# Lesson 4

Describing Distributions

## 4.1: Which One Doesn't Belong: Cracking Glass (5 minutes)

### Warm-up

This warm-up prompts students to compare four distributions. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.

### Launch

Arrange students in groups of 2–4. Display the data displays for all to see. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, ask each student to share their reasoning as to why a particular item does not belong, and together, find at least one reason each item doesn’t belong.

### Student Facing

Four different kinds of glass are hit with a hammer and the length of the longest crack formed is recorded. The process is repeated 150 times for each type of glass.

Which one doesn’t belong?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

Ask each group to share one reason why a particular item does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question asking which one does not belong, attend to students’ explanations and ensure the reasons given are correct.

During the discussion, ask students to explain the meaning of any terminology they use, such as measure of center, measure of variability, bell-shaped, bimodal, or outliers. Also, press students on unsubstantiated claims.

## 4.2: Name That Distribution Shape (15 minutes)

### Optional activity

This activity is optional. It reviews how to describe distributions based on words such as “bell-shaped” and “symmetric.” If students understand these concepts well, this activity may be safely skipped.

In this activity, students take turns with a partner describing distributions of data represented in dot plots. Students trade roles explaining their thinking and listening, providing opportunities to explain their reasoning and critique the reasoning of others (MP3).

### Launch

Arrange students in groups of 2. Tell students that for each dot plot, one partner describes the distributions and explains their reasoning. The partner’s job is to listen and make sure they agree. If they don’t agree, the partners discuss until they come to an agreement. For the next dot plot, the students swap roles. If necessary, demonstrate this protocol before students start working.

*Conversing: MLR2 Collect and Display.*Listen for and collect vocabulary, gestures, and diagrams students use to identify and describe the various distributions. Capture student language that reflects a variety of ways to describe the characteristics of and differences between skewed, symmetric, bell-shaped, uniform, and bimodal distributions. Write students’ words on a visual display and update it throughout the remainder of the lesson. Remind students to borrow language from the display as needed. This will help students read and use mathematical language during their partner and whole-group discussions.

*Design Principle(s): Maximize meta-awareness; Support sense-making*

*Representation: Develop Language and Symbols.*Create a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of symmetric, bell-shaped, bimodal, and skewed distributions. Encourage students to create an icon for each term which they can explain to someone. They can use this as a reference.

*Supports accessibility for: Conceptual processing; Language*

### Student Facing

Take turns with your partner to select a dot plot and describe the distribution.

- For each distribution you describe, use the terms symmetric, skewed, bell-shaped, uniform, and bimodal where appropriate.
- For each distribution your partner describes, listen carefully to their description. If you disagree, discuss your thinking and work to reach an agreement.

- Each student in the class measures the length of their step in centimeters, then walks across the room counting the number of steps to estimate the length of the classroom in centimeters.
- As a test for fitness, students are asked to do as many push-ups as they can without stopping.
- A group of 30 students are asked how many times they eat a meal not made at home each week.
- A company sells small colored erasers in packages of 24 erasers. Thirty packages are inspected, and the number of red erasers in the package are counted.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

The purpose of this discussion is for students to describe various distributions. Here are some questions for discussion.

- “How did you describe the fourth distribution?” (It was really hard to describe, but my partner and I agreed that it was close to uniform and close to being symmetric.)
- “Is approximately symmetric a good way to describe the fourth distribution? Explain your reasoning.” (Yes, it is a good way to describe it because then I know it may not be exactly symmetric, but is very close to being symmetric. The dot plot from the warm-up is also approximately symmetric.)
- “What does it mean to describe a distribution as approximately uniform?” (It means that instead of each value having exactly the same frequency, they all have frequencies that are almost exactly the same.)
- “Draw a dot plot that is approximately bell-shaped. Compare dot plots with a partner.”

## 4.3: Matching Distributions and Statistics (10 minutes)

### Optional activity

This activity is optional. It reviews how to match mean, median, and standard deviation to a distribution. If students understand the roles of measures of center and standard deviation, this activity may be safely skipped.

In this activity, students take turns with a partner matching data represented by dot plots with summary statistics. Students trade roles explaining their thinking and listening, providing opportunities to explain their reasoning and critique the reasoning of others (MP3).

### Launch

Arrange students in groups of 2. Tell students that for each dot plot, one partner describes how they determined which summary statistics match the data in their dot plot. The partner’s job is to listen and make sure they agree. If they don’t agree, the partners discuss until they come to an agreement. For the next dot plot, the students swap roles. If necessary, demonstrate this protocol before students start working.

*Conversing: MLR8 Discussion Supports.*Use sentence frames to support small-group discussion. Display the following for all to see: “_____ and _____ match because . . .”, and “I noticed _____, so I matched . . . .” Encourage students to challenge each other when they disagree. This will help students clarify their reasoning about the relationship between the shape of a distribution and the mean, median, and standard deviation.

*Design Principle(s): Support sense-making; Maximize meta-awareness*

### Student Facing

Take turns with your partner to match a dot plot with the summary statistics for the data shown.

- For each match you find, explain to your partner how you know it’s a match.
- For each match your partner finds, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.

- Mean: 3, median: 2, standard deviation: 2.91
- Mean: 5, median: 5, standard deviation: 2.19
- Mean: 5, median: 5, standard deviation: 3.74
- Mean: 7.13, median: 8, standard deviation: 2.79

### Student Response

For access, consult one of our IM Certified Partners.

### Student Facing

#### Are you ready for more?

Write out the data points from distribution \(A\) in increasing order \(0,0,0,0,0,1,1,1,2,2,2,\dots\). Do the same for distribution \(B\). Form a new distribution \(E\) where the first data point is the first number from your list for \(A\) minus the first number in our list for \(B\). The second data point is the second number from your list for \(A\) minus the second number in our list for \(B\).

- Make a dot plot of the new distribution \(E\).
- Find its mean, median, and standard deviation.
- Which of these statistics would change if we reversed the order of our list for \(B\)?

### Student Response

For access, consult one of our IM Certified Partners.

### Anticipated Misconceptions

Students may not recall the meaning of standard deviation in this situation. Ask students to recall the definition of standard deviation as it is related to how the data are spread. Standard deviation will be more important later in this unit, so it will be important for them to remember its meaning as spread of the distribution.

### Activity Synthesis

The goal of this discussion is for students to understand the relationship between the shape of a distribution and the mean, median, and standard deviation. Here are some questions for discussion:

- “Dot plot A and B have the same mean and median. How did you figure out which one had a greater standard deviation?” (I figured out that the dot plot A had to have a greater standard deviation than dot plot B because more of the values were further from the mean of 5, whereas dot plot B had several values equal to 5 and fewer further from 5. Since the standard deviation is calculated using distances from the mean, it makes sense that dot plot A has a greater standard deviation than dot plot B.)
- “Dot plot C and D are skewed right and skewed left, respectively. What does this tell you about the relationship between the mean and the median for each graph?” (For dot plot C, I expected the median to be less than the mean because it would be less impacted by the values to the right in the dot plot, whereas the mean would be expected to increase because of those values. For dot plot D, I expected the mean to be less than the median because it is disproportionately impacted by the values to the left in the dot plot.)
- "What is the standard deviation measuring? Explain your reasoning.” (The standard deviation is measuring the variability relative to the mean.)

## Lesson Synthesis

### Lesson Synthesis

Display two sets of data and the associated dot plots for all to see:

- Set A: {2,2,3,3,4,4,5,5,6,6}
- Set B: {7,8,8,9,9,9,9,10,10,11}

Here are some questions for discussion:

- “What is the shape of each distribution?” (They are both symmetric. The first one is uniform and the second one is bell-shaped.)
- “What do you notice about the mean and median of each distribution? Explain your reasoning.” (The median is 4 and the mean is 4 for set A while they are both 9 for set B. The mean is equal to the median in both distributions. The symmetry of the data set let me know that they would be equal.)
- “Which data set has a greater standard deviation? Explain your reasoning.” (Set A, because four values are a distance of one from the mean and four values are a distance of two from the mean. In set B, four values are a distance of one from the mean, but only two values are a distance of four from the mean. By looking at the shape you can see that set B has more of its values in the center of the distribution than does set A, so you would expect it to be less variable.)
- “Clare says that if you add 20 to the highest value in the data set that both the mean and the median will increase. Do you agree with Clare? Explain your reasoning.” (I do not agree with Clare. The mean will increase, but the median will stay the same. It stays the same because changing the last number in the data set does not impact the middle number.)

## 4.4: Cool-down - Drawing a Distribution (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.

## Student Lesson Summary

### Student Facing

The distribution of a set of data can be described by its shape. Data can be *symmetric* around the mean or *skewed* to one side or the other.

For example, the first dot plot shows a distribution that is symmetric around the value 10. The second dot plot is skewed to the right since it is not symmetric and the tail is longer on the right side.

A special type of distribution that will be important in this unit is called a *bell-shaped distribution*. Bell-shaped distributions have these properties:

- a single, central peak
- symmetric
- the farther a value is from the center, the less often it appears in the data set

Since the distribution is symmetric, the mean and median are equal.

*Standard deviation* uses the squared distance from each point to the mean to measure the variability of a data set. A greater value for standard deviation means that the values tend to be farther from the mean. A lesser value for standard deviation means that the values tend to be closer to the mean.