Mathematics • Year 8 • Unit 4 • Lesson 4
Dot Plots and Stem-and-Leaf — Mixed Challenge
Pull together everything from Lesson 4: dot plots, stem-and-leaf, back-to-back plots, finding mode/median/range, identifying outliers, and describing distribution shape.
1. Mixed problems — choose the right move
Each question uses a different idea from Lesson 4. Show your working. 3 marks each
1.1 A dot plot for n = 12 shows: 4 (1 dot), 5 (3), 6 (4), 7 (2), 8 (2). Find the mode, median, and range.
1.2 Construct an ordered stem-and-leaf plot from these 10 values: 31, 47, 22, 38, 51, 29, 44, 35, 26, 49. Include a key.
1.3 From the stem-and-leaf 2 | 1 4 7 3 | 0 3 5 5 8 4 | 2 6 (key 2 | 1 = 21), find n, the range, and the median.
1.4 Describe the shape of each distribution in one word (symmetric / skewed left / skewed right):
(a) Most data on the left, long tail to the right.
(b) Most data on the right, long tail to the left.
(c) Balanced bell shape around the middle.
1.5 A back-to-back stem-and-leaf compares two test groups:
9 7 5 | 4 | 1 3 6
8 4 2 | 5 | 0 2 5 7
6 | 1 4
Identify n for each group and the median of each group.
1.6 Decide which display is most appropriate for each data set and justify in one sentence:
(a) Heights of 30 students in centimetres.
(b) Favourite ice-cream flavour of 25 students (categorical).
(c) Test scores out of 100 for 50 students.
2. Find the mistake
Another student attempted this stem-and-leaf problem. Their working is shown below. Exactly one line contains a mistake. Spot it, explain why it's wrong, then re-do the working correctly. 3 marks
Problem: Construct an ordered stem-and-leaf plot for the data 34, 28, 41, 37, 25, 49, 33, 42, 30, then find the median.
Line 1: Stems = 2, 3, 4.
Line 2: 2 | 5 8
Line 3: 3 | 4 7 3 0
Line 4: 4 | 1 2 9
Line 5: n = 9, median position = (9+1)/2 = 5th value.
Line 6: Median = 34 (counting through the leaves as written).
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected line(s) and the correct median.
Stuck? The leaves in stem 3 must be in ascending order. Then re-count to the 5th value.3. Open-ended challenge — compare two groups
This question has many valid answers. 4 marks
3.1 Your job: invent two data sets of 10 values each that allow a sensible back-to-back stem-and-leaf comparison. Possible contexts:
• Boys' vs Girls' long-jump distances (in metres × 100 → cm).
• Class A vs Class B test scores out of 100.
• Morning vs afternoon temperature (°C) for 10 days.
Write up your analysis with the following:
(i) State your two groups and list 10 plausible values for each.
(ii) Construct an ordered back-to-back stem-and-leaf plot with a key.
(iii) Find the median, range, and modal class (if any) for each group.
(iv) Write ONE or TWO sentences comparing the two groups — comment on centre AND spread.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Dot plot
Mode = 6 (4 dots). Median: n = 12 (even) → average of 6th and 7th values. Ordered list: 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8. (6 + 6) ÷ 2 = 6. Range = 8 − 4 = 4.
1.2 — Stem-and-leaf for 10 values
Sorted: 22, 26, 29, 31, 35, 38, 44, 47, 49, 51.
2 | 2 6 9 3 | 1 5 8 4 | 4 7 9 5 | 1. Key: 2 | 2 = 22.
1.3 — Read from stem-and-leaf
n = 3 + 5 + 2 = 10. Min = 21, Max = 46. Range = 25. Median (n = 10, even) = average of 5th and 6th values = (33 + 35) ÷ 2 = 34.
1.4 — Shape descriptors
(a) Skewed right. (b) Skewed left. (c) Symmetric.
1.5 — Back-to-back medians
Left group leaves: stem 4 has 5, 7, 9; stem 5 has 2, 4, 8 → values 45, 47, 49, 52, 54, 58 → n = 6. Median = (49 + 52) ÷ 2 = 50.5.
Right group leaves: stem 4 has 1, 3, 6; stem 5 has 0, 2, 5, 7; stem 6 has 1, 4 → values 41, 43, 46, 50, 52, 55, 57, 61, 64 → n = 9. Median = 5th value = 52.
1.6 — Choose the display
(a) Heights of 30 students → grouped frequency table / histogram (continuous data, n = 30 too many for a dot plot). A stem-and-leaf is also workable.
(b) Favourite ice-cream flavour → bar chart (categorical data — dot plots are only for numeric data).
(c) Test scores out of 100 for 50 students → stem-and-leaf (two-digit numeric data) or grouped frequency / histogram.
2 — Find the mistake
(a) The mistake is on Line 3.
(b) The leaves in stem 3 are in the order they appeared in the raw data (4, 7, 3, 0), not in ascending order. A stem-and-leaf plot must always have leaves ordered smallest to largest within each row.
(c) Corrected Line 3: 3 | 0 3 4 7. Re-count to the 5th value across all rows: 25, 28, 30, 33, 34 — the median is still 34, but only because the data happened to round out that way; the original answer was based on incorrect ordering.
3 — Open-ended (sample solution — Class A vs Class B test scores)
(i) Groups: Class A scores: 42, 48, 51, 54, 56, 58, 63, 65, 68, 72. Class B scores: 45, 50, 55, 58, 60, 62, 65, 68, 71, 78.
(ii) Back-to-back plot (Class A left, outward; Class B right):
8 2 | 4 | 5
8 6 4 1 | 5 | 0 5 8
8 5 3 | 6 | 0 2 5 8
2 | 7 | 1 8
Key: 4 | 2 = 42 (Class A); 4 | 5 = 45 (Class B).
(iii) Class A (n = 10): median = (56 + 58) ÷ 2 = 57; range = 72 − 42 = 30. Class B (n = 10): median = (60 + 62) ÷ 2 = 61; range = 78 − 45 = 33.
(iv) Comparison: Class B performed slightly better on the median (61 vs 57) but had a similar overall spread (range 33 vs 30). Both classes are reasonably consistent with no obvious outliers.
Marking: 1 mark for plausible matched data; 1 mark for correct back-to-back layout with key; 1 mark for accurate medians and ranges; 1 mark for a comparison sentence covering both centre AND spread.