Mathematics • Year 10 • Unit 4 • Lesson 9
Box Plots in the Real World
Apply Lesson 9's box plots to real Year 10 contexts: school timetable, NSW NAPLAN/HSC trial results, weather data, supermarket queue times and inter-class comparisons using parallel box plots. Reinforce the key idea — box plots compare centres, spreads and skewness at a glance.
1. Word problems
Show full quartile and summary work, then write your interpretation in sentences.
1.1 — Maths trial marks. A Year 10 Maths class of 12 students scored: 42, 51, 56, 60, 63, 65, 68, 71, 74, 78, 82, 90.
(a) Find the five-number summary.
(b) Sketch the box plot on a labelled scale 40-100.
(c) Comment on the skewness in one sentence (compare whisker lengths and the median's position inside the box). 3 marks
1.2 — Supermarket queue times. The waiting times (minutes) for 11 shoppers at a checkout were: 1, 2, 2, 3, 4, 5, 5, 6, 8, 10, 15.
(a) Find the five-number summary.
(b) Sketch the box plot.
(c) The supermarket manager wants to know: would adding more registers most help the typical shopper (the middle of the box) or the unlucky few (the upper whisker)? Use your plot to argue your answer. 3 marks
1.3 — Parallel box plots: two PE classes. Two PE classes ran 1.5 km laps. The summaries are:
Class A: min = 5.2, Q1 = 6.0, median = 6.5, Q3 = 7.2, max = 8.5 (minutes).
Class B: min = 4.8, Q1 = 5.6, median = 6.2, Q3 = 7.5, max = 10.0 (minutes).
(a) Sketch parallel box plots on a common scale 4-10.
(b) In two sentences, compare the two classes using BOTH centre and spread (Lesson 9 Learning Intentions). 3 marks
1.4 — Sydney max temperature. Maximum daily temperatures over 14 summer days (°C): 25, 26, 27, 28, 28, 29, 30, 31, 31, 32, 33, 35, 36, 41.
(a) Find the five-number summary.
(b) Sketch the box plot.
(c) Is the 41 °C reading an outlier by the 1.5×IQR rule? Justify with the upper-fence calculation. 3 marks
1.5 — Reading three box plots. Three subjects' trial scores (Maths, English, Science) are summarised:
Maths: median 65, Q1 = 55, Q3 = 75.
English: median 70, Q1 = 65, Q3 = 76.
Science: median 65, Q1 = 50, Q3 = 80.
(a) Which subject had the highest "typical" score?
(b) Which subject had the most consistent middle 50%?
(c) Which subject was most spread out in the middle 50%? 3 marks
2. Explain your thinking
This question is about communication. Use full sentences. 4 marks
2.1 A friend says "I can read every student's mark off a box plot." Using Lesson 9's misconception card (a box plot does NOT show individual values), write a four-sentence reply that (i) names what is wrong with the friend's claim, (ii) lists the EXACT five pieces of information a box plot DOES show, (iii) names one display from earlier in the unit that DOES show individual values, and (iv) finishes with one rule of thumb for when to use a box plot vs that other display.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Maths trial marks
(a) n = 12. Min = 42, max = 90. Median = (65 + 68)/2 = 66.5. Lower 6: 42, 51, 56, 60, 63, 65 → Q1 = (56+60)/2 = 58. Upper 6: 68, 71, 74, 78, 82, 90 → Q3 = (74+78)/2 = 76. Summary: 42, 58, 66.5, 76, 90.
(b) Box 58 to 76, median line 66.5, whiskers 42 and 90 on a labelled 40-100 axis.
(c) Lower whisker = 16 (42 to 58); upper whisker = 14 (76 to 90). Median (66.5) is slightly closer to Q1 (58) than to Q3 (76), and the lower whisker is slightly longer — distribution is mildly negatively (left) skewed, but close to symmetric.
1.2 — Supermarket queue times
(a) n = 11, median = 5 (6th value). Lower 5: 1, 2, 2, 3, 4 → Q1 = 2. Upper 5: 5, 6, 8, 10, 15 → Q3 = 8. Summary: 1, 2, 5, 8, 15.
(b) Box from 2 to 8, median line at 5, whiskers to 1 and 15.
(c) The upper whisker (8 to 15) is much longer than the lower (1 to 2). Adding registers would most help the unlucky few — typical shoppers (the box) wait 2-8 minutes, but the worst cases stretch out to 15 minutes. The positively skewed shape signals the long tail is the problem.
1.3 — PE classes (parallel box plots)
(a) Two box plots on the same 4-10 scale.
(b) Centre: Class B is slightly faster (median 6.2 min vs 6.5 min). Spread: Class A is more consistent (IQR = 7.2 − 6.0 = 1.2 min) than Class B (IQR = 7.5 − 5.6 = 1.9 min); Class B's range (5.2 min) is also wider than Class A's (3.3 min). So Class B is on average a bit faster but much more variable.
1.4 — Sydney max temperature
(a) n = 14, median = (29 + 30)/2 = 29.5. Lower 7: 25, 26, 27, 28, 28, 29, 30 → median of 7 is the 4th = 28 → Q1 = 28. Upper 7: 31, 31, 32, 33, 35, 36, 41 → Q3 = 33. Summary: 25, 28, 29.5, 33, 41.
(b) Box 28 to 33, median 29.5, whiskers 25 and 41 on a 24-42 scale.
(c) IQR = 33 − 28 = 5. Upper fence = 33 + 1.5×5 = 33 + 7.5 = 40.5. Since 41 > 40.5, 41 °C IS an outlier.
1.5 — Three subjects
(a) English had the highest median (70).
(b) English had the smallest IQR (76 − 65 = 11), so the most consistent middle 50%.
(c) Science had the largest IQR (80 − 50 = 30), so the most spread-out middle 50%.
2.1 — Explain your thinking (sample response)
The friend is wrong: the Lesson 9 misconceptions card says a box plot does NOT show individual values — it only summarises the data. The five things a box plot shows are: the minimum, Q1, median, Q3 and maximum (the five-number summary). If the friend wants to see every student's mark, they should look at a dot plot or stem-and-leaf plot, which preserves individual values. Rule of thumb: use a box plot when you need to compare overall shape, centre and spread across groups; use a dot plot or stem-and-leaf when you need to see individual data points.
Marking: 1 mark naming the misconception, 1 for the correct five-number summary, 1 for naming a display that does show individual values, 1 for a clear rule of thumb.