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📖 Lesson 8 ⏱ ~30 min Year 10 · Unit 4 ⚡ +50 XP

Measures of Spread — Range and IQR

Calculate the range and interquartile range to describe the spread of a data set.

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0/5QUESTS

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Worksheet

Worksheet

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Warm-up
Think First
+5 XP each

Q1 · Two classes both have an average test score of 75%. One class has scores from 70–80, the other from 40–100. What does that tell you?

Q2 · Why might we care more about where the "middle 50%" of data sits than just the gap between the highest and lowest values?

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From the lesson
Intentions

Learning Intentions

Know

  • Range = maximum − minimum. IQR = Q3 − Q1, where Q1 is the median of the lower half and Q3 is the median of the upper half.

Understand

  • Why the IQR is a better measure of spread than the range when outliers are present.

Can Do

  • Calculate range and IQR from ordered data and identify outliers using the 1.5 × IQR rule.
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From the lesson
Key Terms

Key Terms

Range — The difference between the maximum and minimum values in a data set.
Quartile — Values that divide ordered data into four equal parts: Q1, Q2 (median), Q3.
Interquartile range (IQR) — Q3 − Q1; the spread of the middle 50% of data.
Outlier rule — Values below Q1 − 1.5×IQR or above Q3 + 1.5×IQR are considered outliers.
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From the lesson
Misconceptions

Misconceptions to Fix

✗

Wrong: The range is not affected by outliers because it only uses two values.

✓

Right: The range IS heavily affected by outliers — it uses the MAXIMUM and MINIMUM, so one extreme value changes the range completely. This is exactly why IQR is preferred for spread.

✗

Wrong: Q1 is the lowest quarter of values.

✓

Right: Q1 is a single number (the value below which 25% of data falls), not a set of values. For {2,4,6,8,10}, Q1=4, not {2,4}.

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From the lesson
Content

Measures of Spread — Range and IQR

Work through the content, activities and worked examples below. Test your understanding with the questions in the Questions phase.

Remember Five-number summary: minimum, Q1, median, Q3, maximum. Box plot: box spans Q1 to Q3, line inside is median, whiskers extend to min and max (or outliers).
HSC Note When sketching a box plot, always label the scale on the horizontal axis. Examiners deduct marks for unlabelled axes.
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From the lesson
Activity
✏ Activity 1 — Sketch a Box Plot

For each five-number summary, sketch a box plot on a number line:

  1. Min 10, Q1 20, Median 30, Q3 40, Max 50
  2. Min 5, Q1 15, Median 25, Q3 35, Max 60
  3. Min 100, Q1 120, Median 150, Q3 180, Max 220 (outlier at 250)
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From the lesson
Worked Example

Worked Example

Step-by-step
Draw a box plot for the following data: 12, 15, 18, 21, 24, 27, 30, 33, 36.
  1. 1
    Step 1: Find the five-number summary. Min = 12, Max = 36, Median = 24, Q1 = 16.5, Q3 = 31.5.
  2. 2
    Step 2: Draw a horizontal scale from 10 to 40.
  3. 3
    Step 3: Draw a box from Q1 (16.5) to Q3 (31.5).
  4. 4
    Step 4: Draw a line inside the box at the median (24). Draw whiskers from the box to min (12) and max (36).
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From the lesson
Revisit

Revisit Your Thinking

Look back at your Think First response. What new understanding do you have now?

Reflect
Revisit your thinking
reflect

Earlier you were asked: What was your first thought on this topic?

Now that you've worked through the lesson, write a fuller answer. What changed in your thinking?

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From the lesson
Multiple Choice

Multiple Choice

Select the best answer for each question.

1 mark For the ordered data 2, 4, 6, 8, 10, 12, 14, the median (Q2) is:

1 mark For 2, 4, 6, 8, 10, 12, 14, Q1 is:

1 mark For 2, 4, 6, 8, 10, 12, 14, the IQR is:

1 mark The range of 5, 10, 15, 20, 100 is:

1 mark A value is considered an outlier if it is:

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From the lesson
Short Answer

Short Answer

Show all working and justify your answers.

1. 4 marks For the data set 8, 12, 15, 18, 20, 22, 25, 28, 30, 35:
(a) Find the five-number summary.
(b) Draw a box plot.
(c) Identify any outliers using the 1.5 × IQR rule.

2. 3 marks Two classes sat the same test. Class A: median 72, IQR 12. Class B: median 68, IQR 20. Compare the performance of the two classes.

3. 2 marks Explain why a box plot is useful for comparing two or more data sets.

Marking guidance: 1 mark each for MCQs. See mark allocations for each short answer question.

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