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

Standard Deviation

Understand standard deviation as a measure of spread and calculate it for small data sets.

Today's hook:
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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 ยท What does it mean for data to be "spread out"? How might you measure that spread with a single number?

Q2 ยท If everyone in a class scored exactly the same mark, what would the "spread" be? What if scores were all over the place?

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From the lesson
Intentions

Learning Intentions

Know

  • Standard deviation measures how far data values typically are from the mean. A small SD means data is clustered; a large SD means data is spread out.

Understand

  • Why standard deviation uses squared differences to ensure all deviations contribute positively to the measure of spread.

Can Do

  • Calculate standard deviation using the formula and interpret it in context.
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From the lesson
Key Terms

Key Terms

Standard deviation โ€” A measure of the average distance of data values from the mean.
Variance โ€” The square of the standard deviation; the average of squared deviations from the mean.
Deviation โ€” The difference between a data value and the mean: x โˆ’ xฬ„.
Population SD โ€” ฯƒ = โˆš(ฮฃ(xโˆ’ฮผ)ยฒ/n) for an entire population.
Sample SD โ€” s = โˆš(ฮฃ(xโˆ’xฬ„)ยฒ/(nโˆ’1)) for a sample from a larger population.
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From the lesson
Misconceptions

Misconceptions to Fix

โœ—

Wrong: If two data sets have the same mean, they are identical.

โœ“

Right: Two data sets can have the same mean but very different spreads. Always compare both centre and spread.

โœ—

Wrong: The standard deviation tells you the centre of the data.

โœ“

Right: Standard deviation measures spread, not centre. The mean or median tells you the centre.

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From the lesson
Content

Standard Deviation

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

Remember When comparing data sets, report: (1) measure of centre (mean or median), (2) measure of spread (range, IQR or standard deviation), and (3) shape (symmetric or skewed).
HSC Note In exam questions, you may be asked to compare two data sets using only a box plot or summary statistics. Always refer to specific values in your comparison.
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From the lesson
Activity
โœ Activity 1 โ€” Compare Data Sets

Compare the following pairs of data sets:

  1. Set A: mean 50, SD 5. Set B: mean 50, SD 15.
  2. Set A: median 60, IQR 10. Set B: median 55, IQR 20.
  3. Set A: symmetric. Set B: positively skewed.
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From the lesson
Worked Example

Worked Example

Step-by-step
Class A has test scores with mean 72 and standard deviation 8. Class B has test scores with mean 68 and standard deviation 12. Compare the two classes.
  1. 1
    Centre: Class A has a higher mean (72 vs 68), so on average Class A performed better.
  2. 2
    Spread: Class A has a smaller standard deviation (8 vs 12), so Class A's scores are more consistent (less spread out).
  3. 3
    Conclusion: Class A performed better on average and had more consistent results. Class B had more variability in scores.
  4. 4
    Context: A student in Class B might score much higher or much lower than the mean, whereas Class A students tend to cluster around 72.
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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 A small standard deviation means:

1 mark The standard deviation can never be:

1 mark If all values in a data set are identical, the standard deviation is:

1 mark Standard deviation is preferred over range because it:

1 mark For the data 2, 4, 6, 8, 10 (mean = 6), the sum of squared deviations is:

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From the lesson
Short Answer

Short Answer

Show all working and justify your answers.

1. 3 marks Team A and Team B both have a mean height of 180 cm. Team A has a standard deviation of 4 cm, while Team B has a standard deviation of 12 cm. Compare the two teams.

2. 3 marks A box plot shows Data Set X with median 50 and IQR 10. Another box plot shows Data Set Y with median 50 and IQR 25. What does this tell you about the two data sets?

3. 2 marks Explain why it is important to consider both centre and spread when comparing data sets.

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

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