Unit 8, Lesson 3
Interpreting Histograms
Accelerated 6
3.1
Warm-up
Dog Show (Part 1)
Instructional Routines
NoneMaterials
NoneActivity Narrative
The purpose of this Warm-up is to connect the analytical work that students have done with dot plots in previous lessons with statistical questions. This activity reminds students that we gather, display, and analyze data in order to answer statistical questions. This work will be helpful as students contrast dot plots and histograms in subsequent activities.
Launch
Arrange students in groups of 2. Give students 1 minute of quiet work time, followed by 2 minutes to share their responses with a partner. Ask students to decide, during a partner discussion, if each question proposed by their partner is a statistical question that can be answered using the dot plot. Follow with a whole-class discussion.
If students have trouble getting started, consider giving a sample question that can be answered using the data on the dot plot (for example, “How many dogs weigh more than 100 pounds?”)
Activity
NoneHere is a dot plot showing the weights, in pounds, of 40 dogs at a dog show.
A dot plot, weight in pounds, labeled 60 to 180 by tens. Starting at 68 with intervals of 2, the number of dots above each increment is 1, 1, 2, 0, 1, 1, 0, 2, 1, 0, 0, 4, 1, 3, 0, 0, 4, 0, 0, 1, 0, 0, 1, 5, 0, 3, 0, 1, 2, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 1.
- Write two statistical questions that can be answered using the dot plot.
- What would you consider a typical weight for a dog at this dog show? Explain your reasoning.
Student Response
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Building on Student Thinking
Activity Synthesis
Ask students to share questions that they agreed were statistical questions that could be answered using the dot plot. If there is time, consider asking students how they would find the answer to some of the statistical questions.
Display the dot plot for all to see. Ask students to share a typical weight for a dog at this dog show and why they think it is typical. Mark their answers on the displayed dot plot. After each student shares, ask the class if they agree or disagree.
Lesson Synthesis
A histogram is a visual representation of data that groups values together in intervals, or “bins,” to combine their frequency. This histogram, for instance, represents the distribution for the weights of some dogs.
- “What could the smallest dog weigh? The largest?” (10 kilograms; up to almost 35 kilograms)
- “What does the bar between 25 and 30 tell you?” (5 dogs weigh between 25 and just under 30 kilograms.)
- “How would you describe a typical weight for this group of dogs?” (The center of the distribution seems to be around 20–25 kilograms. I might choose 22 because it’s in that interval and there seem to be a few more smaller dogs than larger ones.)
- “What can we say about the spread of the dog weights based on this histogram?” (The dogs range in weight from around 10 kilograms up to almost 35 kilograms. It is hard to be precise with a histogram, but those are the outer edges of the bars shown.)
- “In general, what information does a histogram allow us to see? How is it different from a dot plot?” (A bigger picture of the distribution is shown in the histogram, but some of the detail is lost when compared to a dot plot. For example, this histogram does not show the weight of any individual dogs.)
- “What are some advantages of using a histogram instead of a dot plot?” (A histogram is useful for looking at general trends by grouping a range of values together into a single frequency. When the data are very spread out, when there are not very many data points with the same value, or when an overall idea of the distribution is more important than a detailed view, a histogram may be more useful than a dot plot.)
Student Lesson Summary
In addition to using dot plots, we can also represent distributions of numerical data using histograms.
Here is a dot plot that shows the weights, in kilograms, of 30 dogs, followed by a histogram that shows the same distribution.
A dot plot, the numbers 10 through 35, in increments of 5, are indicated. The 30 data values are as follows: 10 kilograms, 1 dot. 11 kilograms, 1 dot. 12 kilograms, 2 dots. 13 kilograms, 1 dot. 15 kilograms, 1 dot. 16 kilograms, 2 dots. 17 kilograms, 1 dot. 18 kilograms, 2 dots. 19 kilograms, 1 dot. 20 kilograms, 3 dots. 21 kilograms, 1 dot. 22 kilograms, 3 dots. 23 kilograms, 1 dot. 24 kilograms, 2 dots. 26 kilograms, 2 dots. 28 kilograms, 1 dot. 30 kilograms, 1 dot. 32 kilograms, 2 dots. 34 kilograms, 2 dots.
A histogram, the horizontal axis is labeled “dog weights in kilograms” and the numbers 10 through 35, in increments of 5, are indicated. On the vertical axis the numbers 0 through 10, in increments of 2, are indicated. The data represented by the bars are as follows: Weight from 10 up to 15, 5. Weight from 15 up to 20, 7. Weight from 20 up to 25, 10. Weight from 25 up to 30, 3. Weight from 30 up to 35, 5.
In a histogram, data values are placed in groups, or “bins,” of a certain size, and each group is represented with a bar. The height of the bar tells us the frequency for that group.
For example, the height of the tallest bar is 10, and the bar represents weights from 20 to less than 25 kilograms, so there are 10 dogs whose weights fall in that group. Similarly, there are 3 dogs that weigh anywhere from 25 to less than 30 kilograms.
Notice that the histogram and the dot plot have a similar shape. The dot plot has the advantage of showing all of the data values, but the histogram is easier to draw and to interpret when there are a lot of values or when the values are all different.
The histogram allows us to learn more about the dog weight distribution and describe its center and spread.
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