Mathematics · 6th Grade · Geometry and Statistics · Weeks 19-27

Measures of Variability: Range and IQR

Students will calculate and interpret the range and interquartile range (IQR) of a data set.

• Common Core State Standards
• CCSS.Math.Content.6.SP.A.2
• CCSS.Math.Content.6.SP.A.3

About This Topic

Measures of variability describe how spread out the values in a data set are. The range (maximum minus minimum) gives a quick but rough sense of spread, while the interquartile range (IQR) measures the spread of the middle 50% of the data. Together, these statistics complement measures of center by helping students understand consistency, predictability, and the presence of outliers.

CCSS 6.SP.A.2 and 6.SP.A.3 require students to calculate and interpret these measures in context. US 6th graders often calculate range correctly but treat it as a single authoritative measure of spread, missing that one extreme value can make a consistent data set look highly variable. The IQR is more resistant to this distortion and becomes especially important in the context of box plots.

Active learning is well-suited here because sorting and ordering data physically makes quartile boundaries visible and intuitive. Comparing two data sets with the same range but different IQRs is a powerful group discussion prompt that builds genuine statistical reasoning.

Key Questions

1. Explain what a large range suggests about the consistency of a data set.
2. Differentiate between range and interquartile range as measures of spread.
3. Analyze how outliers affect the range and IQR of a data set.

Learning Objectives

• Calculate the range and IQR for given data sets.
• Compare the range and IQR of two data sets, explaining differences in spread.
• Analyze the impact of outliers on the range and IQR of a data set.
• Explain what a large range indicates about the consistency of data.

Before You Start

Measures of Center: Mean, Median, Mode

Why: Students need to understand how to find the median and order data to calculate quartiles and the IQR.

Data Representation: Dot Plots and Stem-and-Leaf Plots

Why: Students should be familiar with organizing and visualizing data, which aids in identifying minimum, maximum, and quartiles.

Key Vocabulary

Range	The difference between the maximum and minimum values in a data set. It provides a simple measure of the total spread.
Interquartile Range (IQR)	The difference between the third quartile (Q3) and the first quartile (Q1) of a data set. It measures the spread of the middle 50% of the data.
Outlier	A data point that is significantly different from other observations in the data set. Outliers can heavily influence the range.
Quartiles	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

A large range always means the data is highly variable throughout.

What to Teach Instead

One outlier can produce a large range even if all other values are tightly clustered. The IQR filters out extreme values by focusing on the middle 50%, giving a more reliable picture of typical spread. Side-by-side data sets in group investigations surface this effectively.

Common Misconception

The IQR is just the two numbers at the quartile boundaries.

What to Teach Instead

Students sometimes report Q1 and Q3 separately rather than computing Q3 minus Q1. Connecting IQR calculation to the box in a box plot (the width of the box equals the IQR) gives students a visual check on their arithmetic.

Active Learning Ideas

Inquiry Circle: Range vs. IQR

Groups receive two data sets printed on cards: one tightly clustered except for one outlier, one genuinely spread out. They calculate range and IQR for both, then discuss: which set is more consistent? Which measure reflects that better?

35 min·Small Groups

Think-Pair-Share: Quiz Score Analysis

Present two classes' quiz scores. Both have the same mean, but different IQRs. Pairs discuss which class performed more consistently and what that means for the teacher's interpretation of the results.

20 min·Pairs

Stations Rotation: Box Plot Building

At each station, students order a data set, find the five-number summary (minimum, Q1, median, Q3, maximum), calculate the IQR, and sketch a basic box plot. Rotating through multiple data sets reinforces the process and builds fluency.

45 min·Small Groups

Real-World Connections

• Sports analysts use range and IQR to describe the performance spread of players over a season. For example, a basketball player's points per game might have a small IQR if they are consistently scoring, but a large range if there are a few unusually high or low scoring games.
• Meteorologists calculate the range and IQR of daily temperatures for a city to understand typical weather patterns and predict the likelihood of extreme heat or cold spells. This helps in issuing weather advisories.

Assessment Ideas

Exit Ticket

Provide students with a small data set (e.g., test scores for 5 students). Ask them to calculate the range and IQR. Then, ask: 'What does the range tell you about the spread of these scores? What does the IQR tell you?'

Discussion Prompt

Present two data sets with the same range but different IQRs (e.g., Set A: 1, 2, 3, 4, 10; Set B: 4, 5, 6, 7, 10). Ask: 'Which data set is more consistent in its middle values? How do the range and IQR help you answer this?'

Quick Check

Give students a data set with a clear outlier. Ask them to calculate the range. Then, ask them to remove the outlier and recalculate the range. Prompt: 'How did removing the outlier affect the range? Why is the IQR sometimes a better measure of spread for data with outliers?'

Frequently Asked Questions

What is the difference between range and interquartile range?

Range is simply the difference between the maximum and minimum values in a data set, giving overall spread. IQR is the difference between the third and first quartiles (Q3 - Q1), measuring the spread of only the middle 50% of the data. IQR is less affected by extreme values.

How do you find the interquartile range?

Order your data from least to greatest. Find the median to split the data into two halves. Q1 is the median of the lower half, and Q3 is the median of the upper half. Subtract Q1 from Q3. The result is the IQR.

How does active learning help students understand range and IQR?

Sorting physical data cards and marking off quartile boundaries gives students a hands-on way to see what the IQR actually measures. When groups compare two data sets with identical ranges but very different IQRs, the discussion that follows builds genuine statistical judgment about what 'spread' means.

Why is IQR more useful than range when outliers are present?

Range includes the most extreme values, so a single outlier can make a compact data set look widely spread. IQR ignores the top and bottom 25% of values, so it reflects the typical spread of the bulk of the data regardless of extreme values.