Unit 7, Lesson 4
Describing Distributions (Optional)
Integrated Math 3
4.1
Warm-up
Instructional Routines
Which Three Go Together?Materials
NoneActivity Narrative
This Warm-up prompts students to compare four representations of 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 distributions for all to see. Give students 1 minute of quiet think time, and ask them to indicate when they have noticed three distributions that go together and can explain why. Next, tell students to share their response with their group and then work together to find as many sets of three as they can.
Activity
NoneFour 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 three go together? Why do they go together?
AHistogram. Horizontal axis from 0 to 22 by point 5’s, labeled “length of crack in glass (mm)”. Vertical axis from 0 to 12 by 2’s. Beginning at 0 up to but not including point 5, height of bar at each interval is 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 3, 3, 4, 6, 7, 6, 9, 9, 10, 10, 11, 9, 8, 7, 9, 6, 8, 4, 3, 2, 3, 3, 0, 2, 2, 1, 0.
BDot plot from 0 to 22 by point 5’s. length of crack in glass (mm). Beginning at 0, number of dots above each increment is 2, 0, 0, 0, 0, 0, 0, 2, 2, 2, 2, 4, 5, 5, 5, 6, 7, 8, 9, 10, 12, 10, 11, 9, 10, 7, 6, 4, 2, 2, 1, 2, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2.
CDot plot from 0 to 22 by point 5’s. length of crack in glass (mm). Beginning at 0, number of dots above each increment is 1, 2, 3, 4, 6, 10, 8, 8, 5, 4, 2, 2, 2, 1, 2, 2, 2, 1, 1, 2, 1, 3, 3, 4, 4, 8, 6, 9, 8, 5, 5, 5, 3, 3, 2, 3, 1, 2, 0, 2.
DDot plot from 0 to 22 by point 5’s. length of crack in glass (mm). Beginning at 0, number of dots above each increment is 2, 2, 7, 5, 3, 7, 7, 8, 18, 10, 13, 9, 11, 8, 7, 7, 7, 7, 4, 5, 3.
Student Response
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Building on Student Thinking
Activity Synthesis
Invite each group to share one reason why a particular set of three go together. 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 of which three go together, attend to students’ explanations, and ensure the reasons given are correct.
During the discussion, prompt students to explain the meaning of any statistical terminology they use, such as “mean,” “measure of center,” “dot plot,” “histogram,” or “bell-shaped,” and to clarify their reasoning as needed. Consider asking:
- “How do you know_____?”
- “What do you mean by _____?”
- “Can you say that in another way?”
Lesson Synthesis
The purpose of this discussion is to describe distributions using the shapes and statistics.
Display two sets of data and the associated dot plots:
- 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 median and mean are both 9 for Set B. The mean is equal to the median in both distributions. I knew they would be equal because of the symmetry of the data set.)
- “Which data set has a greater standard deviation? Explain your reasoning.” (Set A, because four values are a distance of 1 from the mean, and four values are a distance of 2 from the mean. In Set B, four values are a distance of 1 from the mean, but only two values are a distance of 4 from the mean. By looking at the shape, we can see that Set B has more of its values in the center of the distribution than does Set A, so we would expect it to be less variable.)
- “Clare says that if you change the highest value in the data set to 20, 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.)
Student Lesson Summary
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 points tend to be farther from the mean. A lesser value for standard deviation means that the points tend to be closer to the mean.