Mathematics · Year 10 · Statistical Measures and Graphs · Spring Term

Measures of Spread: Range and Interquartile Range

Calculating and interpreting range and interquartile range from raw data and frequency tables.

• National Curriculum Attainment Targets
• GCSE: Mathematics - Statistics

About This Topic

Measures of spread, such as range and interquartile range (IQR), describe how data points vary within a dataset. The range subtracts the minimum value from the maximum, offering a quick snapshot but vulnerability to outliers. The IQR measures the spread of the middle 50% of data, from the first quartile (Q1) to the third quartile (Q3). Students calculate both from raw data lists and frequency tables, often alongside box plots for visual comparison.

This topic aligns with GCSE Mathematics Statistics in the UK National Curriculum, building on central tendency measures like median. It emphasises why IQR proves robust for skewed distributions or data with extremes, unlike range, which amplifies outliers. Key skills include interpreting these measures to compare datasets, such as exam scores or reaction times, and explaining impacts on data representation.

Active learning suits this topic well. Sorting physical data cards helps students grasp quartiles kinesthetically, while group analysis of altered datasets reveals outlier effects concretely. Collaborative tasks with class-generated data make abstract calculations relevant and reinforce interpretation through peer explanation.

Key Questions

1. Explain why the interquartile range is a robust measure of spread for skewed data.
2. Differentiate between range and interquartile range in terms of data representation.
3. Analyze how extreme values impact the range versus the interquartile range.

Learning Objectives

• Calculate the range and interquartile range for a given set of raw data.
• Calculate the range and interquartile range from data presented in frequency tables.
• Compare the range and interquartile range of two different datasets, justifying the choice of measure for skewed data.
• Explain how extreme values affect the range and interquartile range, using specific examples.
• Analyze the robustness of the interquartile range compared to the range when dealing with datasets containing outliers.

Before You Start

Calculating the Median

Why: Students need to be able to find the median of a dataset to calculate quartiles, which are essential for the IQR.

Ordering Data

Why: Both range and IQR calculations require data to be ordered from smallest to largest.

Reading Frequency Tables

Why: Students must understand how to interpret frequency tables to calculate statistical measures from grouped data.

Key Vocabulary

Range	The difference between the maximum and minimum values in a dataset. It provides a simple measure of spread but is sensitive to extreme values.
Interquartile Range (IQR)	The difference between the upper quartile (Q3) and the lower quartile (Q1). It represents the spread of the middle 50% of the data and is less affected by outliers.
Lower Quartile (Q1)	The value below which 25% of the data falls. It is the median of the lower half of the dataset.
Upper Quartile (Q3)	The value below which 75% of the data falls. It is the median of the upper half of the dataset.
Outlier	A data point that is significantly different from other observations in the dataset. Outliers can heavily influence the range.

Watch Out for These Misconceptions

Common Misconception

The range is always the best measure of spread.

What to Teach Instead

Range distorts with outliers, while IQR focuses on central data. Group sorting activities let students manipulate datasets, visually seeing how one extreme value skews range but spares IQR, building preference for robust measures through comparison.

Common Misconception

Outliers are included in the interquartile range.

What to Teach Instead

IQR spans Q1 to Q3, excluding extremes by design. Hands-on box plot construction with movable markers helps students position quartiles and isolate outliers, clarifying boundaries via tactile feedback and peer verification.

Common Misconception

Quartiles from frequency tables use the same method as raw data.

What to Teach Instead

Cumulative frequency guides quartile positions in tables. Relay tasks with tables encourage step-by-step practice, where teams check calculations collectively, reducing errors and highlighting table-specific steps through discussion.

Active Learning Ideas

Card Sort: Dataset Ordering

Provide printed cards with data values for two datasets, one with an outlier. In small groups, students sort cards by size, identify min/max for range, then mark Q1/Q3 for IQR. Groups compare results and discuss outlier impact on a shared poster.

35 min·Small Groups

Frequency Table Relay: Team Calculation

Divide class into teams. Each member calculates range or a quartile from a frequency table section, passes to next for IQR. First team with correct box plot wins. Review errors as whole class.

40 min·Small Groups

Outlier Investigation: Pairs Edit

Pairs receive raw data sets, calculate range/IQR, then add/remove outliers and recalculate. They sketch box plots before/after and note changes in a table. Share findings in plenary.

30 min·Pairs

Class Survey: Real Data Analysis

Collect class data on a topic like travel times. Whole class tallies into frequency table. Subgroups compute range/IQR, plot box plots, and present interpretations comparing to national data.

45 min·Whole Class

Real-World Connections

• Sports statisticians use measures of spread to analyze player performance. For example, comparing the range of points scored by two basketball players in a season can be misleading if one player has a few exceptionally high-scoring games, while their IQR might give a better sense of their typical scoring range.
• Financial analysts examine the spread of stock prices to understand market volatility. The range of a stock's price over a year might show its highest and lowest points, but the IQR can reveal the typical trading range, offering a more stable indicator of risk.

Assessment Ideas

Quick Check

Provide students with two small datasets, one with an obvious outlier and one without. Ask them to calculate both the range and IQR for each dataset. Then, ask: 'Which measure of spread, range or IQR, better represents the typical spread of data for the dataset with the outlier? Explain why.'

Exit Ticket

Give students a frequency table showing the number of hours Year 10 students spent on homework per week. Ask them to calculate the IQR. On the back, have them write one sentence explaining why the IQR is a useful measure for this type of data.

Discussion Prompt

Present a scenario: 'A teacher compares the test scores of two classes. Class A has a range of 60 marks and an IQR of 20 marks. Class B has a range of 30 marks and an IQR of 25 marks.' Ask students: 'What can you infer about the distribution of scores in each class? Which class had more consistent performance in the middle 50% of scores, and why?'

Frequently Asked Questions

Why is interquartile range better for skewed data?

IQR resists outliers by measuring middle 50% spread, unlike range which stretches with extremes. In skewed sets like income data, range misrepresents typical variation, but IQR captures core data reliably. Students analysing real skewed datasets via box plots quickly see this advantage, aiding GCSE exam responses on measure selection.

How do outliers affect range versus IQR?

Outliers inflate range dramatically as min/max shift, but IQR remains stable within central quartiles. Editing datasets in pairs reveals this: range jumps from 20 to 80 with one outlier, IQR stays 15. Visual box plots make the difference concrete for interpretation tasks.

How can active learning help teach range and IQR?

Active methods like card sorts and data relays engage kinesthetic learners, making quartile finding intuitive. Groups physically order values, compute measures, and debate changes, turning calculations into discoveries. Class surveys link to real data, boosting retention and skills in GCSE-style analysis over passive worksheets.

How to calculate IQR from a frequency table?

Build cumulative frequency column, find total n, locate Q1 (n/4) and Q3 (3n/4) positions, read values from table or graph. Practice with printed tables in relays ensures accuracy. Students plot resulting box plots to verify, connecting calculation to visualisation for deeper understanding.