Mathematics Standard • Year 12 • Module 8 • Lesson 5
Box Plots and Outliers — Problem Set
Apply five-number summaries, the 1.5 × IQR outlier rule and box plots to realistic Australian data — house prices, athletes, hospital wait times and weather.
Problem 1 — Suburb house prices (outlier hunt)
An estate agent records 12 recent house sales in a Sydney suburb ($,000): 720, 780, 815, 830, 850, 870, 880, 900, 920, 950, 980, 3,800 (a luxury renovation).
Set up: What are we solving for?
(i) Find the five-number summary. 2 marks
(ii) Apply the 1.5 × IQR rule. Is the $3.8 million sale an outlier? 2 marks
(iii) An estate-agent advertisement headlines "Median house price ≈ $880k". State whether this is a fair summary given the outlier, and explain in one sentence why. 2 marks
Stuck? Revisit lesson § The 1.5 × IQR Rule.Problem 2 — Comparing two football teams (side-by-side box plots)
Goals per match across 11 matches for two NRL teams. Team P: 18, 20, 22, 22, 24, 26, 28, 30, 32, 34, 38. Team Q: 6, 12, 16, 20, 24, 26, 28, 32, 36, 42, 60.
Set up: What are we solving for?
(i) Find the five-number summary for each team. 2 marks
(ii) Test whether 60 (Team Q's highest) is an outlier using the 1.5 × IQR rule. 2 marks
(iii) Without drawing the box plot, describe in one sentence each which team is (a) higher-scoring on average and (b) more consistent — using the five-number summary numbers. 2 marks
Stuck? Median compares centre; IQR compares spread.Problem 3 — Hospital wait times (interpret a box plot)
The box plot below summarises waiting times (minutes) at a public-hospital ED for two months.
July box plot: Min = 15, Q1 = 35, Median = 55, Q3 = 90, Max = 240 (with outlier dot at 240).
August box plot: Min = 12, Q1 = 30, Median = 48, Q3 = 70, Max = 180 (with outlier dot at 180).
Set up: What are we solving for?
(i) Compute the IQR for each month. 1 mark
(ii) Verify that 240 (July) and 180 (August) are outliers using the 1.5 × IQR rule. 2 marks
(iii) The hospital director wants to claim "wait times improved in August". Identify two pieces of evidence from the box plots that support this claim, and one piece that complicates it. 3 marks
Stuck? Look at median, IQR and the outlier value separately.Problem 4 — Daily temperatures (raw data → box plot)
Daily peak temperatures (°C) in a NSW town over 11 days: 22, 24, 25, 26, 27, 28, 29, 30, 32, 34, 41.
Set up: What are we solving for?
(i) Find the five-number summary and IQR. 2 marks
(ii) Apply the 1.5 × IQR rule to decide if 41 °C is an outlier. 2 marks
(iii) Sketch the box plot to scale (use 20-45 °C on the x-axis). Show any outlier as a separate dot. 2 marks
Stuck? Whisker should stop at the largest non-outlier; plot outlier as a separate point.Problem 5 — Two athletes (decision making with box plots)
A coach is choosing between two 400 m runners for a relay leg. Times (seconds) over 11 races each:
Runner X: 5-num summary = 49.6, 50.8, 51.4, 52.0, 53.2.
Runner Y: 5-num summary = 48.4, 49.6, 51.2, 52.6, 55.8.
Set up: What are we solving for?
(i) Compute the IQR and range for each runner. 2 marks
(ii) Which runner is faster on average (lower median = faster), and which is more consistent? 2 marks
(iii) The relay leg is critical and the team needs predictable performance. Which runner should the coach pick and why? Reference the IQR in your answer. 2 marks
Stuck? Runner Y has occasional very fast races (48.4) but also slow ones (55.8) — variability matters in a relay.How did this worksheet feel?
What I'll revisit before next class:
Problem 1 — Suburb house prices
(i) n = 12, ordered already. Min = 720, Max = 3,800. Median = (6th + 7th)/2 = (870 + 880)/2 = 875. Lower 6: 720, 780, 815, 830, 850, 870 → Q1 = (815 + 830)/2 = 822.5. Upper 6: 880, 900, 920, 950, 980, 3,800 → Q3 = (920 + 950)/2 = 935. Summary ($,000): 720, 822.5, 875, 935, 3,800.
(ii) IQR = 935 − 822.5 = 112.5. Upper fence = 935 + 1.5(112.5) = 935 + 168.75 = 1,103.75 ($1.1m). 3,800 ≫ 1,103.75 → $3.8m is a strong outlier.
(iii) Quoting the median of $875,000 is the right summary precisely because the $3.8 m outlier would inflate the mean to ≈ $1.18 m; the median ignores the outlier and gives buyers a realistic typical price.
Problem 2 — Two NRL teams
(i) P (n = 11): Min 18, Q1 = 22, Median = 26, Q3 = 32, Max = 38. Summary: 18, 22, 26, 32, 38. Q (n = 11): Min 6, Median = 26, Lower 5 (6,12,16,20,24) → Q1 = 16. Upper 5 (28,32,36,42,60) → Q3 = 36. Summary: 6, 16, 26, 36, 60.
(ii) Q IQR = 36 − 16 = 20. Upper fence = 36 + 1.5(20) = 36 + 30 = 66. 60 ≤ 66 → 60 is NOT an outlier by the 1.5 × IQR rule (close but inside the fence).
(iii) (a) Same median (26) → neither team is higher on average. (b) Team P's IQR (10) is half Team Q's IQR (20) → Team P is much more consistent.
Problem 3 — ED wait times
(i) July IQR = 90 − 35 = 55 min. August IQR = 70 − 30 = 40 min.
(ii) July: upper fence = 90 + 1.5(55) = 172.5; 240 > 172.5 → outlier ✓. August: upper fence = 70 + 1.5(40) = 130; 180 > 130 → outlier ✓.
(iii) Support: (1) Median dropped from 55 to 48 min (typical wait improved). (2) IQR dropped from 55 to 40 min (waits became more consistent). Complication: (3) Both months still have extreme outliers (240 and 180 min); even if "typical" wait improved, a small group of patients still waited a very long time, and the worst case is also outlier-level in August.
Problem 4 — NSW town temperatures
(i) n = 11. Median = 6th value = 28. Lower 5: 22,24,25,26,27 → Q1 = 25. Upper 5: 29,30,32,34,41 → Q3 = 32. Summary: 22, 25, 28, 32, 41. IQR = 32 − 25 = 7.
(ii) Upper fence = 32 + 1.5(7) = 32 + 10.5 = 42.5. 41 ≤ 42.5 → 41 is NOT an outlier (only just inside the fence).
(iii) Box plot: scale 20-45. Box from 25 to 32, vertical median line at 28. Left whisker from 22 to 25; right whisker from 32 to 41 (since 41 is not an outlier, the whisker reaches it). No separate outlier dot.
Problem 5 — Athletes
(i) X IQR = 52.0 − 50.8 = 1.2 s; range = 53.2 − 49.6 = 3.6 s. Y IQR = 52.6 − 49.6 = 3.0 s; range = 55.8 − 48.4 = 7.4 s.
(ii) Medians: X = 51.4 s, Y = 51.2 s → Y is faster on average (by 0.2 s). Y has smaller fastest time but also larger slowest time. X is more consistent (much smaller IQR and range).
(iii) Pick Runner X for the relay leg. Even though Y has a slightly better median, X's IQR is 1.2 s vs Y's 3.0 s — X reliably runs between 50.8 and 52.0 s, while Y's middle 50% of times spans 49.6 to 52.6 s, so Y could run a 52-second leg and cost the team the race even though Y is faster on her best day.