to discuss their work with their partner. Ask students to pause afterwards.

Display the box plot for all to see. Reiterate that a box plot is a way to represent the five-number summary and the overall distribution. Explain:

• “The left and right sides of the box are drawn at the first and third quartiles (Q1 and Q3).”

• “A vertical line inside the box is drawn at the median (Q2).”

• “The two horizontal lines (or 'whiskers') extend from the first quartile to the minimum and from the third quartile to the maximum.”

• “The height of the box does not give additional information about the data, but should be tall enough to distinguish the box from the whiskers.”

Ask students to now draw a box plot on the same grid, above their dot plot. Give students 4–5 minutes to complete the questions. Follow with a whole-class discussion.

Display the dot plot and the box plot for all to see. 

Discuss:

• “How many data values are included in each part of the box plot?” (5 data values in each part.)
• “If you just look at the box plot, can you tell what any of the data values are?” (Only the minimum and the maximum values.)
• “If you just look at the dot plot, can you tell where the median is? Can you tell which values of the data make up the middle half of the data? Can you tell where each quarter of the data values begin and end?” (It is possible to tell, but it is not straightforward; it requires some counting.)

The focus of this activity is on constructing a box plot and understanding its parts, rather than on interpreting it in context. If students seem to have a good grasp of the drawing process and what the parts entail and mean, consider asking them to interpret the plots in the context of the research study. Ask: “Suppose you are the scientist who conducted the research and are writing an article about it. Write 2–3 sentences that summarize your findings, based on your analyses of the dot plot and the box plot.”