Mathematics Standard • Year 12 • Module 8 • Lesson 3
Measures of Spread — Past-Paper Style
Practise HSC Mathematics Standard 2-style writing on range, IQR and standard deviation, including outlier impact and comparative analysis.
1. Short-answer questions
1.1 The scores of 9 students on a quiz are: 5, 6, 7, 8, 8, 9, 9, 10, 10.
(a) Find the range.
(b) Find Q1, Q3 and IQR.
(c) Use a calculator to find σ (1 d.p.). 3 marks Band 3
1.2 Two factories making the same product report mean weight 500 g but different standard deviations: Factory A σ = 3 g; Factory B σ = 9 g.
(a) Which factory's products are more consistent?
(b) Explain in one sentence why a buyer might care about this difference, even though the means are identical. 3 marks Band 3-4
1.3 A data set is 4, 6, 8, 10, 12, 14, 16, 50.
(a) Find the range and the IQR.
(b) Explain in one sentence which measure of spread is more affected by the value 50 and why. 4 marks Band 4
2. Extended response
2.1 A regional council compares the daily peak temperatures (°C) for two coastal towns during the same week.
Town P (sheltered bay): 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27
Town Q (open coast): 18, 19, 22, 23, 25, 25, 27, 28, 30, 32, 35
(a) Calculate the mean for each town (1 d.p.).
(b) Calculate the range and σ for each town (σ to 1 d.p.).
(c) In 2-3 sentences, compare the two towns' temperature distributions using both centre and spread, and recommend which town would be more comfortable for a heat-sensitive elderly resident. 7 marks Band 5-6
Explicit marking criteria
Part (a) — 2 marks
• 1 mark each — correct mean for P and Q.
Part (b) — 3 marks
• 1 mark — correct range for both towns.
• 1 mark — correct σ for Town P.
• 1 mark — correct σ for Town Q.
Part (c) — 2 marks
• 1 mark — compares both centre (means) and spread (σ or range), with numerical reference.
• 1 mark — defensible recommendation linked to spread (P is more stable, fewer extreme hot days).
Your response:
Stuck? Use the calculator's 1-Var Stats / Stats mode to get σ in seconds.How did this worksheet feel?
What I'll revisit before next class:
1.1 — Quiz scores (3 marks)
Sample response. (a) Range = 10 − 5 = 5. (b) n = 9, median = 5th value = 8. Lower 4: 5,6,7,8 → Q1 = 6.5. Upper 4: 9,9,10,10 → Q3 = 9.5. IQR = 9.5 − 6.5 = 3. (c) σ ≈ 1.6 (calculator).
Marking notes. 1 mark each for (a), (b) and (c). A response that quotes the sample s ≈ 1.7 instead of population σ ≈ 1.6 is normally accepted at this level provided the symbol used is consistent with the working.
1.2 — Two factories (3 marks)
(a) Sample response. Factory A (σ = 3 g) is more consistent — its products are more tightly clustered around 500 g than Factory B (σ = 9 g).
(b) Sample response. Even with identical means, Factory B's larger σ means more individual products are noticeably under- or over-weight, which can lead to consumer complaints or product rejected by quality control even when the average is "fine".
Marking notes. 1 mark for (a). 2 marks for (b): one for connecting σ to per-unit consistency, one for naming a concrete consequence (quality control / consumer complaints / shrinkage cost).
1.3 — Range vs IQR with outlier (4 marks)
(a) Sample response. Range = 50 − 4 = 46. n = 8, lower 4: 4, 6, 8, 10 → Q1 = 7. Upper 4: 12, 14, 16, 50 → Q3 = 15. IQR = 15 − 7 = 8.
(b) Sample response. The range is much more affected because it depends directly on the maximum value (50). The IQR only depends on Q1 and Q3 — it ignores the top 25% of the data, so the outlier 50 has no impact on it.
Marking notes. 1 mark for the range. 1 mark for the IQR. 1 mark for naming range as more affected. 1 mark for the reasoning (IQR ignores tails / range uses max).
2.1 — Two towns' temperatures (7 marks): Band-6 sample with annotations
(a) Means.
Town P: Σ = 272, mean = 272/11 = 24.7°C. [1 mark.]
Town Q: Σ = 284, mean = 284/11 = 25.8°C. [1 mark.]
(b) Range and σ.
Town P: range = 27 − 22 = 5°C; σ ≈ 1.6°C (calculator). [1 mark for both ranges. 1 mark for σP.]
Town Q: range = 35 − 18 = 17°C; σ ≈ 5.1°C. [1 mark for σQ.]
(c) Comparison and recommendation.
The two towns have similar averages (24.7°C vs 25.8°C — only about 1°C difference), but Town Q's spread is far larger (range 17 vs 5; σ 5.1 vs 1.6), meaning Q has both cool 18°C days and hot 35°C days while P stays within 22-27°C all week. [1 mark — both centre and spread compared with numbers.]
Recommendation: a heat-sensitive elderly resident should choose Town P. Even though the average temperatures are close, Town Q's much larger spread means more frequent extreme-heat days (e.g. 30°C, 32°C, 35°C), which are the days most dangerous to a heat-sensitive resident. Town P's tight spread means temperature stays moderate even on its hottest days. [1 mark — recommendation tied to spread, not just mean.]
Total: 7/7.
Band descriptors for marker.
Band 3: Correct means and at least one range; σ values missing or wrong; (c) is one-liner. ≈ 2-3 marks.
Band 4: All numerical answers correct; (c) compares only means and ignores spread. ≈ 4-5 marks.
Band 5: All numerical correct and (c) mentions both centre and spread, but recommendation is vague ("it depends"). ≈ 6 marks.
Band 6: Full numerical solution AND a defensible recommendation that explicitly links the choice to spread (extreme-heat days), with numerical references throughout. 7/7.