Ordered stem-and-leaf diagrams
Preserve individual data values in an ordered diagram with a clear place-value key.
Core and Extended. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Order numbers.
- Understand decimal place value.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What does each leaf usually represent?
What you will learn
- Construct stems and ordered leaves.
- Recover values and summary measures from the diagram.
The key tells you how to reconstruct a value
For values 12,14,14,21,23,30, tens can be stems and units leaves. Write 1 | 2 4 4, 2 | 1 3 and 3 | 0, with key 1 | 2 means 12. Repeated values require repeated leaves; each leaf represents one observation.
Order both stems and leaves so the diagram can reveal median positions and extremes. Include empty intervening stems when useful for a consistent scale. A decimal example needs a different explicit key, such as 2 | 5 means 2.5 kg; the same visual marks can otherwise be ambiguous.
Read positions across rows, not row averages
The six-value example has central observations fourteen and twenty-one, so median is 17.5. Mode is fourteen and range is 30 − 12 = 18. Unlike a grouped interval table, this diagram preserves the exact recorded values.
Count leaves to check the sample size, then work through the ordered positions. A stem with many leaves has a high frequency, but the stem itself is not the mode. For back-to-back comparisons, use a shared key and order leaves correctly on each side; the key and individual values remain essential.
Pause and explain
For 12,14,14,21,23,30, what is the median?
Worked example
Construct a stem-and-leaf diagram for 12,14,14,21,23,30 and find the median.
Show the worked solution
- Use stems 1,2,3 with leaves 2 4 4; 1 3; 0.
- State key 1 | 2 = 12.
- The third and fourth values are 14 and 21, giving median 17.5.
Answer 1 | 2 4 4; 2 | 1 3; 3 | 0. Key 1 | 2 = 12; median 17.5.
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Write a stem-and-leaf row for 21,23,23,28 using tens as stems.
Give me a hint
Keep duplicate leaves.
Compare my reasoning
- The common stem is two.
- Ordered leaves are one, three, three, eight.
- Write 2 | 1 3 3 8 with key 2 | 1 = 21.
2 | 1 3 3 8; key 2 | 1 = 21.
Look for these in your work
- Ordered leaves.
- Retained duplicates and key.
A row is 4 | 2 5 5 9, with key 4 | 2 = 4.2 kg. Recover the values and range.
Give me a hint
The leaves represent tenths under this key.
Compare my reasoning
- Values are 4.2,4.5,4.5,4.9 kg.
- Maximum minus minimum is 4.9−4.2.
- Range is 0.7 kg.
4.2,4.5,4.5,4.9 kg; range 0.7 kg.
Look for these in your work
- Used the stated decimal key.
- Calculated spread in kilograms.
A student removes repeated leaves because 'the value is already shown'. Explain the effect.
Give me a hint
Repetitions are frequency information.
Compare my reasoning
- Each repeated leaf stands for a separate observation.
- Removing one changes the sample size and may change averages.
- Keep every observation, including duplicates.
The data frequencies would be lost; retain duplicate leaves.
Look for these in your work
- Recognised individual observations.
- Explained a consequence of deletion.
Common mistakes
- Omitting a key.
- Deleting duplicate leaves.
- Finding median from unordered leaves.
What can you explain now?
What essential label accompanies a stem-and-leaf diagram?
Compare with the explanation
A key showing the value represented by a stem–leaf pair.
Without a key, place value and units may be unclear.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Tomorrow, construct an ordered diagram and reconstruct its original list to verify it.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.