Mathematics Standard • Year 12 • Module 8 • Lesson 4
Displaying Data — Past-Paper Style
Practise HSC Mathematics Standard 2-style writing on histograms, frequency tables, stem-and-leaf plots and distribution shape.
1. Short-answer questions
1.1 A stem-and-leaf plot of test marks reads:
4 | 2 5 8
5 | 1 3 5 6 9
6 | 0 2 2 4 7 8
7 | 1 3 5 8
8 | 0 4
(a) How many students took the test?
(b) What is the median mark?
(c) Describe the shape of the distribution. 3 marks Band 3
1.2 A frequency table records the number of online orders received per day at a bookshop:
| Orders/day | 0-9 | 10-19 | 20-29 | 30-39 | 40-49 |
|---|---|---|---|---|---|
| Frequency | 3 | 12 | 18 | 9 | 3 |
(a) Total number of days recorded.
(b) Estimate the mean orders per day using midpoints. 3 marks Band 3-4
1.3 A clothing store displays customer satisfaction scores (1-10). It shows a histogram with the following frequencies for scores 1 through 10: 1, 2, 2, 3, 5, 8, 15, 22, 18, 4.
(a) Describe the shape of the distribution (symmetric, left-skew, right-skew).
(b) State, with reasoning, whether the mean is likely to be larger or smaller than the median. 4 marks Band 4
2. Extended response
2.1 A NSW high school's PE department recorded the resting heart rate (bpm) of 25 Year 12 students:
52, 56, 58, 60, 62, 62, 64, 65, 66, 68, 68, 70, 70, 72, 72, 74, 75, 76, 78, 80, 82, 84, 85, 88, 95.
(a) Construct a frequency table using class width 10 (start at 50-59).
(b) Draw the corresponding histogram in the space provided.
(c) Describe the shape of the distribution and recommend, with reasoning, the most appropriate display (histogram, stem-and-leaf, dot plot or pie chart) for the PE teacher to present to a parent-teacher night audience. 7 marks Band 5-6
Explicit marking criteria
Part (a) — 2 marks
• 1 mark — correct frequency counts for all 5 classes.
• 1 mark — frequencies sum to 25 (confirms no values lost).
Part (b) — 2 marks
• 1 mark — bars of equal width, no gaps, x-axis labelled "Heart rate (bpm)".
• 1 mark — y-axis "Frequency", correct bar heights matching the table.
Part (c) — 3 marks
• 1 mark — names the shape (e.g. roughly symmetric / slight right-skew).
• 1 mark — recommends a display.
• 1 mark — justification tied to audience (parents) and to the data type (continuous numerical), e.g. histogram makes shape clear at a glance.
Your response:
Stuck on (a)? Sort first, then count how many fall in each class.How did this worksheet feel?
What I'll revisit before next class:
1.1 — Stem-and-leaf (3 marks)
(a) Total leaves = 3 + 5 + 6 + 4 + 2 = 20 students.
(b) Median = (10th + 11th) value / 2. Ordered, the 10th is 62 and the 11th is 64. Median = (62 + 64)/2 = 63.
(c) Roughly symmetric and unimodal, centred around 62-64.
Marking notes. 1 mark each for (a), (b), (c). Common errors: counting stems rather than leaves; picking the median by visual position rather than the (n+1)/2 calculation.
1.2 — Bookshop orders (3 marks)
(a) Total = 3 + 12 + 18 + 9 + 3 = 45 days.
(b) Midpoints: 4.5, 14.5, 24.5, 34.5, 44.5. Σ(f × midpoint) = 3(4.5) + 12(14.5) + 18(24.5) + 9(34.5) + 3(44.5) = 13.5 + 174 + 441 + 310.5 + 133.5 = 1,072.5. Estimated mean = 1,072.5 / 45 = 23.8 orders/day.
Marking notes. 1 mark for total days. 1 mark for using midpoints correctly. 1 mark for correct mean. Quoting "23" without working scores 1 max.
1.3 — Satisfaction histogram (4 marks)
(a) The frequencies grow slowly from 1, 2, 2, 3, 5, 8 then jump to 15, 22, 18, 4 — heavy on the high (right) side with a long thin tail on the low (left) side. Shape: left-skewed (negative skew).
(b) In a left-skewed distribution the long tail of low scores pulls the mean down, so mean < median. The median is roughly 8 (centre of the bulk), the mean would be pulled below that by the small tail at 1-5.
Marking notes. 1 mark for correctly identifying left-skew. 1 mark for direction "mean < median". 1 mark for explaining the role of the low-end tail. 1 mark for an estimate of the median location (or for tight reasoning instead of an estimate).
2.1 — PE heart rates (7 marks): Band-6 sample with annotations
(a) Frequency table.
| Heart rate (bpm) | Frequency |
|---|---|
| 50 – 59 | 2 |
| 60 – 69 | 8 |
| 70 – 79 | 9 |
| 80 – 89 | 5 |
| 90 – 99 | 1 |
Check: 2 + 8 + 9 + 5 + 1 = 25 ✓. [1 mark — correct frequencies. 1 mark — sum confirmed.]
(b) Histogram.
x-axis labelled "Heart rate (bpm)", y-axis "Frequency"; bars of equal width over 50-59, 60-69, …, 90-99; bar heights 2, 8, 9, 5, 1 with no gaps. [1 mark — labels and equal widths. 1 mark — correct bar heights.]
(c) Shape + display recommendation.
The distribution is roughly symmetric with a slight right-skew (the single 95 bpm value extends the upper tail). [1 mark — shape.]
Recommended display for a parent-teacher night: a histogram. [1 mark — recommendation.] Heart rate is a continuous numerical variable, and a histogram immediately shows parents the shape and the typical range (60-79 bpm captures 17 of the 25 students), which is the message the PE teacher wants to communicate — a stem-and-leaf would be more accurate but harder for non-technical parents to read at a glance, and a pie chart would not work for continuous data. [1 mark — justification ties audience + data type to the chosen display.]
Total: 7/7.
Band descriptors for marker.
Band 3: Frequency table with one or two miscounts; histogram unlabelled; (c) is one sentence. ≈ 2-3 marks.
Band 4: Correct table and a workable histogram; (c) names the shape and a display but no audience-tied justification. ≈ 4-5 marks.
Band 5: Full table, correctly labelled histogram, (c) names shape and chosen display with one reason. ≈ 6 marks.
Band 6: Full table and histogram, sophisticated (c) that justifies the chosen display by reference to both audience and data type, and acknowledges alternatives. 7/7.