Lesson 14

In this warm-up, students review ways to interpret and compare data shown on a dot plot. The discussion on each given statement gives the teacher an opportunity to hear how students reason about the median, mean, typical value, spread, balance point, and MAD. This discussion will be helpful in upcoming activities, as students compare median and mean values for different data sets. 

Give students 2 minutes of quiet work time. Follow with a whole-class discussion. 

For each statement, ask students to indicate if they agree or disagree. If all students agree or all students disagree, ask a couple of students to explain their reasoning. If the class is divided on a statement, ask students on both sides to share their reasoning until the class comes to an agreement. As students share, record and display their responses for all to see. If possible, record their reasoning on the dot plots to highlight important terms students use. 

To help facilitate the discussion, consider asking:

• “Who can restate ___’s reasoning in a different way?”
• “Did anyone have the same reasoning but would explain it differently?”
• “Did anyone reason about the statement in a different way?”
• “Does anyone want to add on to _____’s reasoning?”
• “Do you agree or disagree? Why?”

In this lesson, students begin to notice how the distribution of data affects the mean and median of a data set. Using their class height data, they examine how both measures of center are affected when a value that is far from the center is added to a data set. Students find that adding these unusually large or small values pulls the mean up or down while having little or no effect on the median. They begin to see that for data sets with some far-off values, the median might be a better choice for describing a typical value because the sizes of those extreme values (whether very large or very small) do not affect the median as much as they do the mean. 

Consider preparing a large-scale dot plot that shows student height data and displaying it for all to see.  

Arrange students in groups of 3–4. Provide each student with the data on students' heights (collected in the first lesson of the unit) and access to calculators. Give groups 8–10 minutes to complete the first three questions, and then ask them to pause for a brief class discussion before moving on to the last set of questions.

During this discussion, compare the mean that students calculated and the median they found, and solicit students' ideas on why the mean changed more than the median if the tallest person in the world joined the class. (If a class dot plot that shows student heights is made, add the point that represents the tallest person in the world.)

At this time students should begin to see that the value of the additional data point greatly affects the mean because finding the mean entails redistributing data values so that they are all the same (or moving the balance point toward the new data point to keep the distribution balanced). The new data point does not affect the median as much in this case because finding the median entails finding the point in a distribution that divides the data in half, and the numbers around the middle of the distribution are likely very close. 

Give groups another few minutes to complete the rest of the task.

When calculating the mean after a new person joined the class, some students might enter individual heights into a calculator once again, rather than using the sum from their original mean calculation to save time (e.g. by adding 251 to the sum of heights of students and then dividing by a class size that includes one additional student). Urge them to think about how they might use the previous calculation to make the process more efficient. 

Use the whole-class discussion to further explore how unusually high or low values affect the mean and the median. Invite several students to share the new mean and median, and to explain how these measures changed if the world's shortest person joined the class. (If a class dot plot that shows student heights is made, add the point that represents the smallest person in the world.) Discuss:

• “What affect does the smallest person in the world have on the mean? Why?”
• “Which would affect the mean more: the height of the tallest person, or the height of the smallest person? Why?”
• “Suppose a new student who has a height close to the mean joined the class. Would her height affect the mean? Why or why not?”
• “Does adding two values—one unusually high and one unusually low—affect the median? Why or why not?”

Students should see that adding a data point to each end does not change the median much, if at all; the original middle value would still be the middle value as the halfway point is unchanged. Conversely, the mean will change greatly due to values that are extremely greater or less than most of the data.

In the previous activity, students analyzed the effects of unusually high or low values on the mean and median. Here they study distributions (displayed using dot plots and a histogram) for which the mean and median can be the same, close, or far apart, and make conjectures about how the distributions affect the mean and median (MP7). Along the way, students recognize that the mean and median are equal or close when the distribution is roughly symmetrical and are farther apart when the distribution is non-symmetrical. 

Arrange students in groups of 3–4. Provide each group with a cut-up set of cards from the blackline master. Give groups 4–5 minutes to take turns sorting the cards and completing the first two problems. Then, pause the activity to discuss the sorting decisions and observations of the class.

Ask a few groups how they sorted the cards. If not mentioned by students, highlight that in three of the distributions, the mean and median of the data are approximately equal. In the other three distributions, the mean and median are quite different. Discuss:

• “What do you notice about the shape and features of distributions that have roughly equal mean and median?” (They are roughly symmetrical and each have one peak in the middle, with roughly the same number of values to the left and right. They may have gaps, but the gaps are somewhat evenly spaced out.)
• “What about the shape and features of a distribution that has very different mean and median?” (They are not at all symmetrical. They may have one peak, but it is off to one side, or they don't really show any peaks. They may have gaps or data values that are unusually high or low. There is more variability in these data sets.)
• “In the second group, why might the mean and the median be so different?” (The mean is pulled toward the direction of unusually large or small values. The median simply tells us where the middle of the data lies when sorted, so it is not as affected by these values that are far from where most data points are.)

Afterwards, give students another 3–4 minutes to answer and discuss the remaining questions with their group. 

Use the whole-class discussion to reinforce the idea that the distribution of a data set can tell us which measure of center best summarizes what is typical for the data set. Briefly review the answers to the statistical questions, and then focus the conversation on the last questions (how students knew which measure of center to use in each situation). Select a couple of students to share their responses. Discuss:

• “For data sets with non-symmetrical distributions, why does the median turn out to be a better measure of center for non-symmetrical data sets?” (Non-symmetrical data sets often have unusual values that pull the mean away from the center of data. The median is less influenced by these values.)
• “Does it matter which measure we choose to describe a typical value? For example, in Card F, would it matter if we said that a typical age for the people who went on the field trip to D.C. was about 21 years old?” (Yes, it does matter in some cases. In that example, it wouldn’t really make sense to say that 21 years is a typical age because the vast majority of the people on the trip were teenagers.)