Mathematics Standard • Year 12 • Module 8 • Lesson 4
Displaying Data — Problem Set
Apply data-display techniques (frequency tables, histograms, stem-and-leaf, pie charts) to realistic Australian data — sport, transport, retail, weather and demographics.
Problem 1 — Cricket scores (stem-and-leaf)
An under-15 cricket club records its top-12 batters' season scores: 14, 18, 22, 27, 31, 35, 38, 44, 47, 52, 58, 73.
Set up: What are we solving for?
(i) Draw a stem-and-leaf plot using tens as stems and units as leaves. 2 marks
(ii) Find the median, range and modal stem. 2 marks
(iii) Describe the shape of the distribution in one sentence. 1 mark
Stuck? Revisit lesson § Stem-and-Leaf — every value is preserved while the shape is visible.Problem 2 — Train arrival delays (histogram)
A Sydney train line records delay times (minutes) for 50 morning peak trains:
| Delay (min) | 0-4 | 5-9 | 10-14 | 15-19 | 20-24 | 25-29 |
|---|---|---|---|---|---|---|
| Frequency | 18 | 14 | 9 | 5 | 3 | 1 |
Set up: What are we solving for?
(i) Draw the histogram (bars touching, height = frequency). 2 marks
(ii) Describe the shape (symmetric, left-skew, right-skew) and explain in one sentence what that means for commuter experience. 2 marks
(iii) Estimate the percentage of trains delayed by 10 minutes or more. 2 marks
Stuck? Add the frequencies for the 10-14, 15-19, 20-24, 25-29 classes; divide by 50; multiply by 100.Problem 3 — School demographics (pie chart)
A high school has 1,200 students. The Year-level breakdown is: Y7 240, Y8 220, Y9 200, Y10 200, Y11 180, Y12 160.
Set up: What are we solving for?
(i) Calculate the percentage of students in each Year. 2 marks
(ii) Calculate the pie-chart angle (degrees) for Year 12. 1 mark
(iii) A new principal claims the pie chart shows "Year 7 is twice as big as Year 12". Verify this and discuss whether a pie chart or a bar chart is the more useful display for choosing class sizes. 2 marks
Stuck? Pie chart shows proportions; bar chart shows actual counts — useful for class-size planning.Problem 4 — Raw data → frequency table → shape
The number of customers per hour at a chemist over 20 hours: 4, 6, 7, 8, 10, 11, 12, 12, 13, 14, 14, 15, 16, 18, 19, 22, 23, 25, 28, 32.
Set up: What are we solving for?
(i) Group the data into classes of width 5 starting at 0-4. Construct the frequency table. 2 marks
(ii) Describe the shape of the distribution. 1 mark
(iii) The manager wants to know which hours to schedule extra staff. Suggest a sensible cut-off (e.g. "≥ 20 customers/hour") and use the frequency table to estimate how many hours per 20-hour shift this would cover. 2 marks
Stuck? Classes 0-4, 5-9, 10-14, 15-19, 20-24, 25-29, 30-34.Problem 5 — Choosing the right display
You are presenting the following four data sets at a school assembly. For each, state which display you would use and give a one-sentence reason.
Data 1: The favourite genre of music for 200 surveyed students (pop, rock, hip-hop, classical, other).
Data 2: The body temperatures (to 0.1°C) of 12 students measured in PE class.
Data 3: The heights of 400 Year 7 students from across the state.
Data 4: The number of pets in each of 30 surveyed households.
Set up: What are we solving for?
(i) Choose a display for Data 1 and Data 2 with one-sentence justifications. 2 marks
(ii) Choose a display for Data 3 and Data 4 with one-sentence justifications. 2 marks
(iii) Explain in one sentence why a pie chart would be a poor choice for Data 3. 2 marks
Stuck? Match the data type to the display from Q1.2 of the Build worksheet.How did this worksheet feel?
What I'll revisit before next class:
Problem 1 — Cricket scores
(i) 1 | 4 8
2 | 2 7
3 | 1 5 8
4 | 4 7
5 | 2 8
6 |
7 | 3
(ii) Median = (6th + 7th)/2 = (35 + 38)/2 = 36.5. Range = 73 − 14 = 59. Modal stem = 3 | 1 5 8 (3 leaves).
(iii) Roughly symmetric with a single high score (73) creating a slight right-skew.
Problem 2 — Train delays
(i) Histogram bars (left to right) of heights 18, 14, 9, 5, 3, 1. (Drawn freehand; key features: equal widths, no gaps, tallest bar on the left.)
(ii) Right-skewed (positive skew). Most trains run close to on time; a small but real tail of trains run very late, which is what frustrates commuters most.
(iii) Delayed ≥ 10 min = 9 + 5 + 3 + 1 = 18. Percentage = 18/50 = 36%.
Problem 3 — School demographics
(i) Total = 1,200. Percentages: Y7 = 240/1200 = 20%, Y8 = 220/1200 ≈ 18.3%, Y9 = 200/1200 ≈ 16.7%, Y10 ≈ 16.7%, Y11 = 180/1200 = 15%, Y12 = 160/1200 ≈ 13.3%.
(ii) Year 12 angle = 13.33% × 360 = 48° (or 160/1200 × 360 = 48°).
(iii) Y7 has 240 students; Y12 has 160 — Y7 is 1.5× larger, not 2× — the principal is wrong. For planning class sizes, a bar chart is more useful because it shows the actual counts (240 vs 160) needed to allocate teachers and rooms; a pie chart only shows proportions.
Problem 4 — Chemist customers/hour
(i) 0-4: 1 (4). 5-9: 3 (6, 7, 8). 10-14: 7 (10, 11, 12, 12, 13, 14, 14). 15-19: 4 (15, 16, 18, 19). 20-24: 2 (22, 23). 25-29: 2 (25, 28). 30-34: 1 (32). Total = 20.
(ii) Approximately symmetric with slight right-skew — peaks in the 10-14 class, tails out at both ends but the right tail is longer.
(iii) Sensible cut-off: ≥ 20 customers/hour. From the table this covers 2 + 2 + 1 = 5 hours per shift (25% of the shift), so extra staff for roughly one in every four hours.
Problem 5 — Choosing displays
(i) Data 1 (music genres) → bar chart or pie chart — nominal categorical with a small number of named categories. Data 2 (12 temperatures to 0.1°C) → stem-and-leaf plot — small numerical data set where preserving every measurement matters.
(ii) Data 3 (400 heights) → histogram — large continuous data set where shape (likely normal-bell) is the key message. Data 4 (pets per household, 30) → dot plot (or column graph) — small discrete data set with low repeats, where individual counts are easy to see.
(iii) A pie chart shows percentages of a whole, but heights are continuous numerical — there is no natural way to slice a "100% pie" of heights, and the shape (centre and spread) would be invisible. A histogram tells you those things at a glance.