Stem and Leaf Plots

Learn to build and read stem-and-leaf plots, find the median, mode and range from them, and compare two groups with a back-to-back plot.

Intermediate25 min18 problems
Practice Problems

Definition

A stem-and-leaf plot shows every value of a data set while also showing its shape. Each number is split into two parts:
  • The stem is every digit except the last one (for two-digit numbers, the tens digit)
  • The leaf is the last digit (the ones digit)
Stems are written once, in a column from smallest to largest. The leaves of each stem are written in a row to the right, in order from smallest to largest. A key tells the reader how to read one entry.
Here are the quiz scores of 15 students: 52, 58, 61, 63, 67, 67, 70, 74, 75, 78, 81, 84, 86, 90, 95.
StemLeaves52 861 3 7 770 4 5 881 4 690 5\begin{array}{r|l} \text{Stem} & \text{Leaves} \\ \hline 5 & 2~8 \\ 6 & 1~3~7~7 \\ 7 & 0~4~5~8 \\ 8 & 1~4~6 \\ 9 & 0~5 \end{array}
Key: 5∣2=525 \mid 2 = 52
Every leaf is one data value, so the number of leaves is the number of data values (here 15). Turned on its side, the plot looks like a histogram with intervals 50-59, 60-69 and so on, but unlike a histogram you can still read every single score.

Worked Examples

Make a stem-and-leaf plot of these quiz scores: 74, 58, 86, 61, 90, 67, 52, 78, 70, 84, 63, 95, 67, 75, 81.
1

Choose the stems

The smallest score is 52 and the largest is 95, so the tens digits go from 5 to 9 → Stems: 5, 6, 7, 8, 9
2

Write each ones digit as a leaf next to its stem

74 gives leaf 4 on stem 7, 58 gives leaf 8 on stem 5, and so on for all 15 scores → 15 leaves written
3

Put the leaves of every row in order

52 861 3 7 770 4 5 881 4 690 5\begin{array}{r|l} 5 & 2~8 \\ 6 & 1~3~7~7 \\ 7 & 0~4~5~8 \\ 8 & 1~4~6 \\ 9 & 0~5 \end{array}
→ Ordered stem-and-leaf plot
4

Add a key and check the count

Key: 5∣2=525 \mid 2 = 52. Count the leaves: 2+4+4+3+2=152 + 4 + 4 + 3 + 2 = 15 → 15 leaves for 15 scores

Common Mistakes

Writing the leaves in the order the data was given (for example 6 | 7 1 7 3)

Why it's wrong: The plot still shows the shape, but counting to the median or spotting repeated values only works when each row is in order.

Correct: Always rewrite each row with the leaves from smallest to largest: 6 | 1 3 7 7.

Leaving out the key or ignoring it

Why it's wrong: The same entry 4∣74 \mid 7 can mean 47, 4.7 or 470. Without the key the reader has to guess.

Correct: Always write a key such as 4∣7=4.74 \mid 7 = 4.7 seconds and read every value with it.

Skipping a stem that has no leaves

Why it's wrong: If no value is between 30 and 39 and the row for stem 3 is left out, the gap in the data disappears and the shape looks wrong.

Correct: Write every stem from the smallest to the largest, even when a row stays empty.

Reading the left side of a back-to-back plot in the wrong direction (reading 3∣73 \mid 7 as 37)

Why it's wrong: On the left side the leaf stands before the stem, but the stem is still the tens digit.

Correct: Read left leaves outward from the stem: leaf 3 next to stem 7 is 73. The key of a back-to-back plot shows this.

Interactive Visual

Quiz scores of 15 students: sort the leaves

StemLeaf
5
6
7
8
9

Key: 7 | 4 = 74

Tap a leaf to see its value.

Values
15
Median
74
Mode
67
Range
95 − 52 = 43

Back-to-back plot: Class A and Class B

Class AStemClass B
4
5
6
7
8
9

Key: 7 | 2 = 72Left side: 3 | 7 = 73

Tap a leaf to see its value.

Class A

Values
10
Median
63
Mode
64
Range
80 − 48 = 32

Class B

Values
10
Median
75
Mode
none
Range
91 − 57 = 34

Commute times of 14 workers (minutes) with split stems

StemLeaf
2
2*
3
3*

Key: 2 | 1 = 21Split stems: the first row of a stem holds leaves 0 to 4, the row with * holds 5 to 9.

Tap a leaf to see its value.

Values
14
Median
28.5
Mode
27
Range
38 − 21 = 17

Histogram

Same quiz scores grouped by stem (the plot turned on its side)

50-59(2)
60-69(4)
70-79(4)
80-89(3)
90-99(2)

Interactive Sandbox

Expression Calculator

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Practice Problems

18 problems
Problem 1 of 18
Easy
In a stem-and-leaf plot the key is 4∣7=474 \mid 7 = 47. Which value does stem 6 with leaf 3 stand for?

Why It Matters

Stem-and-leaf plots are a quick way to organize a small or medium data set by hand:
  • No data is lost: unlike a histogram or a box plot, every original value can be read back from the plot
  • Shape at a glance: the length of each row shows where the data is crowded and where it is thin
  • Ordering for free: once the leaves are in order, the data is sorted, so the median, the mode and the range are easy to find
  • Comparing groups: a back-to-back plot puts two data sets on the same stems, for example the test scores of two classes or the temperatures of two towns
Sports reporters use them for points per game, doctors for pulse rates, and timetables in many countries still list departure minutes next to each hour, which is exactly the stem-and-leaf idea.

Real World Applications

Points per Game in Basketball

Coaches and sports reporters use stem-and-leaf plots to show a player's scoring over a season without losing any game.

Example:

A player's points in 12 games:

08 912 4 4 720 1 5 633 8\begin{array}{r|l} 0 & 8~9 \\ 1 & 2~4~4~7 \\ 2 & 0~1~5~6 \\ 3 & 3~8 \end{array}
Key: 1∣2=121 \mid 2 = 12 points (the stem 0 holds the one-digit scores 8 and 9).

1Try It Yourself

Use the plot of the player's 12 games above.

What is the median number of points per game?

Step 1: Write the mathematical expression

Write the calculation for the median:

Comparing Temperatures of Two Towns

Weather services compare towns with back-to-back plots: the same stems in the middle, one town on each side.

Example:

Daily high temperatures in °C:

Town AStemTown B9 608 5 3 114 72 020 3 5 7 831 2 4\begin{array}{r|c|l} \text{Town A} & \text{Stem} & \text{Town B} \\ \hline 9~6 & 0 & \\ 8~5~3~1 & 1 & 4~7 \\ 2~0 & 2 & 0~3~5~7~8 \\ & 3 & 1~2~4 \end{array}
Key: 6∣0=66 \mid 0 = 6 °C for Town A, 1∣4=141 \mid 4 = 14 °C for Town B. Town A has 8 days, Town B has 10 days.

2Try It Yourself

Use the back-to-back plot of the two towns above.

By how many degrees is the median of Town B higher than the median of Town A?

Step 1: Write the mathematical expression

Write median B minus median A:

Resting Pulse Rates

School nurses and doctors record pulse rates in stem-and-leaf plots to see at once which values are typical and which are unusual.

Example:

Resting pulse rates of 12 students in beats per minute:

62 5 870 1 4 4 983 690 7\begin{array}{r|l} 6 & 2~5~8 \\ 7 & 0~1~4~4~9 \\ 8 & 3~6 \\ 9 & 0~7 \end{array}
Key: 7∣4=747 \mid 4 = 74 beats per minute.

3Try It Yourself

Use the pulse rate plot above.

What is the range of the pulse rates?

Step 1: Write the mathematical expression

Write the calculation for the range:

Key Takeaways

  • 1A stem-and-leaf plot splits each value into a stem (all digits but the last) and a leaf (the last digit)
  • 2Stems go in a column from smallest to largest, leaves in each row are written in order, and every plot needs a key such as 5∣2=525 \mid 2 = 52
  • 3Each leaf is one data value, so the number of leaves is the number of values and every value can be read back
  • 4Because the data is in order, the median is found by counting leaves, the mode is a repeated leaf in one row and the range is last value minus first value
  • 5A back-to-back plot shares the stems between two groups: leaves on the left are read outward from the stem
  • 6Split stems (leaves 0-4 and 5-9 in separate rows) show the shape better when the data uses only a few stems

Frequently Asked Questions

The stem is 0 and the leaf is 7, so 7 is written as 0∣70 \mid 7. That way one-digit values fit in the same plot as two-digit values.
The stem is 0 and the leaf is 7, so 7 is written as 0∣70 \mid 7. That way one-digit values fit in the same plot as two-digit values.
The last digit is still the leaf and the key says what it means. For 4.7 seconds use 4∣7=4.74 \mid 7 = 4.7; for 125 grams use 12∣5=12512 \mid 5 = 125.
Both show how many values fall in each interval. A histogram shows only the counts, while a stem-and-leaf plot keeps every single value, so you can still find the exact median, mode and range.
When nearly all the data shares only two or three stems, the rows get very long. Splitting each stem into leaves 0-4 and 5-9 doubles the number of rows and shows the shape more clearly.

Glossary

Stem-and-leaf plot
A display that splits each value into a stem and a leaf so that every value is shown and the shape of the data is visible
Stem
All digits of a value except the last one, written once in the left column (in 74 the stem is 7)
Leaf
The last digit of a value, written in the row of its stem (in 74 the leaf is 4)
Key
A note that shows how to read an entry, for example 5∣2=525 \mid 2 = 52 or 5∣2=5.25 \mid 2 = 5.2
Back-to-back plot
Two stem-and-leaf plots that share one column of stems, with one group's leaves on the left and the other's on the right
Split stems
Each stem is used for two rows, the first with leaves 0 to 4 and the second with leaves 5 to 9

Formula Card

Reading an entry (key stem | leaf)

7∣4=7×10+4=747 \mid 4 = 7 \times 10 + 4 = 74

For two-digit data the value is stem times 10 plus leaf. Always check the key: with $7 \mid 4 = 7.4$ the leaf is the tenths digit.

Position of the median

position=n+12\text{position} = \frac{n + 1}{2}

Count leaves from the top until you reach this position. If it ends in .5, take the mean of the two values on either side.

Range

range=last leaf value−first leaf value\text{range} = \text{last leaf value} - \text{first leaf value}

The largest value is the last leaf of the last stem and the smallest is the first leaf of the first stem.

Today's challengeOne problem, three tries

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