Mathematics · Grade 6 · Data, Statistics, and Variability · Term 4

Measures of Variability: Range and Interquartile Range

Calculating and interpreting range and interquartile range to describe the spread of data.

• Ontario Curriculum Expectations
• 6.SP.A.3
• 6.SP.B.5.C

About This Topic

In Grade 6 mathematics, students calculate and interpret measures of variability: the range and interquartile range (IQR). The range subtracts the minimum value from the maximum to show overall data spread. The IQR, calculated as the third quartile (Q3) minus the first quartile (Q1), describes the spread of the middle 50% of data and remains stable despite outliers. These tools help students summarize data sets from surveys or measurements.

This topic aligns with Ontario's Data, Statistics, and Variability expectations in Term 4. Students differentiate range from IQR by analyzing how outliers inflate the range but barely shift the IQR. They answer key questions about data spread, building skills for box plots and informed comparisons in science or social studies.

Active learning benefits this topic because students handle real data. Sorting values, plotting dot plots, and adjusting sets for outliers make calculations meaningful. Collaborative tasks reveal why IQR offers robust insights, turning formulas into practical tools students apply confidently.

Key Questions

1. Explain how the range describes the overall spread of a data set.
2. Differentiate between range and interquartile range in terms of what they measure.
3. Analyze how outliers affect the range compared to the interquartile range.

Learning Objectives

• Calculate the range of a given data set by subtracting the minimum value from the maximum value.
• Determine the first quartile (Q1) and third quartile (Q3) for a data set to find the interquartile range (IQR).
• Compare the range and IQR of two different data sets to explain which measure better represents the spread of the middle 50% of the data.
• Analyze the effect of an outlier on the range and IQR of a data set, explaining why IQR is less affected.

Before You Start

Ordering and Representing Data

Why: Students need to be able to order data from least to greatest and identify minimum and maximum values before calculating the range.

Finding the Median

Why: Calculating quartiles requires finding the median of the entire data set and the medians of the lower and upper halves, which are foundational skills for IQR.

Key Vocabulary

Range	The difference between the highest and lowest values in a data set. It shows the total spread of the data.
Interquartile Range (IQR)	The difference between the third quartile (Q3) and the first quartile (Q1) of a data set. It represents the spread of the middle 50% of the data.
Outlier	A data point that is significantly different from other data points in the set. Outliers can greatly affect the range.
Quartile	Values that divide a data set into four equal parts. Q1 is the median of the lower half, and Q3 is the median of the upper half.

Watch Out for These Misconceptions

Common Misconception

Range shows the typical spread between most data points.

What to Teach Instead

Range only captures extremes and ignores clustering. Hands-on sorting and plotting data sets helps students visualize dense middle areas, clarifying that range overstates everyday variability.

Common Misconception

Outliers affect the interquartile range as much as the range.

What to Teach Instead

IQR focuses on the central half, excluding extremes. Side-by-side calculations with modified data sets in pairs demonstrate this stability, building student confidence in choosing measures.

Common Misconception

Quartiles are the same as mean or median.

What to Teach Instead

Quartiles divide ordered data into quarters; median is Q2. Group tasks ordering real data reinforce positions, preventing confusion through visual number lines.

Active Learning Ideas

Pairs Activity: Outlier Hunt

Provide pairs with data sets like student heights; one set includes an outlier. Pairs order data, calculate range and IQR for both, then graph on number lines. Discuss how the outlier changes each measure.

25 min·Pairs

Small Groups: Survey and Analyze

Groups survey classmates on favorite activities or scores, record 20 data points. Order data to find quartiles, compute range and IQR, and present findings on posters. Compare spreads across groups.

35 min·Small Groups

Whole Class: Data Modification Challenge

Display class-generated data on board. As a class, add or remove outliers, recalculate range and IQR each time. Vote on which measure best describes the group.

30 min·Whole Class

Individual: Sports Data Review

Students select Canadian hockey stats, list top 15 points per game. Compute range and IQR, note outlier effects from star players. Write one sentence summary.

20 min·Individual

Real-World Connections

• Sports statisticians use range and IQR to analyze player performance data. For example, they might compare the range of points scored by two basketball players in a season to see the overall variability, and then use the IQR to understand the typical scoring performance of each player.
• Financial analysts might examine the range and IQR of stock prices for a particular company over a month. The range shows the highest and lowest prices, while the IQR indicates the typical price fluctuation within the middle 50% of trading days, helping to assess risk.

Assessment Ideas

Quick Check

Present students with a small data set (e.g., test scores: 75, 82, 88, 90, 95, 100). Ask them to calculate the range and the IQR. Then, ask: 'Which measure, range or IQR, better describes the typical spread of scores for most students in this group and why?'

Discussion Prompt

Provide two data sets, one with an outlier and one without, that have similar middle values. For example, Data Set A: 5, 10, 12, 15, 18, 20. Data Set B: 5, 10, 12, 15, 18, 50. Ask students to calculate the range and IQR for both. Facilitate a discussion on how the outlier in Data Set B affects the range compared to the IQR, and what this tells us about the data's distribution.

Exit Ticket

Give each student a card with a data set. Ask them to write down the range and the IQR. On the back, have them write one sentence explaining what the IQR tells them about the spread of the data that the range does not.

Frequently Asked Questions

How do you teach range versus interquartile range in Grade 6?

Start with familiar data like test scores. Model range as max minus min, then order data to highlight Q1 and Q3 for IQR. Use visuals like dot plots to compare. Emphasize range's sensitivity to outliers through examples, guiding students to see IQR's focus on typical spread. Practice with varied sets builds fluency.

What are common student errors with range and IQR?

Students often think range reflects average differences or that outliers sway IQR equally. They mix up quartile positions. Address with concrete data manipulation: alter sets and recompute. Peer sharing corrects errors as students explain their steps, solidifying concepts.

How does active learning help students grasp measures of variability?

Active tasks like collecting class data, sorting collaboratively, and testing outlier changes make abstract stats tangible. Students see range balloon with extremes while IQR holds steady, fostering deeper insight. Group discussions connect math to real decisions, like comparing school test spreads, improving retention over worksheets.

What Canadian real-world examples use range and IQR?

Analyze NHL player goals: range shows top-to-bottom span, IQR typical performance. Weather data from Environment Canada tracks temperature spreads; IQR ignores freak cold snaps. Student surveys on Toronto commute times use range for extremes, IQR for common experiences. These tie math to local contexts.