Stemplots (Stem-and-Leaf Plots)

13 min read

FACULTY REFERENCE

Textbook: Moore, McCabe, and Craig, Introduction to the Practice of Statistics (IPS), 9th edition. Location: Chapter 1 (Looking at Data: Distributions), Section 1.2 (Displaying Distributions with Graphs), subsection on quantitative variables (stemplots). Pages: Page numbers pending faculty verification.

A first picture of a wrong turn here is common and shows something worth understanding. There is no rush. Building a stemplot by hand is one of the calmest ways to see what a batch of numbers is doing.

Before You Start: Prerequisite Check

Try This First

Here are the ages, in years, of 12 people at a small family reunion:

$$31,\ 24,\ 38,\ 22,\ 45,\ 33,\ 27,\ 41,\ 36,\ 29,\ 34,\ 48$$

Before reading any further, try this with pencil and paper. Sort the ages into four groups by their tens digit: the twenties, the thirties, the forties, and so on. Write the leftover ones digit next to each group.

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

What a Stemplot Is Doing

A stemplot answers a plain question about a batch of numbers: where do the values pile up, and how spread out are they?

Each observation is split into two parts. The leading digits become a label for a row, and the final digit is dropped into that row. As you add more numbers, each row grows sideways. A long row means many values landed in that range. A short row means few did. The overall outline of the rows is the shape of the distribution.

A stemplot keeps every original data value, so you can still read the exact numbers back out of it. A histogram (which comes later) groups values into bars and loses the individual numbers. For small batches of data, the stemplot gives you the shape and the raw numbers at the same time.

Prerequisite Map

Prerequisite Hub

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Cross-course prerequisites: none.

Quick Reference

Stemplot at a Glance

The Official Definition

Stemplot (stem-and-leaf plot):

A stemplot displays the distribution of a quantitative variable by separating each observation into a stem, consisting of all digits except the final one, and a leaf, the final digit. The stems are written in a vertical column in increasing order, and each leaf is written to the right of its stem. A stemplot keeps the actual data values while showing the shape of the distribution.

Source: Moore, McCabe, Craig, IPS 9th edition, Section 1.2 (Displaying Distributions with Graphs); corroborated by OpenStax Introductory Statistics 2e, 2.1 Stem-and-Leaf Graphs (Stemplots) (openstax.org).

There is no theorem to prove here. A stemplot is a construction procedure plus a reading procedure, so the work of this skill is doing both correctly and reading the result honestly.

How to Build a Stemplot, Step by Step

Step 1: Choose the stems. For two-digit numbers, the stem is the tens digit and the leaf is the ones digit. Write the stems in a vertical column from smallest to largest, including any stem that has no leaves so that gaps in the data stay visible.

Step 2: Write each leaf. For every observation, find its stem row and write the final digit to the right.

Step 3: Order the leaves. Within each row, rewrite the leaves from smallest to largest. This is the ordered stemplot, and it makes the count in each row easy to read.

Step 4: Add a key. State what one stem-leaf pair means, for example “$3 \mid 6$ means 36,” so a reader knows the units.

Step 5: Read the shape. Look at the outline of the rows: where the data pile up (the center), how far they reach (the spread), and whether one tail is longer than the other.

A Worked Example

The following are the numbers of pages in 15 short stories submitted to a class anthology:

$$12,\ 7,\ 23,\ 18,\ 9,\ 21,\ 14,\ 6,\ 25,\ 19,\ 11,\ 8,\ 27,\ 16,\ 22$$

Predict first. Before building anything, predict the shape. The values run from a single page to the high twenties, and several land in the teens, so you might predict a single pile somewhere in the teens with shorter tails on each side. Hold that prediction.

Step 1: stems. The smallest value is 6 and the largest is 27, so the stems are 0, 1, and 2 (using the tens digit; a one-digit number like 6 has a stem of 0).

Step 2: leaves, in the order the data appear.

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

Step 3: order the leaves within each row.

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

Key: 1 | 4 means 14 pages

Step 4: read the shape. The middle row (the teens) holds 6 of the 15 values, the most of any row. The rows on either side are shorter. The data run from 6 to 27 with no gaps.

Compare with your prediction. The single pile landed in the teens, as predicted, and the tails on each side are shorter. The prediction matched the picture. If your prediction had been a pile in the twenties, the stemplot would have corrected you, which is exactly what a display is for.

Splitting Stems

Sometimes one stem holds so many leaves that the shape is hard to read. Splitting a stem into two rows can help: one row for leaves 0 through 4 and one row for leaves 5 through 9.

Consider these 14 quiz scores:

$$50,\ 51,\ 53,\ 55,\ 56,\ 58,\ 59,\ 60,\ 62,\ 63,\ 67,\ 68,\ 70,\ 72$$

With single stems, almost everything crowds into the 5 row:

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

Key: 6 | 2 means 62

Splitting each stem into a low half (leaves 0 to 4) and a high half (leaves 5 to 9) spreads the same data over more rows:

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

Key: 6 | 2 means 62

The split version shows more detail in the shape without changing a single data value. Both displays are correct. One way to see the data is the compact three-row version; another way is the spread-out five-row version, and which one you prefer depends on how much detail the question needs.

Back-to-Back Stemplots

To compare two groups on the same variable, a back-to-back stemplot shares one column of stems. One group’s leaves grow to the left and the other group’s leaves grow to the right. Read the left leaves from the stem outward, so the digit nearest the stem is the smallest.

Here are resting heart rates (beats per minute) for two small groups:

$$\text{Group A}:\ 62,\ 65,\ 68,\ 71,\ 74,\ 75,\ 79$$ $$\text{Group B}:\ 58,\ 60,\ 61,\ 67,\ 72,\ 73$$

Group A          Group B
              | 5 | 8
        8 5 2 | 6 | 0 1 7
      9 5 4 1 | 7 | 2 3

Key: 2 | 6 (left) means 62 for Group A
     6 | 0 means 60 for Group B

The stem 5 row carries Group B’s single value 58 on the right and no leaf on the left, since Group A has nothing in the fifties. Group A’s values sit a little higher overall, and Group B reaches down into the high fifties. The shared stems make the comparison direct.

Common Misconceptions

Practice Problems

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

Key: 6 | 4 means 64 degrees

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

Key: 2 | 6 means 26 books

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

Key: 3 | 4 means 34

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

Key: 3 | 4 means 34

Person 1          Person 2
         8 | 1 | 2 5
       5 2 | 2 | 4 8
       4 1 | 3 | 3

Key: 8 | 1 (left) means 18 for Person 1
     1 | 2 means 12 for Person 2

Mastery Checklist

Novice (Level 1-2):

• I can split a number into a stem and a leaf using the place values
• I can build an ordered stemplot from a small batch of data and add a key
• I can state what the length of a row tells me

Competent (Level 3-4):

• I can read the largest value and the most crowded row off a finished stemplot
• I can split stems when one row is too crowded, without changing any data value
• I can explain why ordering the leaves finishes the display

Proficient (Level 5):

• I can build a back-to-back stemplot to compare two groups
• I can describe the center and the spread of a distribution from its stemplot
• I can explain when a stemplot is a better choice than a histogram and when it is not

Mental Model

Think of each number walking up to a row of mailboxes. The stem is the building number, and the leaf is the apartment number. Every observation drops its leaf into the mailbox for its stem. After everyone has dropped a leaf, the mailbox with the most leaves marks where the data crowd together, and the stack of mailboxes from top to bottom shows how far the data reach. Nothing is summarized away: you can still open any mailbox and read the original numbers back out.

Connections

Looking back:

• Categorical vs Quantitative Variables sets up which variables get a stemplot
• Distribution of a Variable is the idea a stemplot makes visible

Looking ahead:

• Describing a Distribution: Shape, Center, and Spread reads these features off the stemplot
• A histogram (later in Section 1.2) groups data into bars for larger data sets where individual values matter less

Real-world connections:

• Quick exploratory looks at small data sets (test scores, ages, counts) before any computation
• A field-notebook display you can draw by hand without software

Resources

Primary textbook:

• Moore, McCabe, and Craig, Introduction to the Practice of Statistics (IPS), 9th edition, Chapter 1, Section 1.2 (Displaying Distributions with Graphs). Page numbers pending faculty verification.

Open-source companion:

• OpenStax Introductory Statistics 2e, Section 2.1: Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs (openstax.org).

Last updated: 2026-06-16