Students sort different histograms during this activity. A sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP2, MP7).

As students work, encourage them to refine their descriptions of the distributions using more precise language and mathematical terms (MP6).

Students may be familiar with the geometric meaning of symmetry. It is used in a similar way here to describe the shape of a distribution.

Students will have had a chance to discuss the different features of a distribution in small groups. Use the whole-class discussion to prompt students to think about what the features might mean, and whether or how they affect the way we characterize a distribution. Remind students that we have been using the center of a distribution to talk about what is typical in a group. Discuss some of these questions:

• “Look at the histograms that you think show symmetry. When a distribution is approximately symmetrical, where might its center be?” (For a symmetrical or approximately symmetrical distribution the center is usually at the line of symmetry.)
• “Is it easy to find the peaks on a distribution? Are there any histograms that your group debated about?” (Sometimes it is easy to find a peak because there is a much larger bar in the histogram compared to the others. We debated about some of them like Histogram F because we might consider there being 3 peaks or maybe just 1.)
• “Look at the histograms that show gaps. How might a gap (such as that in Histogram K) affect our description of what is typical in a group?” (When there are gaps, there might be 2 groups within the data that are different in an important way. It may be important to separate them to discuss what is typical.)
• “Look at the histograms that have values that are far away from other values. Do unusual values (such as those in Histogram G) affect our description of center and spread? If those unusual values weren't there, would our description of center and spread change?” (They might pull the center closer to these values. The spread appears much greater with these unusual values because they indicate that there is a lot of variability.)

It is okay if there is some disagreement about how the histograms are categorized. Depending on the situation, it may be important to consider things like peaks more or less precisely. For example, if the data in Histogram F are very precise like from a chemistry experiment, it may be important to investigate the 3 possible peaks more closely. If the data are less precise like how people are rating a TV show, then it may be okay to say there is only 1 peak or that the data are fairly spread out and there are no peaks.

Expect students’ answers to be very informal. The goal of the discussion is to raise students' awareness that the shape and features of distributions may affect how we characterize the data. This experience provides a conceptual foundation that would help students make sense of measures of center (mean and median) and measures of spread (mean absolute deviation, interquartile range, and range) later.

Conclude the discussion by inviting 2–3 students to share a norm that they identified in action. Provide this sentence frame to help students organize their thoughts in a clear, precise way:

• “I noticed our norm “” in action today. and it really helped me/my group because .”

In this activity, students draw a bar graph and histogram, and then they describe the distributions shown on each display. Although the two visual displays may appear similar at first glance, there are important distinctions between the representations. Students notice differences in how we might characterize distributions in bar graphs and those in histograms, including how we describe typical values or categories. Along the way, students consolidate their understanding about categorical and numerical data.

Monitor for groups that draw a bar graph with the bars in a different order.

The purpose of the discussion is for students to recognize the differences between histograms and bar graphs.

Invite previously selected groups to share an accurate histogram and bar graphs with bars in different orders. Then, solicit several observations about how the two graphical displays compare. Ask questions such as:

• “How are the bar graphs and histograms alike? How are they different?” (They both use bars to represent the frequency of data. Bar graphs represent categorical data and histograms represent numerical data.)
• “Can we use a bar graph to display the data on travel times? Why or why not?” (No, because travel times are numerical. We could convert the times to categories by classifying different intervals as “short” or “long” and then make a bar graph from that, but this would require an extra step and a discussion about what times are considered in each category.)
• “Can we use a histogram to display the data on methods of travel? Why or why not?” (No, histograms can be used only with numerical data.)

Then, invite students to share their descriptions of the distributions shown on each type of display. Ask questions such as:

• “How are your descriptions of the distribution for travel methods different from those for travel times?” (For travel methods, the description can only comment on individual bars in comparison to other ones. For travel times, we can talk about the shape, center, and spread of the distribution.)
• “Can you talk about the shape of a distribution shown on a bar graph? Can you talk about its center and spread? Why or why not?” (No, because the bars can be rearranged in a bar graph. Which bar is in the center is not important, and the spread doesn’t make sense as a numerical value. We could talk about variability by mentioning how many different bars there are and their relative heights, but it does not make sense to give that as a number.)

Students should recognize that only the distribution of numerical data can be described in terms of shape, center, or spread. We cannot analyze these features for a distribution of categorical data on a bar graph because a bar graph does not use a number line. This means that the bars can be drawn anywhere, in any order, and with any kind of spacing, so shape, center, and spread would have no meaning.