Comparing Data Sets using Box Plots and Histograms

Active learning works well for comparing data sets because students must physically and visually engage with the spread, center, and shape of data to truly understand differences between groups. Moving beyond calculations to interpret visual displays helps students develop a nuanced understanding of variability and distribution.

• Year 10
• Mathematics
• 3 activities
• 25 min–50 min

Learning Objectives

1. Compare the distribution, center, and spread of two or more data sets using summary statistics and visual displays.
2. Critique the suitability of box plots and histograms for representing and comparing different types of data distributions.
3. Analyze visual displays of data to identify potential outliers and assess the symmetry or skewness of data sets.
4. Formulate arguments about differences between populations based on statistical evidence from comparative displays.

Inquiry Circle

⏱ 50 min·Small Groups

Inquiry Circle: The Reaction Time Challenge

Students use an online tool to measure their reaction times (e.g., dominant vs. non-dominant hand). In groups, they create back-to-back box plots of the results and write a 'statistical report' comparing the median and spread of the two groups.

Prepare & details

Explain how visual displays can be used to argue that two populations are significantly different?

Facilitation Tip: During the Collaborative Investigation, circulate and ask student groups to point to the parts of their human box plot that represent the median and quartiles, ensuring physical movement reinforces abstract concepts.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Gallery Walk

⏱ 30 min·Small Groups

Gallery Walk: Skewness and Stories

The teacher posts four different histograms (e.g., house prices, heights, dice rolls). Groups must match each histogram to a 'story' or data source and explain their reasoning based on the shape (symmetric, left-skewed, right-skewed) to the rest of the class.

Prepare & details

Compare the central tendency and spread of two data sets based on their box plots.

Facilitation Tip: For the Gallery Walk, provide sticky notes with sentence stems like 'This skewness suggests...' to scaffold written interpretations of the displays.

Setup: Wall space or tables arranged around room perimeter

Materials: Large paper/poster boards, Markers, Sticky notes for feedback

• Understand
• Apply
• Analyze
• Create
• Relationship Skills
• Social Awareness

Think-Pair-Share

⏱ 25 min·Pairs

Think-Pair-Share: The Outlier Debate

Students are given a data set with one extreme outlier. They individually calculate the mean and median, then pair up to discuss which 'average' is a fairer representation of the group. They must agree on a recommendation for a 'news report' based on their findings.

Prepare & details

Critique the effectiveness of different graphical displays for comparing data sets.

Facilitation Tip: During the Think-Pair-Share debate, assign one student in each pair to argue for the mean and the other for the median, forcing both perspectives to be considered before consensus.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Teaching This Topic

Teach this topic by balancing visual interpretation with hands-on construction of both box plots and histograms. Avoid overemphasizing calculation and instead focus on what each display reveals about the data. Use real-world data sets that naturally lead to questions about center and spread, and encourage students to critique which display better answers their questions. Research shows that students better understand variability when they compare and contrast multiple representations of the same data.

What to Expect

Successful learning is evident when students confidently compare data sets using precise language about median, IQR, spread, and skewness, not just shape or range. They should justify their comparisons with evidence from both box plots and histograms, and recognize when one display reveals information the other does not.

Watch Out for These Misconceptions

Common Misconception

During the Collaborative Investigation, watch for students assuming that a longer box or whisker contains more data points.

What to Teach Instead

Use the human box plot to have students count the number of students in each quartile section. Then, measure the distance between quartiles and discuss how spread relates to variability, not quantity of data.

Common Misconception

During the Think-Pair-Share debate, watch for students defaulting to the mean when comparing skewed data sets.

What to Teach Instead

Provide each pair with a data set like annual incomes and have them calculate both the mean and median. Ask them to explain which measure better represents the 'typical' value and why.

Assessment Ideas

Discussion Prompt

After the Collaborative Investigation, present students with two box plots comparing reaction times from two different age groups. Ask: 'Which group has more consistent reaction times? Justify your answer using the IQR and any outliers.'

Quick Check

During the Gallery Walk, give each student a clipboard with a checklist to evaluate two displays. Items include: 'Is the median clearly marked? Is the IQR labeled or calculated? Does the display help explain skewness?' Collect these to assess understanding of key concepts.

Peer Assessment

After the Think-Pair-Share debate, have students exchange their written comparisons of the two data sets. Each student uses a rubric to evaluate their partner's work on clarity, use of statistics, and reasoning about outliers or skewness.

Extensions & Scaffolding

• Challenge: Provide a data set with a known outlier and ask students to create both a histogram and box plot, then write a paragraph explaining how the outlier affects each display and their interpretation of the data.
• Scaffolding: For students struggling with quartiles, give them a pre-sorted set of data cards to physically divide into four equal groups before plotting.
• Deeper exploration: Introduce students to a bimodal data set and ask them to create both displays, then hypothesize about the underlying cause of the two peaks and design a follow-up data collection to test their hypothesis.

Key Vocabulary

Box Plot	A visual representation of the distribution of data through quartiles. It shows the minimum, first quartile (Q1), median, third quartile (Q3), and maximum values.
Histogram	A graphical display of data where the data is divided into bins (intervals), and the frequency of data points falling into each bin is represented by a bar.
Interquartile Range (IQR)	The difference between the third quartile (Q3) and the first quartile (Q1) of a data set, representing the spread of the middle 50% of the data.
Median	The middle value in a data set when the data is ordered from least to greatest. It is a measure of central tendency.
Outlier	A data point that is significantly different from other data points in a data set. Box plots often use fences to identify potential outliers.