Mathematics • Year 9 • Unit 4 • Lesson 12

Box Plots in the Real World

Use box plots, histograms and stem-and-leaf displays to interpret real situations — race times, hospital wait times, music streaming, exam results and rainfall — then explain a comparison in your own words.

Apply · Real-World Maths

1. Word problems

Each problem uses ideas from Lesson 12 — five-number summary, box plot shape, histograms or stem-and-leaf. Show your working — a single final answer with no working only earns half marks.

1.1 — 100 m race times. A Year 9 PE class records 100 m sprint times (seconds):

14.2, 14.8, 15.0, 15.2, 15.5, 15.8, 16.0, 16.4, 17.0.

(a) Write the five-number summary.
(b) Describe the shape of the box plot (symmetric / left-skewed / right-skewed). 4 marks

Stuck? With n = 9 the median is the 5th value. Then split the four either side into two halves.

1.2 — Hospital wait times. The Australian Institute of Health and Welfare reports waiting times (in minutes) at two emergency departments using box plots:

ED-North: Min 10, Q1 25, Med 40, Q3 65, Max 120.
ED-South: Min 15, Q1 30, Med 45, Q3 55, Max 80.

(a) Which ED has the higher typical wait?
(b) Which ED is more consistent? Justify with IQRs. 4 marks

Stuck? Revisit lesson § "Real-World Anchor — Medicine and Public Health". "Typical" usually means median; "consistent" usually means small IQR.

1.3 — Music streaming. Eight friends record the number of songs streamed in a week:

40, 55, 60, 70, 80, 90, 100, 250.

(a) Use the 1.5 × IQR rule to test whether 250 is an outlier.
(b) If 250 is an outlier, where should the right whisker of the box plot end? 4 marks

Stuck? With n = 8, median = average of 4th and 5th values; then take the median of each half of 4 values.

1.4 — Exam scores stem-and-leaf. A teacher records Year 9 Maths test scores out of 100:

52, 58, 61, 65, 65, 70, 72, 78, 81, 85, 88, 91.

(a) Draw a stem-and-leaf plot using stems 5, 6, 7, 8, 9.
(b) State the median and the mode from your plot. 4 marks

Stuck? The leaf is the ones digit. The mode is the value that appears most often — count the leaves.

1.5 — Rainfall histogram. A weather station groups monthly rainfall (in mm) over a year as:

0-49: 2 months, 50-99: 4 months, 100-149: 3 months, 150-199: 2 months, 200-249: 1 month.

(a) How many months were there of data in total? Confirm the answer makes sense.
(b) Which interval is the modal class? Describe the shape of the histogram (symmetric / right-skewed / left-skewed). 4 marks

Stuck? Modal class = the bar with the highest frequency. Skew "to the right" means the tail goes to higher values.

2. Explain your thinking

This question is about communication, not just answers. Use full sentences. 4 marks

2.1 A classmate looks at two side-by-side box plots and says "Class A is better because its box is bigger." In your own words, explain (i) what a "bigger box" actually represents, (ii) why bigger does not mean better, and (iii) what feature of a box plot you should look at to compare the typical performance of two groups. Refer to the IQR and the median somewhere in your answer.

Stuck? Revisit lesson § "Misconceptions" — box plot height carries no meaning, but box width tells you about spread, not score.

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Answers — Do not peek before attempting

1.1 — Sprint times

n = 9, Min = 14.2, Max = 17.0. Median = 5th value = 15.5. Lower half: 14.2, 14.8, 15.0, 15.2 → Q1 = (14.8 + 15.0) ÷ 2 = 14.9. Upper half: 15.8, 16.0, 16.4, 17.0 → Q3 = (16.0 + 16.4) ÷ 2 = 16.2.
(a) Summary: 14.2, 14.9, 15.5, 16.2, 17.0.
(b) Median sits roughly in the middle of the box (Q3 − median = 0.7, median − Q1 = 0.6) and whiskers are similar lengths → roughly symmetric.

1.2 — Hospital wait times

(a) ED-South median = 45 min > ED-North median = 40 min, so ED-South has the higher typical wait (only just).
(b) ED-North IQR = 65 − 25 = 40 min. ED-South IQR = 55 − 30 = 25 min. ED-South is more consistent — its middle 50% of waits fall in a much narrower band.

1.3 — Music streaming

n = 8, median = (70 + 80) ÷ 2 = 75. Lower half: 40, 55, 60, 70 → Q1 = (55 + 60) ÷ 2 = 57.5. Upper half: 80, 90, 100, 250 → Q3 = (90 + 100) ÷ 2 = 95. IQR = 95 − 57.5 = 37.5.
(a) Upper fence = 95 + 1.5(37.5) = 95 + 56.25 = 151.25. 250 > 151.25, so yes, 250 is an outlier.
(b) Right whisker ends at the largest value still inside the fence, which is 100. 250 is plotted as a separate dot.

1.4 — Exam scores stem-and-leaf

Stem | Leaf
  5  | 2 8
  6  | 1 5 5
  7  | 0 2 8
  8  | 1 5 8
  9  | 1

n = 12, median = (6th + 7th) ÷ 2 = (70 + 72) ÷ 2 = 71. Mode = 65 (appears twice; the only repeat).

1.5 — Rainfall histogram

(a) Total = 2 + 4 + 3 + 2 + 1 = 12 months (one year) — confirms the data is consistent.
(b) Modal class is 50-99 mm (frequency 4). The bars decrease in height as rainfall increases beyond 50-99, so the histogram is right-skewed (tail extends to high rainfall).

2.1 — Explain your thinking (sample response)

A "bigger box" on a box plot means the IQR (the spread of the middle 50% of values) is larger — not that the scores are higher. A class with a wide box has more variable results: some students well above and some well below the typical mark. To compare typical performance, you should look at the position of the median line inside each box. For example, two classes might both have medians of 70, but the class with the narrower box is more consistent — most students cluster around 70 — while the wider box hides students who scored much higher and much lower than 70.

Marking: 1 mark for "box = IQR / middle 50%"; 1 mark for "bigger box = more spread, not higher scores"; 1 mark for "compare medians" to judge typical performance; 1 mark for a clear sentence-based explanation.