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

Measures of Centre โ€” Median and Mode

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

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Core learning
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Worksheet

Worksheet

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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 โ€” like shoe sizes or phone brands โ€” 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

Median โ€” The middle value when data is arranged in order; half the values are above and half below.
Mode โ€” The value that occurs most frequently in a data set.
Bimodal โ€” A data set with two modes.
Outlier-resistant โ€” A 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 โ€” 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 โ€” 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 โ€” 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 โ€” 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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