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๐Ÿ“– Lesson 7 โฑ ~30 min Year 10 ยท Unit 4 โšก +50 XP

Measures of Centre, Median and Mode

Find and interpret the median and mode, and choose the most appropriate measure of centre for different data sets.

Today's hook: When one huge value distorts the average, which 'middle' should you actually trust?
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Core learning
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Worksheet

Worksheet

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

Warm-up
Think First
+5 XP each

Q1 ยท Why might the "middle" score tell a different story than the average in a list of house prices that includes one mansion?

Q2 ยท Can you think of a real situation, like shoe sizes or phone brands, where the most common value is more useful than the average?

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From the lesson
Intentions

Learning Intentions

Know

  • The median is the middle value when data is ordered. The mode is the most frequently occurring value.

Understand

  • When each measure of centre is most appropriate: mean for symmetric data, median for skewed data or outliers, mode for categorical data.

Can Do

  • Calculate median and mode from raw data and frequency tables, and select the best measure for a given context.
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From the lesson
Key Terms

Key Terms

MedianThe middle value when data is arranged in order; half the values are above and half below.
ModeThe value that occurs most frequently in a data set.
BimodalA data set with two modes.
Outlier-resistantA statistic that is not heavily influenced by extreme values.
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From the lesson
Misconceptions

Misconceptions to Fix

โœ—

Wrong: The median is always the middle number you can see in the ordered list.

โœ“

Right: For even n, the median is the AVERAGE of the two middle values, it may not be in the data set. For {1, 3, 5, 7} the median is (3+5)/2 = 4.

โœ—

Wrong: Every data set has exactly one mode.

โœ“

Right: A data set can be bimodal, multimodal, or have no mode at all if all values occur equally often.

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From the lesson
Content

Measures of Centre, Median and Mode

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

Remember Standard deviation formula: s = โˆš[ฮฃ(x โˆ’ xฬ„)ยฒ / (n โˆ’ 1)]. Most calculators have a standard deviation button, learn how to use it.
Exam Tip In exams, you are usually given the standard deviation or asked to calculate it using a calculator. Focus on interpreting the value, not manual calculation.
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From the lesson
Activity
โœ Activity 1, Interpret Standard Deviation

For each pair of data sets, determine which has the greater standard deviation (without calculating):

  1. Set A: 5, 5, 5, 5, 5 vs Set B: 1, 3, 5, 7, 9
  2. Set A: 10, 12, 14, 16, 18 vs Set B: 10, 11, 12, 13, 14
  3. Set A: 2, 4, 6, 8, 10 vs Set B: 2, 5, 6, 7, 10
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From the lesson
Worked Example

Worked Example

Step-by-step
The test scores of 5 students are: 65, 72, 78, 85, 90. Calculate the standard deviation.
  1. 1
    Step 1: Find the mean. (65 + 72 + 78 + 85 + 90) / 5 = 390 / 5 = 78.
  2. 2
    Step 2: Find each deviation from the mean: โˆ’13, โˆ’6, 0, 7, 12.
  3. 3
    Step 3: Square each deviation: 169, 36, 0, 49, 144.
  4. 4
    Step 4: Sum of squared deviations = 398. Divide by (nโˆ’1) = 4. Variance = 398/4 = 99.5.
  5. 5
    Step 5: Standard deviation = โˆš99.5 โ‰ˆ 9.97 (2 d.p.).
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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 median of 3, 5, 7, 9, 11 is:

1 mark The median of 2, 4, 6, 8 is:

1 mark The mode of 2, 3, 3, 5, 5, 5, 7 is:

1 mark For a data set with an extreme outlier, the best measure of centre is:

1 mark For categorical data such as favourite colours, the only appropriate measure of centre is:

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From the lesson
Short Answer

Short Answer

Show all working and justify your answers.

1. 4 marks The heights (in cm) of 6 students are: 155, 160, 165, 170, 175, 180.
(a) Calculate the mean.
(b) Calculate the standard deviation using the formula s = โˆš[ฮฃ(x โˆ’ xฬ„)ยฒ / (n โˆ’ 1)]. Show all working.

2. 3 marks Data set A has mean 50 and standard deviation 5. Data set B has mean 50 and standard deviation 15. Describe what this tells you about the two data sets.

3. 2 marks Explain why it is impossible for a data set to have a negative standard deviation.

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

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