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 = average of 7th and 8th values = (30 + 31)/2 = 30.5. 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 → median of 7 is the 4th = 33 → Q3 = 33. Summary: 25, 28, 30.5, 33, 41.
(b) Box 28 to 33, median line at 30.5, whiskers to 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.