Stem-and-Leaf Diagrams

Draw an ordered stem-and-leaf diagram with a key, read the median, mode and range from it, and read a back-to-back diagram.

In the exam for Cambridge IGCSE Maths 0580 · 37 real past-paper questions below

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What a Stem-and-Leaf Diagram Is

A stem-and-leaf diagram shows every value, in order. Each number is split into two parts.

Part	For 23
Stem: the tens digit	2
Leaf: the units digit	3

Rule

Always give a key, like $2 ∣ 3$ means 23. Without it, the reader cannot tell 23 from 2.3.

Read the key first. Exams change it, and then the stem and leaf mean something new.

Worked example

1
$Key: 0 ∣ 6 = 0.6$ — Here the stem is the units and the leaf is the tenths. So the key is for decimals.
2
$2 ∣ 4 = 2.4$ — Same key, another row. Stem 2 and leaf 4 make 2.4, not 24.
3
$Key: 15 ∣ 2 = 152$ — A stem can have two digits. Here 15 is the hundreds and tens, and the leaf is still the units.
4
$0 ∣ 8 = 8$ — With a key like 1 | 2 = 12, a one-digit value such as 8 goes on stem 0. It has no tens, so its tens digit is 0.

Key idea

The leaves in each row go in order, smallest first. So the whole diagram is a sorted list.

Drawing One From a List

The 2 or 3-mark question: “Draw an ordered stem-and-leaf diagram.” These are 15 test scores.

23, 41, 17, 35, 28, 32, 19, 44, 26, 35, 31, 22, 38, 35, 29

Worked example

1
$stems: 1, 2, 3, 4$ — The smallest score is 17 and the largest is 44. So the tens digits run from 1 to 4. Write each stem once, down the left.
2
$1 ∣ 79$ — Stem 1 takes 17 and 19. Their units digits, 7 and 9, are the leaves, smallest first.
3
$2 ∣ 23689$ — Stem 2 takes 23, 28, 26, 22 and 29. Write their units digits, then sort them so the row is in order.
4
$3 ∣ 125558$ — 35 is in the list three times, so leaf 5 goes in three times. Every value needs its own leaf.
5
$4 ∣ 14$ — Stem 4 takes 41 and 44. Every score now has a place.
6
$Key: 2 ∣ 3 = 23$ — Add a key so the reader knows a leaf is a units digit. Some questions give a mark just for the key.
7
$2 + 5 + 6 + 2 = 15$ — Count the leaves in each row and add. 15 leaves for 15 scores, so nothing is missing or written twice.

Key idea

Marks: a full, ordered diagram scores 2. A diagram with the leaves out of order, or with one leaf missing, scores only 1. Tick each value off the list as you place it, then sort each row.

Common mistake

Leaving out the key, leaving the leaves unordered, or writing a repeated value only once. Count the leaves at the end.

Median, Mode and Range From the Diagram

The 1 or 2-mark question: “Find the median” or “Work out the range”. The diagram is already in order, so you just count along the leaves.

Formula

median position = \frac{n + 1}{2}

$n$ is the number of values, which is the number of leaves. The answer tells you which value to count to. It is a position, not the median itself.

Worked example

1
$n = 15$ — Count the leaves in the diagram you just drew: 2 + 5 + 6 + 2 = 15 values.
2
$\frac{15 + 1}{2} = 8 th value$ — Put n = 15 into the formula. The median is the 8th value in order.
3
$17, 19, 22, 23, 26, 28, 29, 31$ — Count along the leaves from the top row, putting the stem back on each one. The 8th value is 31, so the median is 31.
4
$mode = 35$ — The mode is the value that appears most often. Leaf 5 appears three times on stem 3, and no other value repeats that often.
5
$44 - 17 = 27$ — Range = largest − smallest. The largest is the last leaf of the last row, 44. The smallest is the first leaf, 17.

Common mistake

Reading a leaf on its own. The last leaf, 4, means 44, and the first means 17. Put the stem back on before you work anything out.

Key idea

If no value repeats, there is no mode. If two values tie for most often, both are modes.

Median of an Even Number of Values

Many exam diagrams have an even number of values, like 12, 16 or 20. Then there is no single middle leaf. Here is what to do.

12 values. Key: 1 | 4 means 14

Stem	Leaves
1	4 6 9
2	0 3 5 7 8
3	2 2 6
4	1

Worked example

1
$3 + 5 + 3 + 1 = 12$ — Count the leaves in each row and add. There are 12 values, so n = 12.
2
$\frac{12 + 1}{2} = 6.5$ — 6.5 is not a whole position. It means halfway between the 6th and 7th values.
3
$14, 16, 19, 20, 23, 25, 27$ — Count along the leaves to the 6th and 7th values. They are 25 and 27.
4
$\frac{25 + 27}{2} = 26$ — Add the two middle values and halve. The median is 26, even though 26 is not in the diagram.

Common mistake

Giving 6.5 as the median. 6.5 is the position. The median is the value found there, 26.

More Than, Fractions and Percentages

Exams also ask “How many are more than …?”, then for a fraction or a percentage. The diagram is in order, so the values you want sit together.

Formula

percentage = \frac{count}{total} \times 100

count is how many leaves fit the condition. total is the number of leaves in the whole diagram.

Use the 12 values from the last card. What percentage are more than 30?

Worked example

1
$more than 30$ — More than 30 means 31 and above. A value of exactly 30 would not count.
2
$32, 32, 36, 41$ — Find the first leaf past 30 and count every leaf after it. That is 4 values.
3
$\frac{4}{12}$ — Fraction = count over the total of 12 leaves.
4
$= \frac{1}{3}$ — Divide top and bottom by 4. If the question asks for a fraction, give it in its simplest form.
5
$\frac{4}{12} \times 100 = 33.3%$ — For a percentage, multiply the fraction by 100. 33.333… rounds to 33.3% to 3 significant figures.

Common mistake

Counting the boundary value. “More than 30” and “less than 30” leave out 30. “At least 30” keeps it in.

Key idea

Write the fraction before the percentage. It earns a mark for working even if the last step slips.

Back-to-Back Diagrams

The 2-mark question compares two groups with one diagram. The stems are in the middle. One group's leaves go right, the other group's go left.

Worked example

1
$85 ∣ 3 ∣ 16$ — One row of a back-to-back diagram. The stem 3 is in the middle, one group on each side.
2
$left: 35, 38$ — Left leaves grow outwards from the stem, so read them from the stem: 5, then 8. Stem first gives 35 and 38.
3
$right: 31, 36$ — Right leaves read left to right, as in a normal diagram. Stem first gives 31 and 36.

Common mistake

Reading the left side as 53 and 83. The stem always comes first: 5 next to stem 3 is 35.

Key idea

To compare the two sides, find each median and range, then write two comparisons. See Comparing Two Sets of Data.

Finding Missing Leaves

A harder question hides some leaves as letters and gives you the range, median and mean. Use one fact at a time to find each letter.

10 values, in order. Key: 1 | 2 means 12. Range 25, median 17, mean 17.1

Stem	Leaves
0	4 A
1	2 5 B 8
2	1 3 6 C

Worked example

1
$(20 + C) - 4 = 25$ — Range = largest − smallest. The smallest is the first leaf, 4. The largest is the last leaf, which is 20 + C.
2
$20 + C = 29$ — Add 4 to both sides, so the largest value is alone.
3
$C = 9$ — Take 20 from both sides. The largest value is 29.
4
$\frac{( 10 + B ) + 18}{2} = 17$ — 10 values, so the median is halfway between the 5th and 6th. Count along: the 5th is 10 + B and the 6th is 18.
5
$28 + B = 34$ — Multiply both sides by 2, and add 10 + 18 = 28.
6
$B = 6$ — Take 28 from both sides. The 5th value is 16.
7
$total = 17.1 \times 10 = 171$ — Mean = total ÷ number of values, so total = mean × number of values.
8
$4 + A + 12 + 15 + 16 + 18 + 21 + 23 + 26 + 29 = 171$ — Write every value as a sum, using B = 6 and C = 9.
9
$164 + A = 171$ — Add the numbers you know: they make 164.
10
$A = 7$ — Take 164 from both sides. Check: 7 comes after 4 in row 0, so the leaves are still in order.

Key idea

Write each equation down. Each letter is worth marks on its own, and the working earns marks even if one answer is wrong.

Which Method? Read the Question

Remember: the diagram is a sorted list.

You see	You do
Draw an ordered diagram	stems, leaves in order, a key
Median	count to the middle leaf
Median, even number of values	mean of the two middle values
Mode	the leaf that repeats most, with its stem
Range	last value take away first
More than / less than, fraction, %	count the leaves, then over the total
Letters for missing leaves	use the range, median and mean in turn
Leaves on both sides	left side reads outwards from the stem
Mean of a list	Mean, Median, Mode and Range
Quartiles and IQR (Extended)	Finding the Median and Quartiles
Compare two groups	Comparing Two Sets of Data

Key idea

Before you answer, check the number of leaves matches the number of values.

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