# Analyze data distributions

• Create dot plots, histograms and box plots (S-ID.A.1).
• Use available classroom technology to create histograms and box plots and calculate measures of center and spread (S-ID.A.1).
• Use terms such as “flat,” “skewed,” “bell-shaped,” and “symmetric” to describe data distributions (S-ID.A.2).
• Analyze and compare data sets (S-ID.A.3).

Much of this section reviews ideas developed in grades 6 and 7 (but not 8), in order to allow the teacher to address any gaps in understanding revealed in the pre-unit assessment. If students are not already familiar with terms used to describe distributions (e.g., flat, skewed, bell-shaped, symmetric), then these terms should be introduced. With any task used, students should always be asked to interpret the statistical terms and measures in the context of the data set being described.

The teacher may also opt to use the tasks in this section to demonstrate how to use available technology to create histograms and box plots, and to calculate measures of center and spread, helping students to choose appropriate tools strategically when they analyze data in the future.

1 Speed Trap

WHAT: Are the speed distributions similar for cars traveling northbound and for cars traveling southbound on an interstate highway? Students draw box plots for the two data sets. They use the plots and appropriate numerical summaries of the data sets to write a few sentences comparing the speeds of northbound cars and southbound cars.

WHY: Speed Trap could be used if students need more practice finding quartiles, creating box plots, and analyzing and comparing two data sets. This is also an opportunity to notice the effect of an outlier.

## External Resources

1 Using Frequency Graphs

#### Description

WHAT: In this lesson, students work with different data distributions. An ungraded assessment task is given in class or as homework several days before the lesson, asking students to interpret a continuous frequency graph of monthly spending on cell phones. The lesson begins with an example of a line graph and a bar graph for the same data distribution (scores on tests), intended to illustrate the idea that a frequency graph for a discrete distribution can be approximated by a continuous graph. Students are asked questions that can be answered using these graphs. Then students are given eight continuous distributions of test scores and eight written interpretations, and asked to match the distributions with their descriptions (S-ID.A.2, S-ID.A.3). Students review their work on the assessment as a follow-up (in class or as homework).

WHY: In addition to showing how a continuous graph can be used to approximate a graph for a discrete distribution, the lesson is an opportunity to connect differences in distributions with specific outcomes in a real-world context (MP2). Students work in pairs to construct viable arguments to support their matches of distributions and descriptions (MP3).

2 Dice Activity

#### Description

WHAT: In this lesson, the class creates a frequency graph (in this case a dot plot) of the outcomes of rolling two dice. This initiates a lesson about the terms spread, shape, center, mean, median, mode, and quartile, including definitions and examples of each.

WHY: The graphs generated by the activity provide a context in which to use the terms (spread, shape, center, mean, median, mode, and quartile) in discussing different data distributions and to use a frequency graph to create a box plot (S-ID.A.1).

3 Penny Weight Activity

#### Description

WHAT: In this lesson, the class collects the weights and years from a batch of pennies. Students create frequency graphs (again, dot plots, S-ID.A.1), and analyze them using the terms discussed in the previous lesson. The distribution of the years should be pretty flat but may be skewed toward recent years, assuming a random sample of pennies in circulation, and the distribution of weights should show high frequencies around one value—a steep bell shape.

WHY: This lesson provides additional examples of distributions, allowing students opportunities to use and understand the meaning of terms used to describe and compare the shapes of distributions (S-ID.A.2). It also provides students an opportunity to make strategic of use technology to create histograms and box plots, and to calculate measures of center and spread (MP5).

4 Representing Data Using Box Plots

#### Description

WHAT: This lesson builds on Using Frequency Graphs and uses the same format (pre-lesson ungraded assessment—this time with a box plot, lesson, post-lesson follow-up). In the lesson, students are given eight line graphs (the same eight as above) and eight box plots and asked to match those which could represent the same distribution by analyzing differences in shape, center and spread (S-ID.A.2, S-ID.A.3).

WHY: The purpose of this activity is to provide students with an opportunity to connect two different ways of representing data (line graph and box plot) in the same context (MP2). Students work in pairs to construct viable arguments to support their matches (MP3).