Teacher Guide: 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.

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Printable worksheet

All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

10 questions on Statistical Graphs. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives

Build an ordered stem-and-leaf plot with a key from a list of data
Read individual values, the minimum, the maximum and the number of values from a stem-and-leaf plot
Find the median, mode and range directly from a stem-and-leaf plot
Compare two groups using a back-to-back stem-and-leaf plot
Decide when split stems make a plot easier to read

Prerequisites

• Place value of tens and ones (and tenths for decimal keys)
• Dot plots and frequency tables (lessons Dot Plots and Frequency Tables)
• Histograms and intervals (lesson Histograms)
• Median, mode and range of a data set, and box plots for comparing groups (lesson Box Plots (Box and Whisker))

Discussion Starters

1. Why do you think this display is called a stem-and-leaf plot?
2. What information does a stem-and-leaf plot keep that a histogram loses?
3. Why must the leaves be in order before you look for the median?
4. Could you make a stem-and-leaf plot of 500 values? Would it still be useful?
5. A bus timetable lists the hour once and the departure minutes next to it. How is that like a stem-and-leaf plot?

Common Misconceptions

The stem with the longest row holds the largest values

Remediation: Point out that the length of a row is a count (how many values), not a size. Turn the plot on its side and compare it to a histogram: a tall bar means many values, not large values.

The median is the middle stem or the middle leaf of a row

Remediation: Have students count leaves one by one from the top, crossing them off, until they reach position $\frac{n + 1}{2}$ . Then compare with the middle stem to show they usually differ.

Left leaves of a back-to-back plot are read from left to right, so $3 \mid 7$ becomes 37

Remediation: Ask students to rewrite a few left-side entries as numbers using the key, always starting at the stem and moving outward.

Differentiation Ideas

For Struggling Students:

• Give a template with the stems already written and have students only place the leaves
• Start with 8 to 10 two-digit values and no repeated values
• Let students cross off leaves while counting to the median

For On-Level Students:

• Build plots from unordered lists of 15 to 20 values and find the median, mode and range
• Compare two groups with a back-to-back plot and write one sentence about center and one about spread
• Use a decimal key such as $4 \mid 7 = 4.7$

For Advanced Students:

• Decide when split stems help and redraw a plot with them
• Draw a box plot from a stem-and-leaf plot by finding the five-number summary
• Collect real data (for example heights or sports results) and argue which display shows it best

Standards Alignment

6.SP.B.4 (CCSS.MATH.CONTENT.6.SP.B.4)
Display numerical data in plots on a number line, including dot plots, histograms, and box plots
6.SP.B.5.C (CCSS.MATH.CONTENT.6.SP.B.5.C)
Give quantitative measures of center (median and/or mean) and variability, as well as describe any overall pattern
7.SP.B.4 (CCSS.MATH.CONTENT.7.SP.B.4)
Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations

Lesson Resources

visualDot Plot, Histogram and Box Plot of the Same Data
Compare the quiz scores of the lesson in three other displays and discuss what the stem-and-leaf plot keeps that the others lose
activityClass Reaction Times
Students measure a ruler-drop reaction time, then build a class stem-and-leaf plot on the board with a decimal key
worksheetBack-to-Back Comparison
Two sets of data (for example jump distances of two groups) to put in a back-to-back plot, with questions on medians and ranges

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

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.

\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 \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.

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

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

Put the leaves of every row in order

\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

Add a key and check the count

Key:

5 \mid 2 = 52

. Count the leaves:

2 + 4 + 4 + 3 + 2 = 15

→ 15 leaves for 15 scores

Answer: The plot has stems 5 to 9 with ordered leaves 2 8 / 1 3 7 7 / 0 4 5 8 / 1 4 6 / 0 5 and the key $5 \mid 2 = 52$ .

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 $4 \mid 7$ can mean 47, 4.7 or 470. Without the key the reader has to guess.

Correct: Always write a key such as $4 \mid 7 = 4.7$ 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 $3 \mid 7$ 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.

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:

\begin{array}{r|l} 0 & 8~9 \\ 1 & 2~4~4~7 \\ 2 & 0~1~5~6 \\ 3 & 3~8 \end{array}

Key:

1 \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:

\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 \mid 0 = 6

°C for Town A,

1 \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:

\begin{array}{r|l} 6 & 2~5~8 \\ 7 & 0~1~4~4~9 \\ 8 & 3~6 \\ 9 & 0~7 \end{array}

Key:

7 \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 $5 \mid 2 = 52$
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

What is the stem of a one-digit number like 7?

The stem is 0 and the leaf is 7, so 7 is written as

0 \mid 7

. That way one-digit values fit in the same plot as two-digit values.

How do I show decimals or three-digit numbers?

The last digit is still the leaf and the key says what it means. For 4.7 seconds use

4 \mid 7 = 4.7

; for 125 grams use

12 \mid 5 = 125

What is the difference between a stem-and-leaf plot and a histogram?

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 should I use split stems?

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 $5 \mid 2 = 52$ or $5 \mid 2 = 5.2$
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 \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

\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

\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.

Teacher Guide: Stem and Leaf Plots

Use this lesson with your class

For Teachers

For Struggling Students:

For On-Level Students:

For Advanced Students:

Lesson Content

Definition

Worked Examples

Common Mistakes

Why It Matters

Real World Applications

Points per Game in Basketball

Comparing Temperatures of Two Towns

Resting Pulse Rates

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Reading an entry (key stem | leaf)

Position of the median

Range

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Reading Pie Charts

Dot Plots