Mathematics Standard • Year 12 • Module 8 • Lesson 4
Displaying Data — Skill Drill
Build fluency in constructing and reading frequency tables, histograms, stem-and-leaf plots and dot plots — and naming the distribution shape.
1. Quick recall
Answer each in the space provided. 1 mark each
Q1.1 A histogram bar runs from 150 to 160 cm. The class width is ____________ cm and the midpoint is ____________ cm.
Q1.2 Match each display to the data type it suits best.
Stem-and-leaf → ____________ Histogram → ____________ Dot plot → ____________ Pie chart → ____________
Options: small numerical data sets, large grouped continuous data, small discrete data with low repeats, categorical (proportion of a whole).
Q1.3 Name the three common distribution shapes: ____________, ____________ (tail on right), ____________ (tail on left).
2. Worked example — frequency table → histogram → shape
Heights (cm) of 20 students: 152, 155, 158, 160, 162, 163, 165, 166, 168, 170, 171, 172, 175, 178, 180, 182, 185, 188, 190, 195.
Step 1 — Group with class width 10 cm.
| Class (cm) | Frequency |
|---|---|
| 150 – 159 | 3 |
| 160 – 169 | 6 |
| 170 – 179 | 6 |
| 180 – 189 | 4 |
| 190 – 199 | 1 |
Step 2 — Histogram (sketch). Bars of equal width, no gaps between them, height = frequency.
▮▮▮ ▮▮▮▮▮▮ ▮▮▮▮▮▮ ▮▮▮▮ ▮
Step 3 — Stem-and-leaf (stem = tens, leaves = units).
15 | 2 5 8
16 | 0 2 3 5 6 8
17 | 0 1 2 5 8
18 | 0 2 5 8
19 | 0 5
Conclusion. Roughly symmetric, unimodal, centred around 170 cm. Most students fall in the 160-179 cm range.
3. Faded example — group the raw data
The number of texts sent yesterday by 18 students: 0, 1, 2, 4, 5, 6, 8, 10, 12, 14, 17, 20, 23, 28, 35, 42, 48, 60. Use class width 10. 4 marks
| Class (texts) | Tally | Frequency |
|---|---|---|
| 0 – 9 | ____________________ | ____________ |
| 10 – 19 | ____________________ | ____________ |
| 20 – 29 | ____________________ | ____________ |
| 30 – 39 | ____________________ | ____________ |
| 40 – 49 | ____________________ | ____________ |
| 50 – 59 | ____________________ | ____________ |
| 60 – 69 | ____________________ | ____________ |
Step — Describe the shape: The distribution is ____________-skewed (most students texted few times; a few texted very often).
4. Graduated practice — display and describe
Foundation — one-step reading (4 questions)
| Q | Problem | Answer |
|---|---|---|
| 4.1 1 | A class has width 10, lower boundary 30. The midpoint is ____. | |
| 4.2 1 | A histogram has bars at 0-9, 10-19, 20-29 with heights 5, 12, 8. How many total observations? | |
| 4.3 1 | A distribution has its tail on the right. Name the shape. | |
| 4.4 1 | If a stem-and-leaf row reads "4 | 0 3 7", list the data values. |
Standard — typical HSC difficulty (6 questions)
4.5 Group these test scores into classes of width 10 (starting 50-59): 52, 55, 58, 61, 63, 65, 68, 71, 74, 76, 78, 81, 85, 88, 92, 95. Make a frequency table. 2 marks
4.6 Draw a back-to-back stem-and-leaf for two classes' marks. Class A: 62, 65, 67, 70, 71, 74, 75, 78, 82, 85. Class B: 55, 58, 60, 62, 65, 68, 70, 71, 74, 78. (Use stems 5, 6, 7, 8.) 2 marks
4.7 A histogram shows the bars (left to right): 4, 8, 12, 15, 9, 3. Describe the shape (symmetric / left-skew / right-skew). 1 mark
4.8 A pie chart shows favourite drinks at a cafe: coffee 45%, tea 25%, juice 15%, water 10%, other 5%. What angle (in degrees) represents tea? 2 marks
4.9 Draw a dot plot for: 3, 5, 5, 6, 6, 6, 7, 7, 8, 8, 9. Then state the mode and describe the shape. 2 marks
4.10 A stem-and-leaf has rows 3|2 4 9, 4|0 1 5 7 8, 5|2 3 6, 6|1. Find n (total values) and the median. 2 marks
Extension — read and interpret (2 questions)
4.11 A histogram of exam marks (class width 10) has frequencies: 50-59 (5), 60-69 (12), 70-79 (15), 80-89 (10), 90-99 (3). (a) How many students took the exam? (b) Estimate the median by finding the class containing the (n+1)/2-th value. 3 marks
4.12 The same data is shown as a histogram and as a stem-and-leaf plot. State one advantage of the histogram and one advantage of the stem-and-leaf for this data set. 3 marks
5. Self-check the easy 3
How did this worksheet feel?
What I'll revisit before next class:
Q1.1 — Class width and midpoint
Width = 160 − 150 = 10 cm. Midpoint = (150 + 160) / 2 = 155 cm.
Q1.2 — Match displays
Stem-and-leaf → small numerical data sets. Histogram → large grouped continuous data. Dot plot → small discrete data with low repeats. Pie chart → categorical (proportion of a whole).
Q1.3 — Distribution shapes
Symmetric, right-skewed (positive skew, tail on right), left-skewed (negative skew, tail on left).
Q3 — Faded example (texts)
Frequencies: 0-9 (7), 10-19 (4), 20-29 (3), 30-39 (1), 40-49 (2), 50-59 (0), 60-69 (1). Total = 18. Distribution is right-skewed (positive skew) — most students sent few texts; a few students sent many.
Q4.1–4.4 — Foundation
4.1: midpoint = (30 + 40)/2 = 35. 4.2: total = 5 + 12 + 8 = 25. 4.3: right-skewed (positive skew). 4.4: data = 40, 43, 47.
Q4.5 — Frequency table (test scores)
50-59: 3 (52, 55, 58). 60-69: 4 (61, 63, 65, 68). 70-79: 4 (71, 74, 76, 78). 80-89: 3 (81, 85, 88). 90-99: 2 (92, 95). Total = 16.
Q4.6 — Back-to-back stem-and-leaf
Class A | Stem | Class B
| 5 | 5 8
7 5 2 | 6 | 0 2 5 8
8 5 4 1 0 | 7 | 0 1 4 8
5 2 | 8 |
(Class A leaves read right-to-left from the stem; Class B leaves read left-to-right.)
Q4.7 — Shape
Bar heights 4, 8, 12, 15, 9, 3 rise then fall, peak slightly right of centre, slightly longer left tail (4, 8, 12) than right (9, 3). Best description: roughly symmetric (very slight left-skew is acceptable).
Q4.8 — Pie slice for tea
Tea = 25% of 360° = 0.25 × 360 = 90°.
Q4.9 — Dot plot
3: •
5: ••
6: •••
7: ••
8: ••
9: •
Mode = 6 (3 dots, the tallest column). Shape: roughly symmetric, unimodal.
Q4.10 — n and median from stem-and-leaf
Data: 32, 34, 39, 40, 41, 45, 47, 48, 52, 53, 56, 61. n = 12. Median = (6th + 7th)/2 = (45 + 47)/2 = 46.
Q4.11 — Histogram exam marks
(a) Total = 5 + 12 + 15 + 10 + 3 = 45 students.
(b) Median position = (45+1)/2 = 23rd value. Cumulative: 5, 17, 32, 42, 45. The 23rd value falls in the 70-79 class. So estimated median ≈ 74-75 marks (i.e. in the 70-79 group).
Q4.12 — Histogram vs stem-and-leaf
Histogram advantage: shows the shape and frequency of large data sets at a glance — easy to compare bars visually. Stem-and-leaf advantage: retains every original value, so you can still find the exact median, mode and individual data points after plotting.