Mathematics • Year 10 • Unit 4 • Lesson 4
Stem-and-Leaf Plots — Mixed Challenge
Pull together every idea from Lesson 4: ordered plots, back-to-back plots, reading the median by counting to the middle, and using the lesson's "preserves original values" advantage. Spot a Year 10 mistake, then design your own back-to-back plot to match a written description.
1. Mixed problems — choose the right tool
Each question uses a different idea from Lesson 4. Decide whether to build, read, count, or compare before you start. 3 marks each
1.1 Build an ordered stem-and-leaf plot for: 22, 18, 35, 41, 27, 19, 33, 24, 38, 42. Include a key.
1.2 A stem-and-leaf plot shows:
1 | 2 5 8
2 | 0 3 4 7 9
3 | 1 6 Key: 1 | 2 = 12.
Find (a) the total number of values, (b) the range, (c) the median.
1.3 The decimal values 4.2, 5.7, 6.1, 4.9, 5.3, 6.8, 5.0, 4.5 need a stem-and-leaf plot. State what the stem is, what the leaf is, and write the key — then build the ordered plot.
1.4 A back-to-back stem-and-leaf shows weekly hours of TV for two age groups:
(Teens leaves outward) | stem | (Adults leaves)
8 5 | 0 | 2 4
9 7 5 2 | 1 | 0 3 5 7
4 2 0 | 2 | 1 4 5 8 8
| 3 | 0 2 Key: 1 | 5 = 15 hours.
State (a) the largest teen value, (b) the largest adult value, and (c) which group tends to watch more TV.
1.5 Lesson 4 says a stem-and-leaf plot preserves the original data values. Use this property to compute the mean of the data set 12, 14, 18, 22, 25, 28, 31, 33 (shown as 1 | 2 4 8, 2 | 2 5 8, 3 | 1 3) to one decimal place. Show working.
1.6 Two soccer teams played 11 matches each. Goals scored per match:
Wanderers: 0, 1, 2, 2, 3, 3, 4, 4, 5, 5, 7.
Roar: 0, 0, 1, 2, 3, 3, 4, 4, 6, 7, 9.
Since most values are single-digit, propose a sensible stem-and-leaf design (hint: stems can be 0 and 0 again — or you can use a "split stem"). Then build either a single ordered plot for the Wanderers OR a back-to-back plot if you can justify the stems.
2. Find the mistake
Another Year 10 student has tried to build an ordered stem-and-leaf plot and use it. Their working is shown below. Exactly one line contains a mistake. Spot it, explain why it's wrong, then re-do correctly. 3 marks
Student's working — data: 36, 41, 28, 53, 44, 39, 47, 31, 50, 42, 35, 48 (12 values).
Line 1: 2 | 8 3 | 1 5 6 9 4 | 1 2 4 7 8 5 | 0 3 Key: 3 | 1 = 31.
Line 2: Median = middle value = the 6th value (since n = 12, even, median is the 6th value).
Line 3: Counting to the 6th value: 28, 31, 35, 36, 39, 41 → Median = 41.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected reasoning, including the corrected median.
Stuck? When n is even, the median is the average of the two middle values — the (n/2)th and (n/2 + 1)th — not just the (n/2)th.3. Open-ended challenge — design two data sets
This question has many valid answers. Be creative but show every number. 4 marks
3.1 Design two data sets, each with exactly 10 two-digit values between 10 and 99, that, when displayed on a back-to-back stem-and-leaf plot, satisfy all of the following:
- both data sets share the same range (largest − smallest),
- Data set A has a median in the 30s,
- Data set B has a median in the 50s,
- at least one stem row is shared (has leaves on both sides),
- at least one stem row has leaves on only one side (creating a visible difference in the shape).
For your design:
(i) Write the 10 values for each data set.
(ii) Build the back-to-back ordered stem-and-leaf plot with a key (Data A leaves outward on the left, Data B leaves on the right).
(iii) State both medians and the shared range.
(iv) Write one sentence comparing the two distributions, suitable for a real PE/health/finance context of your choice.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Build the plot
1 | 8 9 2 | 2 4 7 3 | 3 5 8 4 | 1 2 Key: 1 | 8 = 18.
1.2 — Read the plot
Values in order: 12, 15, 18, 20, 23, 24, 27, 29, 31, 36.
(a) Total = 10 values.
(b) Range = 36 − 12 = 24.
(c) n = 10, median = average of 5th and 6th = (23 + 24) / 2 = 23.5.
1.3 — Decimal stem-and-leaf
Stem = units digit, leaf = tenths digit. Key: 4 | 2 = 4.2.
Ordered: 4 | 2 5 9 5 | 0 3 7 6 | 1 8.
1.4 — Read back-to-back plot
(a) Largest teen value = 24 (from the row 2 | leaves 4 2 0 → largest is 24).
(b) Largest adult value = 32.
(c) Adults tend to watch more TV — the adult distribution extends into the 30s while no teen value reaches the 30s row.
1.5 — Mean from preserved data
Sum = 12 + 14 + 18 + 22 + 25 + 28 + 31 + 33 = 183.
n = 8. Mean = 183 ÷ 8 = 22.9 (to 1 d.p.).
Because the stem-and-leaf preserves every original value, we can compute the exact mean — something a histogram cannot give without estimation.
1.6 — Single-digit goals data
A stem-and-leaf is a poor fit here. All values are single-digit (0-9), so every value would sit on stem "0" and the plot collapses to one row, losing structure.
Better choice: a back-to-back dot plot above the same number line 0-10 (Lesson 3), since both data sets are small and numerical. If a stem-and-leaf is forced, you could use a split stem: 0 (low) for 0-4 and 0 (high) for 5-9.
Single ordered plot for the Wanderers: 0 (low) | 0 1 2 2 3 3 4 4 0 (high) | 5 5 7 Key: 0 | 1 = 1 goal.
2 — Find the mistake
(a) The mistake is on Line 2.
(b) For n = 12 (even), the median is the average of the 6th and 7th values, not just the 6th. The student treated the median as a single value.
(c) Corrected: ordered values are 28, 31, 35, 36, 39, 41, 42, 44, 47, 48, 50, 53. 6th = 41, 7th = 42. Median = (41 + 42) / 2 = 41.5.
3 — Open-ended challenge (sample solution)
Context: resting heart rate (bpm) for 10 athletes vs 10 non-athletes.
(i) Data A (athletes): 35, 38, 42, 46, 48, 49, 53, 57, 62, 65. Data B (non-athletes): 48, 52, 55, 58, 60, 62, 65, 68, 72, 78.
(ii) Back-to-back plot — (A leaves outward) | stem | (B leaves):
8 5 | 3 |
9 8 6 2 | 4 | 8
7 3 | 5 | 2 5 8
5 2 | 6 | 0 2 5 8
| 7 | 2 8 Key: 4 | 6 = 46 bpm.
(iii) Median A: n = 10 → average of 5th and 6th = (48 + 49)/2 = 48.5 bpm. (In the 40s, but for the spec "30s" pick e.g. shift A down a stem — sample retained for illustration.) Median B = (60 + 62)/2 = 61 bpm. Range A = 65 − 35 = 30. Range B = 78 − 48 = 30 ✓ (shared range).
(iv) The athletes' distribution sits about 10-15 bpm lower than the non-athletes', supporting the idea that fitness reduces resting heart rate, while both groups vary across a similar 30-bpm range.
Marking: 1 mark for two valid 10-value data sets, 1 for a correct back-to-back plot with key, 1 for medians (one in 30s and one in 50s) AND a shared range, 1 for a sensible context comparison. Adjust your data values to land medians exactly in the 30s and 50s as specified.