Stem-and-Leaf Diagrams –

Stem-and-leaf diagrams are tested on GCSE Maths papers at Foundation and Higher tier and appear across AQA, Edexcel and OCR specifications. They are a neat way to display raw data while keeping every individual value visible — unlike bar charts or histograms, no information is lost. Exam questions ask you to draw them, read off the median and mode, calculate the range, and compare two data sets using back-to-back diagrams. This guide covers all of these skills with worked examples and common pitfalls. For the full specification overview, see our complete GCSE Maths topics list.

What Is a Stem-and-Leaf Diagram?

A stem-and-leaf diagram splits each data value into a stem (all digits except the last) and a leaf (the final digit). The stems are listed in a column, and the leaves are written in ascending order next to their stem. A key must always be included to show what the stem and leaf represent.

Back-to-Back Stem-and-Leaf Diagrams

A back-to-back diagram places two data sets on the same stems — one set's leaves go to the right and the other's go to the left (in reverse order). This makes direct comparison easy.

Key Formulas

Step-by-Step Method

1. Choose the stems — usually the tens digits (e.g. 1, 2, 3 for data from 10 to 39).
2. Write the stems in a vertical column in ascending order.
3. Add the leaves in order from smallest to largest next to the correct stem.
4. Write a key — e.g. "3 | 5 means 35".
5. To find the median, count to the middle value using the ordered leaves.
6. To find the mode, look for the most frequently occurring leaf value.
7. To find the range, subtract the smallest value from the largest.

Worked Example 1 — Foundation Level

Question: Draw an ordered stem-and-leaf diagram for these test scores: 34, 28, 41, 35, 29, 47, 33, 38, 42, 31, 36, 45, 27, 39, 44.

Working:

Key: 2 | 7 means 27.

n = 15. Median position = (15 + 1) ÷ 2 = 8th value. Counting through the leaves: 27, 28, 29, 31, 33, 34, 35, 36. Median = 36.

Answer: Stem-and-leaf diagram as shown. Median = 36.

Worked Example 2 — Higher Level

Question: The back-to-back stem-and-leaf diagram below shows sprint times (seconds) for two groups.

Group A leaves | Stem | Group B leaves 9 7 5 | 11 | 2 4 8 6 4 3 | 12 | 1 5 7 8 7 2 1 | 13 | 0 3 6 9 5 | 14 | 2 8

Key: 5 | 11 means 11.5s (Group A), 11 | 2 means 11.2s (Group B).

Compare the two groups.

Working:

Group A: values range from 11.5 to 14.5. Median (8th of 11 values) = 12.6. Range = 14.5 − 11.5 = 3.0s.

Group B: values range from 11.2 to 14.8. Median (7th of 12 values) = 12.75. Range = 14.8 − 11.2 = 3.6s.

Answer: Group A has a slightly lower median (12.6s vs 12.75s), suggesting Group A is slightly faster on average. Group B has a larger range (3.6s vs 3.0s), meaning their times are more spread out and less consistent.

Worked Example 3 — Exam Style

Question: From the stem-and-leaf diagram in Example 1, find (a) the mode, (b) the range, (c) the interquartile range.

Working:

(a) Each value appears once — there is no mode (or you could say all values are equally frequent).

(b) Range = 47 − 27 = 20.

(c) n = 15. Q1 position = (15 + 1) ÷ 4 = 4th value = 31. Q3 position = 3(15 + 1) ÷ 4 = 12th value = 44. IQR = 44 − 31 = 13.

Answer: (a) No mode. (b) 20. (c) 13.

Common Mistakes

• Unordered leaves. Leaves must be written in ascending order — an unordered diagram will lose marks.
• Missing key. Without a key, the examiner cannot interpret your diagram. Always include one.
• Incorrect median position. For listed data use (n + 1) ÷ 2, not n ÷ 2 (that formula is for grouped cumulative frequency).
• Back-to-back errors. For the left-hand data set, leaves should increase as you move towards the stem (i.e. read from right to left).

Exam Tips

• Write an unordered diagram first if that is easier, then rewrite the leaves in order — but make sure your final version is ordered.
• For back-to-back diagrams, label which side is which clearly.
• If asked to compare, always make one statement about an average (usually median) and one about spread (range or IQR), using the context of the data.
• For more on averages and spread, see mean, median, mode and range. For grouped data, see frequency tables and grouped data.

Practice Questions

Q1 (Foundation): Draw a stem-and-leaf diagram for: 15, 22, 18, 31, 27, 24, 19, 33, 26, 21. Include a key and find the median.

Q2 (Foundation): From Q1, find the range and mode.

Q3 (Higher): Two data sets are shown in a back-to-back diagram. Set X has median 54 and range 28. Set Y has median 61 and range 18. Compare the two sets.

Practise stem-and-leaf diagrams and more for free on GCSEMathsAI.

Related Topics

• Mean, Median, Mode and Range
• Comparing Data Sets
• Frequency Tables and Grouped Data

Summary

A stem-and-leaf diagram displays raw data by splitting values into stems and leaves, keeping every data point visible. Leaves must be in ascending order and a key is essential. You can read the median, mode and range directly from the diagram. Back-to-back stem-and-leaf diagrams allow direct comparison of two data sets — always comment on an average and a measure of spread. Use (n + 1) ÷ 2 to find the median position for listed data.