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Stem-and-leaf diagrams

Stem-and-leaf diagrams

calendar_month 2025-09-05
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  • Unit 1: Probability
  • Unit 2: Data Collection
  • Unit 3: Interpreting and discussing results

🎯 In this topic you will

  • Draw and interpret stem-and-leaf diagrams
 

🧠 Key Words

  • mean
  • median
  • mode
  • range
  • stem-and-leaf diagram
Show Definitions
  • mean: The average of a set of numbers, found by adding them together and dividing by how many numbers there are.
  • median: The middle value in a set of numbers when they are arranged in order.
  • mode: The value that appears most often in a set of data.
  • range: The difference between the largest and smallest values in a set of data.
  • stem-and-leaf diagram: A way of organizing numbers to show their distribution by place value.
 

A stem-and-leaf diagram is a way of showing data in order of size.

When you draw a stem-and-leaf diagram, make sure:

  • you write the numbers in order of size from smallest to largest
  • you write a key to explain the numbers
  • you keep all the numbers in line vertically and horizontally
2 3   4   7   8
3 0   6   7   7   8
4 1   9   9

Key: 2 | 3 means 23 cm

 
Worked example

Here are the temperatures, in °C, recorded in 20 cities on one day.

9 19 26 35 6 17 32 21 30 16 14 16 18 29 27 8 25 32 21 32

a. Draw an ordered stem-and-leaf diagram to show this data.
b. How many cities had a temperature over 28 °C?
c. Use the stem-and-leaf diagram to work out
    i. the mode ii. the median iii. the range of the data.

Answer:

a. Key: 1 | 9 means 19°C

Unordered stem–and–leaf

0 | 9 6 8
1 | 9 7 6 4 6 8
2 | 6 1 9 7 5 1
3 | 5 2 0 2 2

Ordered stem–and–leaf

0 | 6 8 9
1 | 4 6 6 7 8 9
2 | 1 1 5 6 6 7 9
3 | 0 2 2 2 5

b.6 cities (29, 30, 32, 32, 32, 35 are over 28 °C).

c i. mode $= 32^\circ\text{C}$

c ii. median $= 21^\circ\text{C}$ (average of the 10th and 11th values, both 21)

c iii. range $= 35-6 = 29^\circ\text{C}$

How to draw an ordered stem-and-leaf diagram. Write a key. Use tens digits as stems (0–3) and ones digits as leaves. First list leaves as they appear, then rewrite each row in ascending order so stems line up vertically and leaves align neatly.

To answer questions: “over 28 °C” means leaves 9 and above in stem 2 and all leaves in stem 3. The mode is the most frequent value (32 appears most). For 20 values, the median is the mean of the 10th and 11th items. Range is highest minus lowest.

 

🧠 PROBLEM-SOLVING Strategy

Draw & Interpret Stem-and-Leaf Diagrams

Organise raw numbers fast, keep place value clear, and read off mode/median/range with minimal calculation.

  1. Choose stems & leaves.
    • Common: tens = stem, ones = leaf (e.g., 32 → 3 | 2).
    • For decimals, put whole part on the stem (e.g., 4.7 → 4 | 7 and add a Key).
  2. Draft, then order.
    • First make an unordered diagram: list leaves in the order given.
    • Then rewrite each row with leaves in ascending order → the ordered diagram.
  3. Write a clear Key.
    • Example: “2 | 3 means 23 cm” or “1 | 6 means 16 points”.
  4. Keep alignment tidy.
    • Stems in one vertical column; leaves spaced evenly. Don’t mix units in the leaves.
  5. Read statistics quickly.
    • Mode: longest row segment (most frequent leaf).
    • Median (n values): locate the middle position(s) ((n+1)/2 or average of n/2 & n/2+1).
    • Range = largest − smallest (use first & last values on diagram).
    • Counts above/below a threshold: scan stems/leaves crossing that value.
  6. Check & interpret.
    • Do you have all data values? Any impossible leaves?
    • Comment on shape: clusters, gaps, outliers, skew.
Mini examples
• Data: 14, 16, 16, 17, 18, 19 → Diagram stems 1; leaves 4 6 6 7 8 9 → Mode = 16, Median = (third & fourth) = (16+17)/2 = 16.5, Range = 19−14 = 5.
• “Over 28” with stems 2 & 3: count leaves ≥ 9 on stem 2 plus all leaves on stem 3.
Common slips
  • Missing/ambiguous Key (unclear place value or units).
  • Unordered leaves → hard to find median/mode quickly.
  • Mixing scales (e.g., using hundreds as stems for some values, tens for others).
  • Counting “≥ / >” thresholds incorrectly around boundaries.
Quick references
• Position of median (n values): k = (n+1)/2
• Range: max − min
• For decimals: make the Key explicit (e.g., “2 | 7 = 2.7 kg”)
 

❓ EXERCISE 16.3

1. Shen listed the playing times, to the nearest minute, of some CDs. He recorded the results in a stem-and-leaf diagram.

Key 4 | 5 means 45 minutes
4 | 5  5  7  9
5 | 0  2  5  6  8  9
6 | 1  2  4  6  7

a) How many CDs did Shen list?

b) What is the shortest playing time?

c) How many of the CDs had a playing time longer than 60 minutes?

d) Work out

i) the mode    ii) the median    iii) the range of the data.

👀 Show answer

a) 15 CDs (there are 4 + 6 + 5 leaves).

b) 45 minutes (leaf 5 on stem 4).

c) 5 CDs (all on stem 6: 61, 62, 64, 66, 67).

d)

i) Mode: 45 minutes (appears twice; all others once).

ii) Median: 56 minutes (8th value of 15 in order).

iii) Range: 67 − 45 = 22 minutes.

 

🧠 Think like a Mathematician

Task: Analyse a stem-and-leaf diagram showing the number of cups of coffee sold each day in one month. Work out missing details and answer questions about the data.

Stem-and-leaf diagram:

11 | 2 4 6 8
12 | 0 0 1 2 4 5 6 9
13 | 1 4 4 5 8 9
14 | 2 7 7 7
15 | 0 1 1 3 6 8

Key: 12 | 4 means 124 cups of coffee.

Questions:

  1. a i) Which month does the stem-and-leaf diagram represent?
  2. a ii) What is the largest number of cups of coffee sold on one day?
  3. a iii) What is the modal number of cups of coffee sold?
  4. b i) Were you able to answer all of the questions in part a?
  5. b ii) What is missing from the stem-and-leaf diagram?
  6. b iii) Can you still answer part a, even though something is missing?
👀 show answer
  • a i) The diagram has 28 data values, so it represents February in a leap year (28 days).
  • a ii) The largest value is 158 cups.
  • a iii) The mode is 127 (appears three times).
  • b i) Yes, you can answer the questions, but with caution.
  • b ii) The diagram is missing a title and key information about the month/year it represents.
  • b iii) Yes, you can still work out the answers to part a from the numbers alone, but the context (which month) has to be inferred from the number of data values.
 

❓ EXERCISES

3. These are the file sizes, in kilobytes (kB), of 30 files on Greg’s computer.

101 128 117 109 154 139 166 155 117 145 135 162 117 168 125
131 140 160 151 125 152 108 139 130 165 158 103 130 110 148

a) Draw an ordered stem-and-leaf diagram to show this data.

b) How many of the files are larger than 150 kB?

c) Which average, the mode or the median, better represents this data? Explain why.

d) Greg works out that the range in his file sizes is 65 kB. Is Greg correct? Explain your answer.

👀 Show answer

a) Ordered stem-and-leaf (tens as stems, ones as leaves). Key: 10 | 1 means 101 kB.

10 | 1  3  8  9
11 | 0  7  7  7
12 | 5  5  8
13 | 0  0  1  5  9  9
14 | 0  5  8
15 | 1  2  4  5  8
16 | 0  2  5  6  8

b) 10 files are larger than 150 kB (151, 152, 154, 155, 158, 160, 162, 165, 166, 168).

c) The median is a better measure of average for this set because the data are spread across a wide range and the mode (117 kB) is not representative of the centre. The median is 137 kB (the mean of the 15th and 16th ordered values, 135 and 139).

d) Greg is not correct. The range is max − min = 168 − 101 = 67 kB, not 65 kB.

 

🧠 Think like a Mathematician

Task: Decide if Opaline’s method is correct and suggest the best way to calculate the mean from a stem-and-leaf diagram.

Scenario: Opaline counted the number of birds in her garden each day for 10 days. She recorded the results in a stem-and-leaf diagram:

0 | 4 8 9
1 | 2 4 9
2 | 0 2 3 7

Key: 0 | 4 means 4 birds.

Opaline’s method: She worked out the mean of each row separately, then averaged those results. She got 15 birds as the overall mean.

Questions:

  1. a) Is Opaline’s method correct? Explain your answer.
  2. b) What is the best method to use to work out the mean from a stem-and-leaf diagram?
👀 show answer
  • a) Opaline’s method is not correct. She has averaged the three rows instead of all 10 data values. This gives each row equal weight, even though one row has 4 values and another has 3.
  • The correct mean is: Values = 4, 8, 9, 12, 14, 19, 20, 22, 23, 27. Sum = 158. Mean = 158 ÷ 10 = 15.8 birds.
  • b) The best method: - List out all the values from the stem-and-leaf diagram. - Add them up. - Divide by the total number of values. This ensures every data point is counted correctly.
  • Conclusion: Opaline’s answer (15) was close, but her method is flawed. The accurate mean is 15.8 birds.
 

❓ EXERCISES

5. Ashish counted the number of cars passing his school between 8.30 a.m. and 9 a.m. each day for 12 days. The stem-and-leaf diagram shows his results. Key: 0 | 1 means 1 car.

0 | 1  2  3  6  7  7  9
1 | 7
2 | 0  4  4  4

a) Use the stem-and-leaf diagram to work out

i) the mode    ii) the median    iii) the mean of the data.

b) Which average, the mode, median or mean, best represents this data? Explain why.

👀 Show answer

a)

i) Mode: 24 (appears three times).

ii) Median: order gives 1,2,3,6,7,7,9,17,20,24,24,24 → median is average of 6th and 7th = (7+9)/2 = 8.

iii) Mean: sum = 144, number of days = 12 → mean = 12 cars.

b) The median (8) best represents a typical day because the mean is pulled up by the three high values (24) and the mode 24 is not typical of most days.

6. The students in class 8B took a test (out of 40). The stem-and-leaf diagram shows their scores. Key: 1 | 8 means 18.

0 | 6  8  8  9  9
1 | 6
2 | 5  6  8  8  9
3 | 0  1  2  3  5  6  7  8  8
4 | 0  0  0

a) What percentage of the students had a score greater than 32?

b) What fraction of the students had a score less than 25%?

c) Any student scoring less than 40% must re-sit the test. How many students do not have to re-sit the test?

👀 Show answer

Total students = 23.

a) Scores > 32 are 33, 35, 36, 37, 38, 38, 40, 40, 40 → 9 students → percentage ≈ 39%.

b) 25% of 40 is 10. Scores < 10 are 6, 8, 8, 9, 9 → 5/23 of the class.

c) 40% of 40 is 16. Students with scores < 16 are the same 5 listed above, so students who do not re-sit = 23 − 5 = 18.

 

⚠️ Be careful!

  • Write a clear key: e.g., “2 | 3 means 23 cm”. Without a key, numbers can be misread (e.g., $2|3$ ≠ $2.3$ unless stated).
  • Order the leaves: within each stem, arrange leaves in ascending order. Don’t leave them in the original (unsorted) order.
  • Keep alignment neat: stems in one vertical column, leaves spaced evenly. Misalignment makes medians/modes easy to miscount.
  • Use consistent stems: choose stems (e.g., tens) and stick to them—even if some stems have no leaves (show an empty row or note it).
  • One digit per leaf: for two-digit data use tens|ones. For 3-digit data, pick stems as hundreds or tens consistently (e.g., 12|3 = 123 if stem = tens).
  • Handle repeats correctly: show duplicates with repeated leaves (e.g., three 32s → stem 3 | 2 2 2).
  • Units & context: include units in the key; don’t mix units (minutes with seconds, °C with °F).
  • “Over” vs “at least”: “over 28” means $\gt 28$; don’t count the leaf 8 on stem 2 (i.e., 28) by mistake.
  • Medians for even $n$: median is the average of the $n/2$ and $(n/2+1)$th ordered values, not one of them.
  • Range & outliers: range = $\max - \min$. Check you used the true smallest/largest values from the diagram.
  • Back-to-back plots: when comparing two groups, mirror the leaves about the stems and keep scales identical.
 

📘 What we've learned — Stem-and-Leaf Diagrams

  • What it is: A way to list data in order while keeping every original value. Each number = stem (leading digits) + leaf (last digit).
  • Always include a key: e.g., 2 | 3 means 23 (with units if needed).
  • How to draw: choose stems → write leaves as they appear (unordered) → rewrite each row in ascending order → keep columns aligned → add totals (optional).
  • Reading from the diagram:
    • Mode = most frequent leaf in a row (overall most frequent value).
    • Median = middle value (or mean of the two middles) after counting along the ordered leaves.
    • Range = largest − smallest (read last and first values).
    • Mean = add all values represented and divide by the count.
  • Counting conditions: “> 28” means stems/leaves strictly above 28; “≥ 28” includes the leaf 8 on stem 2.
  • Design tips: pick sensible stems (e.g., tens), use back-to-back diagrams to compare two groups, show duplicates by repeating leaves.
  • Common slips: missing key, not ordering leaves, misreading units, forgetting that each leaf is a whole data value.
Quick example: Data 6, 8, 9, 14, 16, 16, 17, 18, 19, 21 →
0 | 6 8 9
1 | 4 6 6 7 8 9
2 | 1    Key: 1 | 4 means 14
Mode = 16, Median = average of 5th & 6th (=16), Range = 21 − 6 = 15.
Checklist: Key ✓ Ordered leaves ✓ Aligned stems/leaves ✓ Units noted ✓ Mode/Median/Range correct ✓