Measures of Spread (Range and IQR)

Active learning builds intuition for measures of spread because calculating range and IQR from raw data feels abstract without hands-on handling. Students see how outliers inflate range but barely shift IQR when they sort physical cards or adjust datasets themselves, turning numbers into visible patterns.

• Secondary 3
• Mathematics
• 4 activities
• 25 min–45 min

Learning Objectives

1. Calculate the range and interquartile range (IQR) for both ungrouped and grouped data sets.
2. Explain why the IQR is a more robust measure of spread than the range when outliers are present.
3. Compare and contrast the strengths and weaknesses of the range and IQR in describing data variability.
4. Analyze the impact of extreme values on the range and IQR for a given data set.

Collaborative Problem-Solving

⏱ 35 min·Small Groups

Data Card Sort: Quartile Construction

Provide printed data cards with numbers from a dataset. Students in small groups arrange cards in order, mark median, then split for Q1 and Q3 to compute IQR and range. They sketch box plots and note changes if an outlier card is added.

Prepare & details

Explain why the interquartile range is often a better measure of spread than the total range.

Facilitation Tip: During Data Card Sort: Quartile Construction, place duplicate values in the deck to force students to discuss how to handle ties when finding medians and quartiles.

Setup: Groups at tables with problem materials

Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Collaborative Problem-Solving

⏱ 25 min·Pairs

Outlier Simulation: Dataset Tweaks

Give pairs two identical datasets, one with an outlier. Pairs recalculate range and IQR for both, then graph box plots to compare spreads. Discuss which measure better represents typical variability.

Prepare & details

Analyze how outliers affect the range and interquartile range.

Facilitation Tip: For Outlier Simulation: Dataset Tweaks, provide sliders on a shared spreadsheet so the whole class can watch IQR change (or stay stable) as they drag outlier values.

Setup: Groups at tables with problem materials

Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Collaborative Problem-Solving

⏱ 40 min·Whole Class

Grouped Data Challenge: Frequency Tables

Distribute frequency tables for grouped data like heights. Whole class follows steps to find cumulative frequencies, locate quartiles via interpolation, compute IQR and range. Share findings on board.

Prepare & details

Compare the strengths and weaknesses of range and interquartile range as measures of spread.

Facilitation Tip: In Grouped Data Challenge: Frequency Tables, give one group a table with uneven class widths to spark debate about weighting midpoints.

Setup: Groups at tables with problem materials

Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Collaborative Problem-Solving

⏱ 45 min·Small Groups

Real-World Data Hunt: Class Survey

Students collect class data on study hours individually, input into shared spreadsheet. In small groups, compute measures of spread, identify outliers, and interpret for the group.

Prepare & details

Explain why the interquartile range is often a better measure of spread than the total range.

Facilitation Tip: During Real-World Data Hunt: Class Survey, ask students to predict which measure (range or IQR) will be larger before they calculate, then compare predictions to results.

Setup: Groups at tables with problem materials

Materials: Problem packet, Role cards (facilitator, recorder, timekeeper, reporter), Problem-solving protocol sheet, Solution evaluation rubric

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Teaching This Topic

Teach quartile calculation by having students physically split data cards into halves and quarters before writing formulas. Avoid teaching IQR as a formula alone; connect it to box plots early so students see how Q1, median, and Q3 define the middle 50 percent. Research shows that students grasp outliers better when they create skewed datasets themselves rather than just observing pre-made examples.

What to Expect

Students confidently compute range and IQR, explain why IQR resists outliers, and choose the better measure for skewed or inconsistent data. They justify decisions in small groups, showing they grasp both calculation steps and conceptual trade-offs between measures.

Watch Out for These Misconceptions

Common Misconception

During Data Card Sort: Quartile Construction, watch for students who include the overall median in both lower and upper halves when calculating Q1 and Q3.

What to Teach Instead

Guide them to exclude the overall median from both halves, then find medians of the remaining lower and upper portions to get Q1 and Q3 respectively.

Common Misconception

During Outlier Simulation: Dataset Tweaks, watch for students who assume adding any extreme value will change the IQR.

What to Teach Instead

Have them drag a slider from 1 to 100 and observe that IQR only shifts when the new value crosses Q1 or Q3 boundaries.

Common Misconception

During Grouped Data Challenge: Frequency Tables, watch for students who treat class midpoints as actual data points when calculating range.

What to Teach Instead

Prompt them to compare the grouped range to the raw data range once they reconstruct an approximate dataset from frequencies.

Assessment Ideas

Quick Check

After Data Card Sort: Quartile Construction, provide two small ordered datasets (one with an outlier) and ask students to calculate range and IQR on a whiteboard. Then circulate and note who correctly chooses IQR for the outlier dataset.

Discussion Prompt

During Outlier Simulation: Dataset Tweaks, pause the simulation and ask groups to explain which class (A or B) in the scenario has more consistent scores, citing their calculated range and IQR values.

Exit Ticket

After Real-World Data Hunt: Class Survey, collect students’ exit tickets showing range and IQR for their collected data. Read one sentence explanations aloud to the class and highlight the clearest justifications before dismissal.

Extensions & Scaffolding

• Challenge: Ask early finishers to design a dataset where range is five times IQR, then trade with a partner to verify each other’s work.
• Scaffolding: Provide a partially sorted data set and a template box plot so students only fill in quartiles and spread measures.
• Deeper exploration: Have students collect a larger real-world dataset (e.g., daily temperatures for a month) and compare range and IQR across different months or locations.

Key Vocabulary

Range	The difference between the maximum and minimum values in a data set. It provides a quick, but sometimes misleading, measure of spread.
Interquartile Range (IQR)	The difference between the third quartile (Q3) and the first quartile (Q1). It measures the spread of the middle 50% of the data, making it less sensitive to outliers.
Quartiles	Values that divide a data set into four equal parts. Q1 is the median of the lower half, Q2 is the overall median, and Q3 is the median of the upper half.
Outlier	A data point that is significantly different from other observations in the data set. Outliers can heavily influence the range but have little effect on the IQR.