Mathematics · Year 9 · Statistics and Probability · Term 4

Measures of Spread (Range, IQR)

Students will calculate and interpret the range and interquartile range (IQR) as measures of data spread.

• ACARA Content Descriptions
• AC9M9ST01

About This Topic

Data variation and box plots provide Year 9 students with a powerful way to compare different datasets. By using the 'five-point summary' (minimum, lower quartile, median, upper quartile, and maximum), students can see not just the 'average' of a group, but how spread out or consistent the data is. This is a vital skill for interpreting everything from sports statistics to climate data and school performance.

In the Australian Curriculum, this unit focuses on the visual representation of distribution. Box plots (or box-and-whisker plots) allow for easy side-by-side comparisons of two populations. This topic comes alive when students can use 'human box plots' to physically represent their own class data, such as reaction times or heights. Students grasp this concept faster through collaborative analysis where they must 'tell the story' of the data, identifying which group is more consistent and why.

Key Questions

1. What does the spread of the interquartile range tell us about the consistency of a data set?
2. Compare the range and IQR as measures of spread, highlighting their advantages and disadvantages.
3. Predict how adding an outlier affects the range and IQR of a data set.

Learning Objectives

• Calculate the range and interquartile range (IQR) for a given data set.
• Compare and contrast the range and IQR as measures of data spread, identifying their strengths and weaknesses.
• Analyze how the addition of an outlier impacts the range and IQR of a data set.
• Explain the meaning of the IQR in terms of data consistency and distribution.

Before You Start

Calculating the Median and Mode

Why: Students need to be able to find the median to understand quartiles and the IQR.

Ordering Data Sets

Why: Finding the minimum, maximum, and quartiles requires data to be arranged in ascending order.

Key Vocabulary

Range	The difference between the maximum and minimum values in a data set. It provides a simple measure of the total spread of the data.
Interquartile Range (IQR)	The difference between the upper quartile (Q3) and the lower quartile (Q1) of a data set. It represents the spread of the middle 50% of the data.
Quartiles	Values that divide a data set into four equal parts. Q1 is the 25th percentile, Q2 is the median (50th percentile), and Q3 is the 75th percentile.
Outlier	A data point that is significantly different from other observations in the data set. Outliers can heavily influence the range but have less impact on the IQR.

Watch Out for These Misconceptions

Common Misconception

Students often think that a longer 'box' or 'whisker' means there are more pieces of data in that section.

What to Teach Instead

This is a very common error. They need to understand that each of the four sections of a box plot always contains exactly 25% of the data. A longer section just means the data in that quarter is more 'spread out'. The 'Human Box Plot' activity is perfect for showing that the same number of people can be spread over a larger space.

Common Misconception

Confusing the median with the mean.

What to Teach Instead

Students often want to add everything up and divide. Box plots are strictly about 'position' and 'ordering'. Using a small dataset and physically finding the middle person helps reinforce that the median is the 'middle value', not the 'calculated average'.

Active Learning Ideas

Simulation Game: The Human Box Plot

The class collects data (e.g., number of siblings). Students line up in order. Five students are chosen to represent the 'five-point summary' and hold signs. The rest of the class uses a long rope to create the 'box' and 'whiskers' around them. This makes the quartiles physically visible.

40 min·Whole Class

Inquiry Circle: The Great AFL/NRL Stats Duel

Groups are given the scores from two different sports teams over a season. They must calculate the five-point summary, draw two box plots on the same scale, and then write a 'sports report' comparing the two teams' consistency and performance. This applies stats to a popular Australian context.

50 min·Small Groups

Gallery Walk: Box Plot Detectives

Display several box plots representing different 'mystery' datasets (e.g., daily temperatures in Darwin vs. Hobart). Students move in pairs to match the 'story' to the plot based on the median and the spread (IQR). This builds data interpretation skills.

35 min·Pairs

Real-World Connections

• Sports statisticians use measures of spread like IQR to analyze player performance consistency. For example, they might compare the consistency of points scored by two basketball players over a season, noting how much their scores typically vary.
• Financial analysts examine the spread of stock prices to understand market volatility. A wide range or IQR in daily stock prices might indicate higher risk for investors compared to a stock with a narrower spread.

Assessment Ideas

Quick Check

Provide students with two small data sets, one with an obvious outlier. Ask them to calculate the range and IQR for both sets. Then, ask: 'Which measure of spread is more affected by the outlier, and why?'

Discussion Prompt

Pose the question: 'Imagine you are comparing the test scores of two Year 9 classes. Class A has a range of 40 and an IQR of 15. Class B has a range of 30 and an IQR of 20. Which class is more consistent in its scores, and how do you know?'

Exit Ticket

Give students a data set and ask them to calculate the range and IQR. On the back, have them write one sentence explaining what the IQR tells them about the spread of the middle half of the data.

Frequently Asked Questions

What are the five points needed for a box plot?

You need the Minimum (lowest value), the Lower Quartile (Q1), the Median (Q2), the Upper Quartile (Q3), and the Maximum (highest value). These five points divide your data into four equal-sized groups.

What is the Interquartile Range (IQR) and why does it matter?

The IQR is the 'length of the box' (Q3 minus Q1). It tells you how spread out the middle 50% of the data is. A small IQR means the data is very consistent; a large IQR means it's very varied. It's often more useful than the full range because it ignores outliers.

When is a box plot better than a bar graph?

Box plots are much better when you want to compare the 'spread' and 'skew' of two or more different groups at once. They allow you to see at a glance which group has a higher median and which group is more consistent, without getting bogged down in individual data points.

How can active learning help students understand box plots?

Active learning, like the 'Human Box Plot', is the most effective way to teach quartiles. When students see that the 'box' can be wide or narrow even though it contains the same number of classmates, the concept of 'density vs. spread' becomes intuitive. Collaborative tasks like the 'Stats Duel' also give them practice in 'reading' the data to make a persuasive argument, which is the real-world purpose of statistics.