Box Plots and Five-Number Summary

Active learning works because students need to handle data directly to see how relationships between variables take shape. When they gather their own class data or debate spurious claims, abstract concepts like correlation and causation become concrete through their own reasoning and observations.

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

Learning Objectives

1. Calculate the five-number summary (minimum, first quartile, median, third quartile, maximum) for a given data set.
2. Construct a box plot accurately from a calculated five-number summary.
3. Analyze a box plot to identify the range, interquartile range, and potential outliers.
4. Compare the distribution and skewness of two or more data sets represented by box plots.
5. Explain the relationship between the visual elements of a box plot and the underlying data distribution.

Inquiry Circle

⏱ 50 min·Small Groups

Inquiry Circle: The Great Class Data Collection

Students work in groups to collect two pieces of data from their peers (e.g., arm span vs. height). They plot this on a shared digital scatter plot and use a 'string' or digital tool to find the line of best fit, discussing whether their data shows a strong or weak relationship.

Prepare & details

Explain how a box plot visually represents the five-number summary.

Facilitation Tip: During The Great Class Data Collection, have students rotate through stations to collect multiple data points so everyone contributes to the full data set.

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

Formal Debate

⏱ 30 min·Pairs

Formal Debate: Spurious Correlations

The teacher provides 'crazy' correlations (e.g., ice cream sales vs. shark attacks). Students must work in pairs to identify the 'hidden variable' (e.g., summer heat) and debate why these two things are correlated but not causal.

Prepare & details

Analyze how to identify outliers using the interquartile range.

Facilitation Tip: For Spurious Correlations, assign roles so debaters must cite specific data examples from their posters during rebuttals.

Setup: Two teams facing each other, audience seating for the rest

Materials: Debate proposition card, Research brief for each side, Judging rubric for audience, Timer

• Analyze
• Evaluate
• Create
• Self-Management
• Decision-Making

Gallery Walk

⏱ 40 min·Small Groups

Gallery Walk: Prediction Posters

Groups are given a scatter plot with a line of best fit. They must create a poster that uses the line to make one 'safe' prediction (interpolation) and one 'risky' prediction (extrapolation), explaining the dangers of the latter. The class rotates to critique the 'riskiness' of the predictions.

Prepare & details

Design a box plot for a given data set and interpret its skewness.

Facilitation Tip: In the Gallery Walk, place a sticky note pad at each poster so viewers can post immediate questions or insights for the creators to review after the walk.

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

Teaching This Topic

Teachers should start with familiar contexts students can relate to, like height vs. arm span, so the data feels personal. Avoid rushing to formulas; instead, have students estimate correlations by eye before calculating. Research shows that students grasp causation best when they actively test claims against their data, not just hear explanations.

What to Expect

Successful learning looks like students confidently constructing five-number summaries and box plots, explaining why correlation does not imply causation, and using evidence from their own data to support arguments. They should also recognize the difference between strong, weak, positive, and negative correlations in real contexts.

Watch Out for These Misconceptions

Common Misconception

During The Great Class Data Collection, watch for students who label any downward trend as 'negative correlation' without considering the context or strength of the relationship.

What to Teach Instead

Ask them to quantify the trend by estimating the slope of the line of best fit and to explain whether the relationship is strong or weak using their data.

Common Misconception

During Spurious Correlations, watch for students who assume that any two variables moving together must have a cause-and-effect link.

What to Teach Instead

Have them use the debate format to present counterexamples and examine third variables or coincidences in their data sets.

Assessment Ideas

Quick Check

After The Great Class Data Collection, provide each student with a small data set (e.g., shoe size vs. height) and ask them to calculate the five-number summary and draw a box plot. Collect these to check for accuracy in calculations and plotting.

Peer Assessment

During the Gallery Walk, have students rotate in pairs and use a simple rubric to assess two prediction posters, focusing on the clarity of the correlation and the strength of the evidence presented.

Discussion Prompt

After Spurious Correlations, pose the question: 'How can you tell if a correlation might be due to chance rather than a real relationship?' Lead a discussion where students connect their examples to the idea of outliers and variability in data.

Extensions & Scaffolding

• Challenge: Ask students to find a real-world data set online that appears correlated, then use their five-number summary skills to analyze it and write a short report on whether causation is plausible.
• Scaffolding: Provide partially completed box plots and five-number summaries for students to finish, or offer a data set with fewer than ten values to reduce complexity.
• Deeper: Introduce the concept of residual analysis by having students calculate how far points are from their line of best fit and discuss what those distances reveal about the data’s fit.

Key Vocabulary

Five-Number Summary	A set of five key values that describe the distribution of a data set: minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum.
Box Plot	A graphical representation of the five-number summary, showing the median, quartiles, and range of a data set. It visually displays the spread and central tendency of the data.
Interquartile Range (IQR)	The difference between the third quartile (Q3) and the first quartile (Q1) (IQR = Q3 - Q1). It represents the spread of the middle 50% of the data.
Outlier	A data point that is significantly different from other data points in a data set. In box plots, outliers are often identified using a rule based on the IQR.
Skewness	A measure of the asymmetry of a probability distribution. In a box plot, skewness can be inferred by the position of the median within the box and the lengths of the whiskers.