Measures of Spread: Range and IQR

Active learning works for this topic because students must physically compare datasets and visualize spread to grasp why range and IQR tell different stories. When learners calculate and plot values themselves, the abstract concept of ‘typical spread’ becomes concrete and memorable.

• 9th Grade
• Mathematics
• 4 activities
• 15 min–35 min

Learning Objectives

1. Calculate the range and interquartile range (IQR) for a given dataset.
2. Construct a box plot using the five-number summary (minimum, Q1, median, Q3, maximum).
3. Compare the spread and consistency of two datasets using their box plots and IQR values.
4. Identify potential outliers in a dataset using the 1.5xIQR rule.
5. Explain why the median is a more appropriate measure of center than the mean for skewed distributions, using US household income as an example.

Inquiry Circle

⏱ 35 min·Small Groups

Inquiry Circle: Compare Two Distributions

Give groups a dataset of household incomes for two different US cities. Each group creates box plots for both datasets, calculates the IQR for each, identifies any outliers using the 1.5xIQR rule, and presents a two-sentence statistical comparison to the class that goes beyond just stating the numbers.

Prepare & details

Explain what the width of the box in a box plot tells us about data consistency.

Facilitation Tip: During Collaborative Investigation: Compare Two Distributions, circulate to ensure groups calculate Q1 and Q3 correctly by ordering data first and splitting at the median.

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

⏱ 25 min·Small Groups

Gallery Walk: Reading Box Plots

Post six box plots representing different real-world datasets such as commute times, hourly wages, and test score distributions. Students rotate to each, write the five-number summary, and answer one context-specific question about spread posted below each graph.

Prepare & details

Construct how we mathematically define an outlier using the 1.5xIQR rule.

Facilitation Tip: During Gallery Walk: Reading Box Plots, post sticky notes with key vocabulary (IQR, median, outliers) to anchor student language as they interpret each plot.

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

⏱ 15 min·Pairs

Think-Pair-Share: Which Measure Tells a Better Story?

Present a dataset of US athlete salaries with one extreme outlier. Students individually calculate the range and IQR, then discuss with a partner which measure better represents the typical team's salary spread and why. Pairs share their reasoning with the class, and the teacher connects the discussion to real statistical reporting practices.

Prepare & details

Justify why the median is often preferred over the mean in reporting US household income.

Facilitation Tip: During Think-Pair-Share: Which Measure Tells a Better Story?, listen for pairs who connect spread to real-world implications, such as income inequality or test score consistency.

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

Human Barometer

⏱ 20 min·Whole Class

Whole Class Discussion: The 1.5xIQR Rule

Work through a class example applying the outlier rule step by step, then show a borderline case where a value lands very close to the fence. The class debates whether the context should influence how the borderline value is treated, building the statistical judgment that distinguishes mechanical application from genuine understanding.

Prepare & details

Explain what the width of the box in a box plot tells us about data consistency.

Facilitation Tip: During Whole Class Discussion: The 1.5xIQR Rule, model marking outliers on a number line before moving to box plots to separate the rule from the graphing step.

Setup: Open space for students to form a line across the room

Materials: Statement cards, End-point labels (Agree/Disagree), Optional: recording sheet

• Understand
• Analyze
• Evaluate
• Self-Awareness
• Social Awareness

Teaching This Topic

Teachers succeed by pairing calculations with visuals—students need to see how a single outlier stretches the range while leaving the IQR nearly unchanged. Avoid rushing to box plots before students internalize the five-number summary; build fluency with medians and quartiles on raw data first. Research shows that students who construct box plots by hand understand their components better than those who only use digital tools.

What to Expect

Successful learning looks like students confidently distinguishing between range and IQR, identifying outliers using the 1.5xIQR rule, and explaining why two datasets with the same mean can have different spreads. Students should also justify their choices when selecting which measure better represents a dataset’s variability.

Watch Out for These Misconceptions

Common Misconception

During Collaborative Investigation: Compare Two Distributions, watch for students who declare the dataset with the larger range as ‘more spread out’ without comparing the IQR.

What to Teach Instead

Ask groups to calculate both range and IQR for their datasets, then present side-by-side comparisons to highlight when the range is misleading due to an outlier.

Common Misconception

During Gallery Walk: Reading Box Plots, watch for students who assume Q2 in a five-number summary is different from the median.

What to Teach Instead

Have students label Q2 = median on every box plot they examine until the equivalence becomes automatic in their labeling.

Common Misconception

During Collaborative Investigation: Compare Two Distributions, watch for students who believe box plots display every data point.

What to Teach Instead

Show a small dot plot and its corresponding box plot side by side, then ask students to generate a second dataset with different values that produces the same box plot to reveal what is hidden.

Assessment Ideas

Quick Check

After Collaborative Investigation: Compare Two Distributions, collect one dataset per group, have them calculate range, IQR, and identify outliers using the 1.5xIQR rule on an index card to submit before moving to the next activity.

Discussion Prompt

After Gallery Walk: Reading Box Plots, facilitate a whole-class discussion using the prompt: ‘Which class had more consistent scores in the middle 50%? Justify using the box plots from the gallery.’

Exit Ticket

After Whole Class Discussion: The 1.5xIQR Rule, ask students to sketch a number line with a box plot showing an outlier and label the outlier using the 1.5xIQR rule in one sentence.

Extensions & Scaffolding

• Challenge students to modify a dataset so its range increases but its IQR stays the same.
• For students who struggle with quartiles, provide a color-coded number line with medians marked to scaffold splitting the data into halves and quarters.
• Deeper exploration: Have students research a real dataset (e.g., city temperatures, sports salaries) and justify which measure of spread best describes its variability, citing their calculations.

Key Vocabulary

Five-Number Summary	A set of five key values that describe the distribution of a dataset: minimum, first quartile (Q1), median, third quartile (Q3), and maximum.
Interquartile Range (IQR)	The difference between the third quartile (Q3) and the first quartile (Q1) of a dataset; it represents the spread of the middle 50% of the data.
Box Plot	A graphical representation of the five-number summary, displaying the distribution of data through quartiles and identifying potential outliers.
Outlier	A data point that is significantly different from other observations in a dataset, often identified using the 1.5xIQR rule.