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📖 Lesson 6 ⏱ ~30 min Year 10 · Unit 4 ⚡ +50 XP

Measures of Centre, Mean

Calculate and interpret the mean as a measure of central tendency, including from frequency tables.

Today's hook: A billionaire walks into a café and now everyone inside is a millionaire 'on average'. Why can the mean lie?
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Core learning
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From the lesson
Worksheet

Worksheet

Use the worksheet to complete this lesson in your book or digitally.

Warm-up
Think First
+5 XP each

Q1 · If five friends spent $12, $15, $10, $20 and $18 on lunch, how would you find what a "typical" person spent?

Q2 · What happens to the average if one person spent $100 while everyone else spent around $15? Is the average still a good description?

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From the lesson
Intentions

Learning Intentions

Know

  • The mean is the sum of all values divided by the number of values. For frequency tables, mean = sum of (value × frequency) / total frequency.

Understand

  • Why the mean is affected by outliers and may not represent a typical value in skewed distributions.

Can Do

  • Calculate the mean from raw data and from frequency tables.
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From the lesson
Key Terms

Key Terms

MeanThe arithmetic average: sum of all values divided by the number of values.
Frequency table meanSum of (each value × its frequency) divided by total frequency.
Outlier effectExtreme values pull the mean in their direction, making it unrepresentative.
SumThe total obtained by adding all values together.
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From the lesson
Misconceptions

Misconceptions to Fix

✗

Wrong: Adding a constant to every value does not change the mean.

✓

Right: Adding a constant k to every value increases the mean by exactly k. If every score rises by 5, the mean rises by 5.

✗

Wrong: The mean from a frequency table is calculated by dividing the sum of values by the number of rows.

✓

Right: Divide the sum of (value × frequency) by the TOTAL frequency (n), not by the number of distinct values or rows.

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From the lesson
Content

Measures of Centre, Mean

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

Remember Range = maximum − minimum. IQR = Q3 − Q1. The IQR is more robust because it ignores outliers.
HSC Note When comparing two data sets, always report both a measure of centre (mean or median) AND a measure of spread (range or IQR).
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From the lesson
Activity
✏ Activity 1, Calculate Range and IQR

For each data set, find the range and interquartile range:

  1. 4, 8, 12, 16, 20, 24, 28
  2. 5, 10, 15, 20, 25, 30, 100
  3. 2, 4, 6, 8, 10, 12, 14, 16, 18
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From the lesson
Worked Example

Worked Example

Step-by-step
Find the range and interquartile range of: 8, 12, 15, 18, 21, 24, 28, 32, 36.
  1. 1
    Step 1: Order the data (already ordered). n = 9.
  2. 2
    Step 2: Range = 36 − 8 = 28.
  3. 3
    Step 3: Q1 position = (9+1)/4 = 2.5. Q1 = average of 2nd and 3rd values = (12 + 15)/2 = 13.5.
  4. 4
    Step 4: Q3 position = 3(9+1)/4 = 7.5. Q3 = average of 7th and 8th values = (28 + 32)/2 = 30. IQR = 30 − 13.5 = 16.5.
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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 The mean of 5, 7, 9, 11, 13 is:

1 mark For the data set 2, 3, 3, 5, 7, 100, the mean is:

1 mark From a frequency table, the mean is calculated by:

1 mark If every value in a data set is increased by 5, the mean:

1 mark The mean of 4, 8, x, 12 is 8. The value of x is:

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From the lesson
Short Answer

Short Answer

Show all working and justify your answers.

1. 4 marks The daily maximum temperatures (°C) for two weeks are: 22, 24, 25, 23, 26, 28, 30, 29, 27, 25, 24, 23, 22, 21.
(a) Find the range.
(b) Find the interquartile range.
(c) On one day, the temperature was 38°C instead of 28°C. Explain how this would affect the range and the IQR.

2. 3 marks Data set A has range 20 and IQR 8. Data set B has range 20 and IQR 15. Compare the spread of the two data sets.

3. 2 marks Explain why the IQR is preferred over the range when comparing data sets that may contain outliers.

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

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