Stem-and-Leaf Diagrams

Each value split into a stem and a leaf: a diagram that shows the shape of the data and keeps every value, for one set of data or two side by side.

What You Should Already Know

A dot diagram (lesson 12.1) shows every value as a dot on a number line. When the values are spread over a wide range, such as test marks from 23 to 45, its line becomes long and most columns hold only one dot.
Place value: in 47 the tens digit is 4 and the ones digit is 7; in 142 the first two digits, 14, are the number of tens.

Stems and Leaves

A stem-and-leaf diagram splits each value into two parts. The leaf is the last digit. The stem is all the digits before it. Values with the same stem are written in one row, so the rows show how the data are spread, and every value can still be read back.

Value	Stem	Leaf	Written
47	4	7	4 \| 7
8	0	8	0 \| 8
142 cm	14	2	14 \| 2
4.7 s (leaf = tenths)	4	7	4 \| 7

Look at the first and last rows: 47 and 4.7 are both written 4 | 7. Only the key tells the reader which is meant, so a stem-and-leaf diagram always has one.

Stem-and-leaf diagram

Each value is split into a stem (all its digits but the last) and a leaf (its last digit). Every leaf is a single digit.

The stems are written in order, smallest at the top, in a column to the left of a vertical line. Every stem from the smallest to the largest is written, even one with no leaves.

The leaves of each stem are written in a row to the right of the line, in order from smallest to largest, equally spaced so that the rows can be compared by length.

A key with the unit shows how to read the diagram: "Key: 14 | 2 means 142 cm".

Drawing a Stem-and-Leaf Diagram

Steps

1. Find the smallest and largest values, and so the first and last stems.

2. Write every stem from the first to the last in a column, and draw the line.

3. Go through the data in the order given and write each leaf in its stem's row.

4. Redraw the diagram with the leaves of each row in order from smallest to largest.

5. Add the key, and check that the number of leaves is the number of values.

Example 1: Drawing from data that are not in order

The marks, out of 50, of 20 pupils in a Mathematics test were: 23, 35, 41, 28, 33, 45, 38, 27, 42, 35, 31, 44, 29, 36, 40, 33, 38, 26, 42, 35. Draw a stem-and-leaf diagram of the marks.

Solution

The smallest mark is 23 and the largest is 45, so the stems are the tens digits 2, 3 and 4.

Going through the marks in order: 23 puts 3 in row 2, 35 puts 5 in row 3, 41 puts 1 in row 4, and so on.

Now put each row's leaves in order and check the count: $5 + 9 + 6 = 20$ leaves.

The finished diagram is the second one above, with its leaves in order and the key "2 | 3 means 23 marks".

Example 2: Three-digit values and an empty stem

The masses, in grams, of 15 mangoes were: 212, 245, 238, 251, 229, 266, 204, 248, 233, 287, 259, 241, 225, 263, 236. (a) Draw a stem-and-leaf diagram of the masses. (b) How many mangoes have a mass of at least 250 g?

Solution

(a) The values have three digits, so the stem is the first two digits and the leaf the last: 212 is 21 | 2.

The smallest mass is 204 g (stem 20) and the largest is 287 g (stem 28), so the stems are 20, 21, 22, …, 28.

No mango has a mass from 270 g to 279 g, but stem 27 is still written, with no leaves. Leaving it out would put 287 g straight under the 260s, as if it were close to them.

Check: $1 + 1 + 2 + 3 + 3 + 2 + 2 + 0 + 1 = 15$ leaves.

(b) At least 250 g: the leaves of stems 25, 26, 27 and 28: 251, 259, 263, 266, 287. That is 5 mangoes.

(a) Drawn above, with stems 20 to 28 and the key "20 | 4 means 204 g". (b) 5 mangoes.

Reading a Stem-and-Leaf Diagram

Each leaf is one value, read with the key. The smallest value is the first leaf of the top row and the largest is the last leaf of the bottom row. Count leaves to answer "how many".

Example 3: Reading decimal data

The stem-and-leaf diagram shows the times, in seconds, taken by 16 pupils to run 100 m. (a) What were the fastest and the slowest times? Find the range. (b) How many pupils took less than 15 s? (c) What percentage of the pupils took at least 16 s?

Solution

The key says 13 | 2 means 13.2 s: the stem is the whole number of seconds and the leaf is the tenths.

(a) The fastest is the first leaf: 13 | 2, that is 13.2 s. The slowest is the last leaf: 17 | 9, that is 17.9 s. Range $= 17.9 - 13.2 = 4.7$ s.

(b) Less than 15 s: every leaf on stems 13 and 14, $2 + 4 = 6$ pupils.

(c) At least 16 s: every leaf on stems 16 and 17, $3 + 2 = 5$ pupils. Percentage $= \frac{5}{16} \times 100\% = 31.25\%\text{.}$

(a) Fastest 13.2 s, slowest 17.9 s; range 4.7 s. (b) 6 pupils. (c) 31.25%

Example 4: A missing leaf

The stem-and-leaf diagram shows the masses, in kg, of the 12 members of a school wushu team. One leaf is shown as $k$ . (a) The range of the masses is 23 kg. Find $k$ . (b) How many members have a mass of at least 50 kg?

Solution

(a) The lightest member is 38 kg. The heaviest is the leaf on stem 6, which is $60 + k$ kg.

Range $=$ largest $-$ smallest: $(60 + k) - 38 = 23$ , so $22 + k = 23$ and $k = 1$ .

(b) At least 50 kg: the 4 leaves on stem 5 and the 1 leaf on stem 6, $4 + 1 = 5$ members.

(a) $k = 1$ (the heaviest member is 61 kg). (b) 5 members.

Back-to-Back Stem-and-Leaf Diagrams

Two sets of data can share one column of stems: one set's leaves go to the right of the stems and the other's to the left. On the left the leaves are read outwards from the stem, so the smallest leaf is next to the stem, and a left leaf of 5 next to stem 3 is 35. Each side has its own key.

Example 5: Comparing two classes

The back-to-back stem-and-leaf diagram shows the marks, out of 50, of 15 pupils of Class 2A and 15 pupils of Class 2B in the same test. (a) Write down the lowest and highest marks of each class. (b) How many pupils in each class scored at least 40? How many scored below 30? (c) Which class did better in the test? Give a reason.

Solution

(a) Class 2A (read outwards on the left): the lowest is 21 (leaf 1 on stem 2) and the highest 47. Class 2B: the lowest is 18 and the highest 45.

(b) At least 40, the leaves on stem 4: Class 2A has 5 (40, 41, 43, 44, 47) and Class 2B has 2 (42, 45).

Below 30, the leaves on stems 1 and 2: Class 2A has 3, Class 2B has $2 + 6 = 8$ .

(c) Class 2A's leaves are bunched on stems 3 and 4 (12 of its 15 marks are 30 or more), while more than half of Class 2B's marks (8 of 15) are below 30.

(a) Class 2A: 21 and 47; Class 2B: 18 and 45. (b) At least 40: 5 in Class 2A, 2 in Class 2B; below 30: 3 in Class 2A, 8 in Class 2B. (c) Class 2A did better: 12 of its 15 marks are 30 or more, against 7 of 15 in Class 2B.

Split Stems

When a row has many leaves, each stem can be written twice: the first row takes the leaves 0 to 4 and the second the leaves 5 to 9. The diagram then shows the data in steps of 5 instead of 10, and the shape can show more.

Example 6: Splitting the stems

The stem-and-leaf diagram shows the heights of 40 pupils. (a) Which stem has the most leaves? (b) Redraw the diagram with split stems. Which interval of 5 cm holds the most heights?

Solution

(a) The rows have $3$ , $14$ , $16$ and $7$ leaves. Stem 16 has the most: more heights lie from 160 cm to 169 cm than in any other interval of 10 cm.

(b) Each stem is written twice: leaves 0 to 4 on the first row, 5 to 9 on the second.

The rows now have $2$ , $1$ , $4$ , $10$ , $8$ , $8$ , $3$ and $4$ leaves. The longest row is the second row of stem 15: heights from 155 cm to 159 cm.

The split shows what the first diagram hid: the 160s are spread evenly over two rows of 8, while the 150s are crowded into their upper half.

(a) Stem 16 (16 leaves). (b) With split stems, 155 cm to 159 cm holds the most heights (10 pupils).

Spotting Errors

Example 7: What is wrong?

Diagram A shows the numbers of sit-ups done by 8 pupils: 18, 12, 15, 26, 21, 24, 43, 40. Diagram B shows the masses of 6 apples: 142, 147, 151, 156, 158, 163 g. What is wrong with each diagram? Draw each correctly.

Diagram A:

Diagram B:

Solution

Diagram A: the leaves are not in order; stem 3 is missing, so 40 and 43 look as if they follow straight after the 20s; and there is no key.

Diagram B: the leaves have two digits. A leaf is only the last digit, so the stems are 14, 15 and 16. As drawn, every value sits in one long row and the diagram shows nothing of the shape.

Diagram A: leaves out of order, stem 3 left out and no key. Diagram B: leaves of two digits; the stems should be 14, 15 and 16.

Common Mistakes

Leaving the leaves in the order of the data. In the finished diagram each row is in order from smallest to largest.
Leaving out the key, or giving it without the unit. Without a key, 4 | 7 could be 47 or 4.7.
Leaving out a stem with no leaves. Every stem from the smallest to the largest is written.
Writing a leaf of two digits (1 | 42). The leaf is the last digit only: 142 is 14 | 2.
Spacing the leaves unevenly, so that a long row looks short. Every leaf takes the same width.
Reading the left side of a back-to-back diagram the wrong way: a left leaf 5 next to stem 3 is 35, not 53.

Practice Questions

Try each question first, then open the solution to check your working.

Question 1

Write each value as stem | leaf. (a) 63 (b) 5 (c) 158 cm (d) 7.4 kg, with the leaf as the tenths. (e) A key says "12 | 5 means 125 cm". What does 13 | 0 mean?

ShowHide solution

(a) The leaf is the last digit, 3; the stem is 6: 6 | 3.

(b) 5 has no tens, so its stem is 0: 0 | 5.

(d) The stem is the whole number and the leaf the tenths: 7 | 4.

(e) The stem gives the tens of centimetres and the leaf the ones: 130 cm.

(a) 6 | 3 (b) 0 | 5 (c) 15 | 8 (d) 7 | 4 (e) 130 cm

Question 2

The waiting times, in minutes, of 18 commuters at a bus stop were: 7, 12, 25, 18, 9, 14, 31, 22, 16, 11, 28, 15, 19, 8, 24, 13, 20, 17. (a) Draw a stem-and-leaf diagram of the times. (b) How many commuters waited at least 20 minutes?

ShowHide solution

(a) The times go from 7 to 31, so the stems are 0, 1, 2 and 3. A time under 10 minutes has stem 0: 7 is 0 | 7.

Record the leaves in the order given, then sort each row, and add the key.

Check: $3 + 9 + 5 + 1 = 18$ leaves.

(b) The leaves on stems 2 and 3: $5 + 1 = 6$ commuters.

(a) Drawn above, with the key "0 | 7 means 7 minutes". (b) 6 commuters.

Question 3

The stem-and-leaf diagram shows the waiting times at a polyclinic. (a) How many patients are shown? (b) What was the longest waiting time? (c) How many patients waited less than 15 minutes?

ShowHide solution

(a) Count the leaves: $3 + 6 + 5 + 2 = 16$ patients.

(b) The last leaf of the last row: 3 | 5, which is 35 minutes.

(a) 16 (b) 35 minutes (c) 5 patients

Question 4

The numbers of visitors to a museum on 12 days were: 341, 312, 368, 304, 329, 349, 318, 371, 336, 325, 362, 344. (a) Draw a stem-and-leaf diagram of the data. (b) On how many days were there more than 340 visitors?

ShowHide solution

(a) The values have three digits, so the stems are the first two digits: 30 to 37. No day had 350 to 359 visitors, but stem 35 is still written.

(b) More than 340: 341, 344, 349, 362, 368 and 371, which is 6 days.

(a) Drawn above, stems 30 to 37 with stem 35 empty, key "30 | 4 means 304 visitors". (b) 6 days.

Question 5

The stem-and-leaf diagram shows the masses, in kg, of 15 babies born at a hospital on one day. (a) How many babies had a mass of less than 3.0 kg? (b) Find the range of the masses. (c) What percentage of the babies had a mass of at least 3.5 kg?

ShowHide solution

(a) The leaves on stem 2: 2.6, 2.8 and 2.9 kg, which is 3 babies.

(b) The lightest is 2.6 kg and the heaviest 4.2 kg. Range $= 4.2 - 2.6 = 1.6$ kg.

(c) At least 3.5 kg: 3.5, 3.6, 3.8, 3.9, 4.0 and 4.2, which is 6 babies. Percentage $= \frac{6}{15} \times 100\% = 40\%\text{.}$

(a) 3 (b) 1.6 kg (c) 40%

Question 6

The stem-and-leaf diagram shows the heights, in cm, of 11 plants in a school garden. One leaf is shown as $p$ . (a) The range of the heights is 35 cm. Find $p$ . (b) How many plants are shorter than 25 cm?

ShowHide solution

(a) The tallest plant is 46 cm. The shortest is the first leaf of stem 1: $10 + p$ cm.

$46 - (10 + p) = 35$ , so $36 - p = 35$ and $p = 1$ .

(b) Shorter than 25 cm: 11, 15, 18, 20, 23 and 23, which is 6 plants.

(a) $p = 1$ (b) 6 plants

Question 7

The back-to-back stem-and-leaf diagram shows how many push-ups 12 girls and 12 boys did in one minute. (a) What number does the leaf 9 next to stem 1 on the girls' side stand for? (b) How many girls and how many boys did at least 30 push-ups? (c) Find the range for each group. (d) Which group generally did more push-ups? Give a reason.

ShowHide solution

(a) On the left, a leaf is read with the stem after it: leaf 9 and stem 1 make 19 push-ups.

(b) At least 30, the leaves on stems 3 and 4: girls 3 (30, 32, 35); boys $5 + 2 = 7$ .

(d) Most of the boys' leaves lie on stems 3 and 4, and most of the girls' on stems 1 and 2: 7 of the 12 boys did 30 or more, against 3 of the 12 girls.

(a) 19 push-ups (b) 3 girls and 7 boys (c) girls 23, boys 27 (d) The boys: 7 of them did at least 30 push-ups, against 3 girls.

Question 8

The ages of 20 members of a badminton club are: 21, 22, 24, 25, 26, 27, 27, 28, 29, 31, 32, 33, 33, 34, 36, 38, 41, 43, 47, 49. (a) Draw a stem-and-leaf diagram with split stems. (b) Which interval of 5 years holds the most ages?

ShowHide solution

(a) Each stem is written twice: leaves 0 to 4 on the first row, 5 to 9 on the second.

(b) The rows have $3$ , $6$ , $5$ , $2$ , $2$ and $2$ leaves. The longest is the second row of stem 2: ages 25 to 29.

(a) Drawn above. (b) 25 to 29 years (6 members).

Question 9

Mei Ling drew this diagram of the points scored by a basketball team in 10 games: 34, 38, 41, 45, 45, 52, 67, 69, 61, 63. Find her mistakes and draw the diagram correctly.

ShowHide solution

The leaves of stems 3 and 6 are not in order, and there is no key.

Her leaves on stems 3 and 6 are out of order, and she has no key. The correct diagram is drawn above.

Question 10

A stem-and-leaf diagram of the marks of a class has stems 4 to 8. Stems 4, 5, 7 and 8 have 3, 7, 9 and 4 leaves. There are 30 pupils in the class. (a) How many leaves has stem 6? (b) What fraction of the class scored from 70 to 89? Give it in its lowest terms.

ShowHide solution

(a) The leaves shown on the other stems: $3 + 7 + 9 + 4 = 23$ . Every pupil has one leaf, so stem 6 has $30 - 23 = 7$ leaves.

(b) Marks from 70 to 89 are the leaves on stems 7 and 8: $9 + 4 = 13$ . The fraction is $\frac{13}{30}$ , already in its lowest terms.

(a) 7 leaves (b) $\frac{13}{30}$

Key Points

The leaf is the last digit of a value; the stem is all the digits before it (142 is 14 | 2; 8 is 0 | 8).
Stems go in order down the left, every one from the smallest to the largest, even a stem with no leaves.
Leaves go in order from the stem outwards, equally spaced, one digit each.
Every diagram has a key with the unit: it decides whether 4 | 7 is 47 or 4.7.
A stem-and-leaf diagram keeps every value and shows the shape of the data; count leaves to answer "how many".
In a back-to-back diagram the left leaves are read outwards from the stem; each side has its own key.
Split stems (leaves 0 to 4, then 5 to 9) show the data in steps of 5.