Mathematics • Year 10 • Unit 4 • Lesson 8
Range and IQR in the Real World
Apply Lesson 8's measures of spread to real Year 10 contexts: NSW HSC test results, Sydney commute times, weekly grocery bills, and a school sports carnival. Practise the rule of always reporting BOTH centre and spread when comparing two data sets.
1. Word problems
Show full quartile working. Bare answers earn half marks.
1.1 — Sydney commute times. A student records her morning bus commute times (minutes) over 10 school days: 24, 26, 27, 25, 28, 30, 26, 27, 29, 55. The 55-minute day was a bus breakdown.
(a) Find the range of all 10 days.
(b) Find the IQR of all 10 days.
(c) Apply the 1.5×IQR rule to verify whether 55 is an outlier. 3 marks
1.2 — HSC test marks. A class of 11 students sat a Maths Advanced trial. Marks were: 38, 52, 55, 60, 63, 68, 72, 75, 78, 82, 89.
(a) Find the median, Q1, Q3 and IQR.
(b) The school reports the "middle 50%" of marks. What range of marks does that cover, and what does the IQR value tell a student about how spread out the middle group is? 3 marks
1.3 — Two PE classes comparison. Two Year 10 PE classes did the beep test. Class A's level scores were 6, 7, 7, 8, 8, 8, 9, 9, 10, 11. Class B's were 4, 5, 7, 8, 8, 9, 10, 11, 12, 13.
(a) Find the range and IQR of each class.
(b) Which class has more consistent fitness levels? Justify using BOTH measures of spread (Lesson 8 HSC Note: always report centre AND spread when comparing). 3 marks
1.4 — Weekly grocery bills. A family records the weekly grocery bill for 8 weeks ($): 215, 198, 240, 225, 210, 230, 219, 380. The $380 week was for a birthday party.
(a) Find the range and IQR.
(b) Apply the 1.5×IQR rule to test whether the $380 week is an outlier.
(c) If the family wants to budget for a "typical" week, would they use range or IQR? Why? 3 marks
1.5 — Sports carnival reaction times. Reaction times (in milliseconds) for 9 sprinters at the school start line were: 165, 172, 180, 175, 168, 188, 170, 195, 178.
(a) Order the data and find the range, Q1, Q3, IQR.
(b) The coach wants to identify the most variable sprinter "block of three". Quote the IQR value and say in one sentence what it means about the middle group of athletes. 3 marks
2. Explain your thinking
This question is about communication. Use full sentences. 4 marks
2.1 A news article says "House prices range from $400 k to $4 m in this suburb, so it's wildly unaffordable." Using Lesson 8's misconception card ("range IS heavily affected by outliers"), write a four-sentence reply that (i) identifies why "range alone" is a misleading summary, (ii) names a better statistic from Lesson 8 to use for spread, (iii) defines that statistic in one sentence using "middle 50%", and (iv) gives a clear rule of thumb a Year 10 student can use to spot when a journalist is using range to exaggerate.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Bus commute times
Ordered: 24, 25, 26, 26, 27, 27, 28, 29, 30, 55. n = 10.
(a) Range = 55 − 24 = 31 min.
(b) Lower half: 24, 25, 26, 26, 27. Upper half: 27, 28, 29, 30, 55. Q1 = 26, Q3 = 29. IQR = 3 min.
(c) Upper fence = 29 + 1.5×3 = 29 + 4.5 = 33.5. Since 55 > 33.5, 55 IS an outlier.
1.2 — HSC trial marks
n = 11. Median (Q2) = 68 (6th value). Lower half (5 values, excluding median): 38, 52, 55, 60, 63. Upper half: 72, 75, 78, 82, 89. Q1 = 55 (3rd of lower half). Q3 = 78. IQR = 23 marks.
(b) The middle 50% covers marks from 55 to 78. The IQR of 23 tells a student that the middle group is moderately spread out — the strongest "middle" student got 23 marks more than the weakest "middle" student.
1.3 — Beep test comparison
Class A (ordered): 6, 7, 7, 8, 8, 8, 9, 9, 10, 11. Range = 5. Q1 = 7, Q3 = 9, IQR = 2.
Class B (ordered): 4, 5, 7, 8, 8, 9, 10, 11, 12, 13. Range = 9. Q1 = 7, Q3 = 11, IQR = 4.
Class A has smaller range AND smaller IQR — Class A is more consistent. Both spread measures agree, satisfying the Lesson 8 HSC note to report centre and spread.
1.4 — Grocery bills
Ordered: 198, 210, 215, 219, 225, 230, 240, 380. n = 8.
(a) Range = 380 − 198 = $182. Q1 = (210 + 215)/2 = 212.5. Q3 = (230 + 240)/2 = 235. IQR = $22.50.
(b) Upper fence = 235 + 1.5×22.5 = 235 + 33.75 = 268.75. 380 > 268.75, so $380 IS an outlier.
(c) For a typical week, use the IQR ($22.50). The range ($182) is blown up by the party week and over-estimates normal week-to-week variation.
1.5 — Sprinters' reaction times
Ordered: 165, 168, 170, 172, 175, 178, 180, 188, 195. n = 9, median = 175 (5th value). Lower half: 165, 168, 170, 172. Upper half: 178, 180, 188, 195. Q1 = (168+170)/2 = 169. Q3 = (180+188)/2 = 184.
Range = 195 − 165 = 30 ms. IQR = 184 − 169 = 15 ms.
(b) The IQR of 15 ms means the middle group of sprinters' reaction times spans 15 ms — moderately consistent (the very fastest and very slowest don't count).
2.1 — Explain your thinking (sample response)
Saying "house prices range from $400 k to $4 m" relies only on the extreme cheapest and most expensive sale — the Lesson 8 misconception card says exactly this: the range IS heavily affected by outliers, so a single mansion can blow it up. A fairer measure of spread is the interquartile range (IQR). The IQR equals Q3 − Q1, the spread of the middle 50% of house prices — it ignores the extreme top and bottom. Rule of thumb: if a news article only mentions range, ask "what does the middle look like?" — request the IQR or the median price.
Marking: 1 mark naming the issue (range), 1 for naming IQR, 1 for definition using "middle 50%", 1 for clear rule of thumb.