Back-to-back stem-and-leaf diagrams
Back-to-back stem-and-leaf diagrams
- Unit 1: Probability
- Unit 2: Data Collection
- Unit 3: Interpreting and discussing results
🎯 In this topic you will
- Draw and interpret back-to-back stem-and-leaf diagrams
🧠 Key Words
- back-to-back stem-and-leaf diagram
- stem-and-leaf diagram
Show Definitions
- back-to-back stem-and-leaf diagram: A diagram that compares two sets of data using a shared stem, with one set shown on the left and the other on the right.
- stem-and-leaf diagram: A way of organizing numbers to display their distribution by place value.
You already know how to use ordered stem-and-leaf diagrams to display one set of data. You can use a back-to-back stem-and-leaf diagram to display two sets of data.
In a back-to-back stem-and-leaf diagram, you write one set of data with its ‘leaves’ to the right of the stem. Then you write the second set of data with its ‘leaves’ to the left of the stem. Both sets of numbers count from the stem, so you write the second set of numbers ‘backwards’.
🔎 Reasoning Tip
Remember, when you draw an ordered stem-and-leaf diagram, you should:
- Write the numbers in order of size, smallest nearest the stem
- Write a key to explain what the numbers mean
- Keep all the numbers in line, vertically and horizontally
❓ EXERCISE 15.3
1. The ages of 16 people in two different clothes shops, A and B, are shown:
| Shop A | ||||||||
|---|---|---|---|---|---|---|---|---|
| 9 | 30 | 18 | 12 | 8 | 29 | 23 | 16 | |
| 24 | 14 | 31 | 17 | 21 | 17 | 10 | 19 |
| Shop B | ||||||||
|---|---|---|---|---|---|---|---|---|
| 36 | 33 | 29 | 25 | 8 | 32 | 35 | 19 | |
| 24 | 36 | 30 | 19 | 31 | 27 | 32 | 27 |
a) Copy and complete the unordered back-to-back stem-and-leaf diagram to show this data. (Use ones as leaves; stems are tens.)
b) Copy and complete the ordered back-to-back stem-and-leaf diagram to show this data.
c) Check your stem-and-leaf diagram is correct by comparing it with another learner’s diagram. If your diagrams are not the same, try to find the mistake.
d) Use your stem-and-leaf diagram to answer these questions:
i) Which shop has the younger shoppers?
ii) Which shop has the older shoppers?
e) Make one conclusion about the types of clothes sold in the two different shops.
👀 Show answer
a) & b) Back-to-back stem-and-leaf diagrams required (unordered then ordered).
d)
i) Shop A has the younger shoppers (many teens and early 20s, plus 8–10).
ii) Shop B has the older shoppers (several in the 30s; only one single-digit age).
e) A reasonable conclusion: Shop A likely sells styles for younger customers (teen/young adult), while Shop B targets adults (late 20s–30s), which fits the age distributions.
🧠 Think like a Mathematician
Task: Compare ice-cream sales at the Beach and City car parks using a back-to-back stem-and-leaf diagram and summary measures.
Data (14 days each)
- Beach: 56, 46, 60, 47, 57, 46, 62, 60, 57, 45, 61, 46, 59, 62
- City: 68, 54, 45, 45, 56, 30, 69, 39, 42, 45, 59, 68, 47, 34
a) Back-to-back stem-and-leaf (stem = tens digit; leaves are ones)
Beach Stem City
3 | 0 4 9
7 6 6 6 5 4 | 2 5 5 5 7
9 7 7 6 5 | 4 6 9
2 2 1 0 0 6 | 8 8 9
b) For each set, find i) mode ii) median iii) range
- Beach (sorted: 45,46,46,46,47,56,57,57,59,60,60,61,62,62)
- Mode: 46
- Median: average of 7th & 8th = (57+57)/2 = 57
- Range: 62 − 45 = 17
- City (sorted: 30,34,39,42,45,45,45,47,54,56,59,68,68,69)
- Mode: 45
- Median: average of 7th & 8th = (45+47)/2 = 46
- Range: 69 − 30 = 39
c) Compare & comment
- Typical sales are higher at the Beach (median 57 vs 46).
- The City has far more variability (range 39 vs 17) with several very low days (30–34) but also a slightly higher maximum (69).
d) Antonino says sales are better at the City car park. Do you agree?
Disagree. Although City has the single highest day, the Beach has a higher median and consistently higher middle values, so overall performance is better at the Beach.
❓ EXERCISES
3. The stem-and-leaf diagram shows the times taken by the students in a Stage 9 class to run 100 m.
Key: For the boys’ times, 1 | 1 means 15.1 seconds. For the girls’ times, 15 | 9 means 15.9 seconds.
a) For each set of times, work out
i) the mode ii) the median iii) the range iv) the mean.
b) Compare and comment on the times taken by the boys and the girls to run 100 m.
c) Zara says: “The girls are faster than the boys, as their mode is higher.”
Do you agree? Explain your answer.
👀 Show answer
a)
Boys (15 values): mode = 17.4 s; median = 16.3 s; range = 18.0 − 15.1 = 2.9 s; mean ≈ 16.6 s.
Girls (16 values): mode = 16.8 s; median ≈ 17.6 s; range = 19.9 − 15.9 = 4.0 s; mean ≈ 17.7 s.
b) Boys are generally faster: both their median (16.3 s) and mean (≈16.6 s) are lower than the girls’ (≈17.6 s and ≈17.7 s). The boys’ times are also less spread out (smaller range).
c) Do not agree. A “higher” mode means a larger time, which is slower, not faster. Also, mode is not the best measure to compare running times here. The boys have lower median and mean times, so they are faster overall. (Girls’ mode 16.8 s is actually lower than the boys’ mode 17.4 s.)
4. The stem-and-leaf diagram shows the mass of 12 desert hedgehogs in two different locations.
Keys: Location A: 4|39 → 394 g. Location B: 38|0 → 380 g.
Use the diagram to answer the questions. (Count the leaves on the relevant rows.)
a) What fraction of the hedgehogs from each location had a mass less than 400 g?
b) What percentage of the hedgehogs from each location had a mass greater than 415 g?
c) Which location, A or B, had the most variation in the mass of the hedgehogs?
d) Work out the mean and median mass of hedgehogs for each location.
e) Which location, A or B, do you think has more food available for the hedgehogs to eat? Explain your answer.
👀 Show answer
How we read the diagram: each leaf is a single hedgehog. Masses below $400$ g are on the 38–39 rows; masses above $415$ g are the leaves $16$–$19$ on the 41 row and all of the 42 row.
a) Location A: leaves < $400$ g = 3 out of 12 → fraction $\dfrac{3}{12}=\dfrac{1}{4}$.
Location B: leaves < $400$ g = 8 out of 12 → fraction $\dfrac{8}{12}=\dfrac{2}{3}$.
b) Location A: masses $>415$ g = 3 out of 12 → $ \dfrac{3}{12}\times 100\% = 25\%$.
Location B: masses $>415$ g = 0 out of 12 → $0\%$.
c) The ranges are similar: Location A (about $394$–$425$ g) and Location B ($380$–$413$ g) both span roughly $31$ g. So the variation is about the same (A is heavier overall, B includes lighter masses).
d) Location A: median lies between the 6th and 7th ordered values (both around low–mid $410$ g) → median $\approx 410$–$412$ g; mean is a little above $410$ g (heavier values in the $420$’s raise it slightly).
Location B: median lies between the 6th and 7th values (around $395$–$398$ g) → median $\approx 396$–$397$ g; mean $\approx 396$ g. (Exact medians/means depend on the exact leaves you read from the diagram.)
e)Location A. The masses there are generally higher (more hedgehogs at $410$–$425$ g), suggesting better feeding conditions and more available food.
🧠 Think like a Mathematician
Task: Analyse the number of website hits for Website A and Website B using a back-to-back stem-and-leaf diagram and measures of consistency.
Data (21 days each)
- Website A: 141, 152, 134, 161, 130, 153, 142, 130, 158, 159, 145, 133, 145, 147, 145, 148, 153, 155, 146, 160, 152
- Website B: 134, 129, 145, 156, 145, 128, 138, 160, 136, 146, 154, 146, 157, 145, 148, 158, 169, 157, 168, 155, 167
a) Back-to-back stem-and-leaf (tens = stem)
Website A Stem Website B
12 | 8 9
0 0 13 | 4 6 8
3 4 5 5 5 7 8 14 | 2 5 5 5 5 5 6 6 6 7 8
1 2 3 6 7 8 15 | 2 4 5 5 7 7 8
2 3 6 16 | 0 7 8 9
1 0 17 |
b) Compare and comment
- Website A: range = 161 − 130 = 31, median ≈ 146, mode = 145, values clustered tightly.
- Website B: range = 169 − 128 = 41, median ≈ 154, mode = 145, values more spread out, with higher maximums.
- Website A is more consistent; Website B shows greater variation and some higher spikes.
c) Marcus’s claim: “Website A is better because it’s more consistent.”
Partly agree. Website A is indeed more consistent (smaller range, tighter cluster). However, Website B has a higher median and larger peak values, so it may be considered “better” if higher traffic days are more important than consistency.
⚠️ Be careful!
- Two clear keys: give a key for each side (Group A and Group B) with units (e.g., “2 | 3 = 23 marks”).
- Same stems for both groups: choose stems (e.g., tens) that cover all values; include empty stems if needed to keep scale continuous.
- Ordering direction: on the right side, leaves increase left→right; on the left side, leaves increase towards the stem (so they appear right→left).
- Line up perfectly: one central stem column; keep spacing consistent so medians and comparisons are easy to read.
- One digit per leaf: don’t bunch digits (e.g., write three 26s as 2 | 6 6 6), and show repeats explicitly.
- No mixing of units: don’t combine minutes with seconds or cm with mm; state units in the keys.
- Sort each side: make each group’s leaves ordered; unordered leaves make modes/medians easy to misread.
- Back-to-back ≠ merged: don’t compute one mean/median from both sides combined unless that’s the intention.
- Compare like with like: use the same stems and formatting when comparing ranges, medians, and shapes (clusters/gaps) across groups.
- Edge cases: for 3-digit data, pick stems consistently (hundreds or tens) and stick with it (e.g., 12 | 3 = 123 if stem=tens).
- Read “greater than” correctly: “> 28” excludes 28; count only leaves strictly above.
- Check totals: count leaves on each side to match the stated sample sizes before calculating averages.