Unit 1, Lesson 4
The Shape of Distributions
Algebra 1
4.1
Warm-up
Which Three Go Together: Distribution Shape
Instructional Routines
Which Three Go Together?
Materials
None
Activity Narrative
This is the first Which Three Go Together routine in the course. In this routine, students are presented with four items or representations and asked: “Which three go together?" and "Why do they go together?”
Students are given time to identify a set of three items, explain their rationale, and refine their explanation to be more precise or find additional sets. The reasoning here prompts students to notice common mathematical attributes, look for structure (MP7), and attend to precision (MP6), which deepen their awareness of connections across representations.
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.
Before students begin, consider establishing a small, discreet hand signal that students can display when they have an answer they can support with reasoning. Signals might include a thumbs-up or a certain number of fingers that tells the number of responses they have. Using such subtle signals is a quick way to see if students have had enough time to think about the problem. It also keeps students from being distracted or rushed by hands being raised around the class.
Launch
Arrange students in groups of 2–4. Display the dot plots for all to see. Give students 1 minute of quiet think time, and ask them to indicate when they have noticed three dot plots that go together and can explain why. Next, tell students to share their response with their group, and then together to find as many sets of three as they can.
Activity
NoneWhich three go together? Why do they go together?
Activity Synthesis
Invite each group to share one reason why a particular set of three goes 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 that the reasons given are correct.
During the discussion, prompt students to explain the meaning of any statistical terminology they use, such as “symmetric,” “skewed,” “uniform,” “bimodal,” and “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?”
Student Lesson Summary
We can describe the shape of distributions as symmetric, skewed, bell-shaped, bimodal, or uniform. The dot plot, histogram, and box plot shown here represent the distribution of the same data set. This data set has a symmetric distribution.
In a symmetric distribution, the mean is equal to the median and there is a vertical line of symmetry in the center of the data display. The histogram and the box plot both group data together. Since histograms and box plots do not display each data value individually, they do not provide information about the shape of the distribution to the same level of detail that a dot plot does. This distribution, in particular, can also be called bell-shaped. A bell-shaped distribution has a dot plot that takes the form of a bell with most of the data clustered near the center and fewer points farther from the center. This makes the measure of center a very good description of the data as a whole. Bell-shaped distributions are always symmetric or close to it.
The dot plot, histogram, and box plot shown here represent a skewed distribution.
In a skewed distribution, one side of the distribution has more values farther from the bulk of the data than the other side has. This results in the mean and median not being equal. In this skewed distribution, the data are skewed to the right because most of the data are near the 8 to 10 interval, but there are many points to the right. The mean is greater than the median. The large data values to the right cause the mean to shift in that direction while the median remains with the bulk of the data. So, the mean is greater than the median for distributions that are skewed to the right. In a data set that is skewed to the left, a similar effect happens but to the other side. Again, the dot plot provides a greater level of detail about the shape of the distribution than do either the histogram or the box plot.
A uniform distribution has the data values evenly distributed throughout the range of the data. This causes the distribution to look like a rectangle.
In a uniform distribution, the mean is equal to the median because a uniform distribution is also a symmetric distribution. The box plot does not provide enough information to describe the shape of the distribution as uniform, though the even length of each quarter does suggest that the distribution may be approximately symmetric.
A bimodal distribution has two very common data values seen in a dot plot or histogram as distinct peaks.
Sometimes, a bimodal distribution has most of the data clustered in the middle of the distribution. In these cases, the center of the distribution does not describe the data very well. Bimodal distributions are not always symmetric. For example, the peaks may not be equally spaced from the middle of the distribution, or other data values may disrupt the symmetry.
Have feedback on the curriculum?
Help us improve by sharing suggestions or reporting issues.