Mathematics Standard • Year 12 • Module 8 • Lesson 3
Measures of Spread — Problem Set
Apply range, IQR and standard deviation to realistic Australian contexts — comparing classes, factories, batsmen, suburbs and quality control.
Problem 1 — Two classes, same mean (compare spread)
Two Year 12 Maths Standard classes both score a mean of 75 on the same test. Class A has SD = 4; Class B has SD = 14.
Set up: What are we solving for?
(i) Describe in 1-2 sentences what the difference in SD tells you about the two classes. 2 marks
(ii) The teacher needs to plan revision. Which class is likely to benefit more from differentiated (multi-level) work and why? 2 marks
(iii) A parent says "the two classes are equal because the means are equal". Briefly evaluate this statement. 2 marks
Stuck? Revisit lesson § The Average Distance from the Mean.Problem 2 — Quality control (factory bolt diameters)
A factory makes bolts. The target diameter is 12.00 mm. Quality control measures 10 bolts each shift. This week, the SD across all measured bolts was 0.15 mm.
Set up: What are we solving for?
(i) The customer specification requires SD ≤ 0.20 mm. Does this batch meet specification? 1 mark
(ii) Next week, one new operator runs the machine and the SD jumps to 0.45 mm (mean still 12.00 mm). Explain in one sentence what this change means for the customer. 2 marks
(iii) Why is SD a better quality-control measure than range here, given that one or two faulty bolts per shift is normal? 2 marks
Stuck? Think about what "SD doubled" looks like in terms of bolts within ±0.40 mm of the target.Problem 3 — Comparing batting consistency
Two cricketers play 11 innings each. Their scores are:
Batter X: 22, 28, 32, 35, 38, 42, 45, 48, 52, 55, 58
Batter Y: 0, 5, 12, 18, 35, 42, 55, 62, 70, 82, 138
Set up: What are we solving for?
(i) Calculate the mean for each batter. 1 mark
(ii) Use a calculator to find σ for each batter (1 d.p.). State the range for each. 2 marks
(iii) The selectors want a reliable middle-order batter. Which one should they pick and why? Reference the SD in your answer. 2 marks
Stuck? Same mean ≠ same consistency. The lower SD means lower variability around the mean.Problem 4 — Suburb commute times (IQR vs range)
An online survey of 11 residents asks their daily commute (one-way, minutes): 18, 22, 25, 28, 30, 32, 35, 38, 42, 45, 110 (one person works very far from home).
Set up: What are we solving for?
(i) Calculate the range and the IQR. 2 marks
(ii) The council wants to advertise "average commute time" plus a measure of spread. Which measure of spread should they use (range, IQR or SD)? Justify in one sentence. 2 marks
(iii) If the 110-min commuter moves house and now commutes 35 min, recalculate range and IQR. Which one changed more, and what does this show about robustness? 2 marks
Stuck? Remove the 110, slot 35 in the ordered list, and recompute Q1/Q3.Problem 5 — Two suppliers (mean and SD both matter)
A bakery is choosing between two flour suppliers. Each delivers 1 kg bags. The bakery weighs 30 bags from each supplier and finds:
Supplier P: mean = 1.003 kg, σ = 0.008 kg.
Supplier Q: mean = 1.012 kg, σ = 0.030 kg.
Set up: What are we solving for?
(i) Which supplier delivers bags closer on average to 1.000 kg? Which is more consistent? 2 marks
(ii) The bakery values consistency above being "generous". State which supplier they should pick and explain in 1-2 sentences, referring to both mean and σ. 2 marks
(iii) Explain why looking only at the mean would have led to a worse decision. 2 marks
Stuck? Picture two histograms: same shape, one narrower.How did this worksheet feel?
What I'll revisit before next class:
Problem 1 — Same mean, different SD
(i) Class A is much more tightly clustered around 75 (most students within a few marks). Class B is much more spread out — students range from very low to very high.
(ii) Class B benefits more from differentiated work, because the large SD means students are at very different levels (some need extension, others need fundamentals); a single revision lesson would miss most students.
(iii) The parent is wrong. Equal means do not imply equal classes — Class B has the same average only because very high and very low marks cancel out. Spread tells a very different story to centre.
Problem 2 — Quality control
(i) SD 0.15 mm < 0.20 mm spec, so yes — the batch meets specification.
(ii) SD tripled even though the mean is still on target. Bolts now vary much more around 12.00 mm, so far more bolts are too thin or too thick — many will fail the customer's tolerance check.
(iii) Range depends only on the worst two bolts of the shift and would jump every time a single defective bolt appeared. SD averages the deviation of all bolts from the mean, so it gives a stable picture of overall process variability and is less misled by one bad bolt.
Problem 3 — Batting consistency
(i) X: Σ = 455, mean = 455/11 ≈ 41.4. Y: Σ = 519, mean = 519/11 ≈ 47.2.
(ii) X: range = 58 − 22 = 36; σ ≈ 11.4. Y: range = 138 − 0 = 138; σ ≈ 39.6.
(iii) Pick Batter X. Although Y has a slightly higher mean, Y's σ is over three times X's — Y's scores swing from 0 to 138 — so X is far more reliable as a middle-order option, contributing 22-58 runs most innings rather than the all-or-nothing pattern of Y.
Problem 4 — Commute times (IQR vs range)
(i) Range = 110 − 18 = 92 min. n = 11, median = 6th value = 32. Lower 5: 18,22,25,28,30 → Q1 = 25. Upper 5: 35,38,42,45,110 → Q3 = 42. IQR = 42 − 25 = 17 min.
(ii) Use the IQR — it is not distorted by the single 110-min outlier and gives a fair sense of the middle 50% of residents' commutes.
(iii) New data (ordered): 18,22,25,28,30,32,35,35,38,42,45. Range = 45 − 18 = 27 min (was 92 — dropped by 65). New Q1 = 25, Q3 = 38, IQR = 13 min (was 17 — dropped by 4). The range changed massively while IQR barely moved — confirming IQR is the more robust measure.
Problem 5 — Flour suppliers
(i) Closer to 1.000 kg on average: Supplier P (mean 1.003 vs 1.012). More consistent: Supplier P (σ 0.008 vs 0.030).
(ii) Choose Supplier P. P's mean is essentially on the 1 kg target (only 3 g over) and its σ is nearly 4× smaller, meaning almost every bag is within a few grams of target — exactly what a bakery wants for predictable recipes.
(iii) Looking at mean alone, Supplier Q seems more "generous" (extra 12 g per bag on average). But the high σ means many of Q's bags are well above or below 1 kg — including some clearly under-weight bags, which the bakery would have to throw out or supplement, costing more in waste than the apparent extra flour is worth.