Section 6.3: Stem-and-Leaf Plots

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Timothy Cooley
Orange Coast College

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Learning Objectives

Draw a stem and leaf plot for a data set

Another type of graphical display is a stem-and-leaf plot. Stem-and-leaf plots offer several unique advantages that make them valuable tools for data analysis. Unlike other graphical displays such as histograms or bar charts that summarize data into bars or categories, a stem-and-leaf plot maintains all the original data points. This dual nature being both a graph and a data list makes it particularly useful for exploratory data analysis.

Definition: Stem-and-Leaf Plot

A stem-and-leaf plot (also called a stem-and-leaf diagram, stem-and-leaf graph, and/or stemplot) is a data display method that organizes numerical data while preserving the actual data values. It provides a quick visual representation of the distribution's shape while allowing you to see every individual data point.

Each data value is split into two parts:

Stem: The leading digit(s)
Leaf: The trailing digit (usually the last digit)

The stems are listed vertically, and the leaves are listed horizontally next to their corresponding stem.

Making a Stem-and-Leaf Plot

Step 1 — Organize your data: Arrange your data values in order from smallest to largest. This isn't strictly necessary, but it makes the process easier and ensures your final plot is organized.
Step 2 — Decide on stems and leaves: Determine how to split each number:
- Stem: Usually the leading digit(s) — the tens place for two-digit numbers
- Leaf: Usually the last digit — the ones place for two-digit numbers
Step 3 — List all possible stems vertically: Write the stems in a column from smallest to largest, even if some stems won't have any leaves. This shows where gaps exist in the data
Step 4 — Add the leaves: Go through your sorted data and place each leaf next to its corresponding stem. Write leaves in order from smallest to largest. (If any leaf or leaves are too long to your disliking, you have the option to use to lines per stem, if necessary).
Step 5 — Add a key: Always include a key that explains how to read the plot. This is essential for interpretation.

✅ A Stem-and-Leaf Plot is BEST used for:

✓ Small to moderate data sets (roughly 15-100 values)
✓ Initial exploratory data analysis
✓ When you need to preserve exact data values
✓ Quick manual analysis without technology
✓ Comparing two small groups
✓ Teaching introductory statistics concepts

❌ A Stem-and-Leaf Plot is NOT IDEAL for:

✗ Very large data sets (becomes cluttered and hard to read)
✗ Data with many decimal places (too complex)
✗ Formal presentations (histograms look more professional)
✗ When only the distribution shape matters (histogram is cleaner)
✗ Comparing more than two groups

The stem-and-leaf plot is a "best of both worlds" tool: it gives you the visual insight of a graph while maintaining the precision of a data list. It's particularly valuable in the early stages of data analysis when you want to understand your data quickly without losing any information. While modern technology has made more sophisticated graphs easier to create, the stem-and-leaf plot remains a practical, efficient, and instructive tool for understanding distributions.

Example #6.3.1 🤔

In the NFL, the 2013 Denver Broncos set the record for both the most points scored in a season and most passing yards in a season, at 606 and 5,572, respectively, both the all-time NFL records. In that year, the following points scored per game is the following:

49, 41, 37, 52, 51, 35, 33, 45, 28, 27, 31, 35, 51, 20, 37, 34

Draw a stem-and-leaf plot for the data.

✅ Solution:

Step 1 — Organize your data: 20, 27, 28, 31, 33, 34, 35, 35, 37, 37, 41, 45, 49, 51, 51, 52
Step 2 — Decide on stems and leaves: Stems are the tens place and the leaves are the ones place (last digit) of any two-digit number in the data set.
Step 3 — List all possible stems vertically: Write the stems in a column from smallest to largest, even if some stems won't have any leaves. This shows where gaps exist in the data.
Step 4 — Add the leaves: Go through your sorted data and place each leaf next to its corresponding stem. Write leaves in order from smallest to largest.

$\begin{array}{c|lc}2&078\\3&1345577\\4&159\\5&112\\\end{array}$

Step 5 — Add a key: Key: 2|0 = 20

Example #6.3.2 🤔

A meteorologist recorded the high temperatures (in °F) for 20 days:

68, 72, 75, 71, 69, 74, 78, 82, 79, 76, 73, 75, 77, 80, 81, 74, 76, 78, 79, 83

Draw a stem-and-leaf plot for the data.

✅ Solution:

Step 1 — Organize your data: 68, 69, 71, 72, 73, 74, 74, 75, 75, 76, 76, 77, 78, 78, 79, 79, 80, 81, 82, 83
Step 2 — Decide on stems and leaves: Stems are the tens place and the leaves are the ones place (last digit) of any two-digit number in the data set.
Step 3 — List all possible stems v
ertically: Write the stems in a column from smallest to largest, even if some stems won't have any leaves. This shows where gaps exist in the data.
Step 4 — Add the leaves: Go through your sorted data and place each leaf next to its corresponding stem. Write leaves in order from smallest to largest.

$\begin{array}{c|lc}6&89\\7&12344556678899\\8&0123\\\end{array}$

Step 5 — Add a key: Key: 6|8 = 68

Example #6.3.3 🤔

The ages of 20 customers at a local movie theater was recorded. The data set is as follows:

62, 55, 19, 15, 43, 28, 27, 16, 41, 57, 21, 21, 25, 46, 48, 19, 29, 23, 40, 20

Draw a stem-and-leaf plot for the data.

✅ Solution:

Step 1 — Organize your data: 15, 16, 19, 19, 20, 21, 21, 23, 25, 26, 27, 28, 29, 40, 41, 43, 46, 48, 55, 57
Step 2 — Decide on stems and leaves: Stems are the tens place and the leaves are the ones place (last digit) of any two-digit number in the data set.
Step 3 — List all possible stems vertically: Write the stems in a column from smallest to largest, even if some stems won't have any leaves. This shows where gaps exist in the data.
Step 4 — Add the leaves: Go through your sorted data and place each leaf next to its corresponding stem. Write leaves in order from smallest to largest.

$\begin{array}{c|lc}1&5699\\2&01135789\\4&01368\\5&57\\6&2\\\end{array}$

Step 5 — Add a key: Key: 1|5 = 15.

Example #6.3.4 🤔

In the 2024-2025 National Hockey League (NHL) season, the overall standings in points for all 32 teams is displayed below:

Pacific Division	Pts	Central Division	Pts	Metropolitan Division	Pts	Atlantic Division	Pts
Vegas Golden Knights	110	Winnipeg Jets	116	Washington Capitals	111	Toronto Maple Leafs	108
Los Angeles Kings	105	Dallas Stars	106	Carolina Hurricanes	99	Tampa Bay Lightning	102
Edmonton Oilers	101	Colorado Avalanche	102	New Jersey Devils	91	Florida Panthers	98
Calgary Flames	96	Minnesota Wild	97	Columbus Blue Jackets	89	Ottawa Senators	97
Vancouver Canucks	90	St. Louis Blues	96	New York Rangers	85	Montréal Canadiens	91
Anaheim Ducks	80	Utah Hockey Club	89	New York Islanders	82	Detroit Red Wings	86
Seattle Kraken	76	Nashville Predators	68	Pittsburgh Penguins	80	Buffalo Sabres	79
San Jose Sharks	52	Chicago Blackhawks	61	Philadelphia Flyers	76	Boston Bruins	76

Draw a stem-and-leaf plot for the data.

✅ Solution:

Step 1 — Organize your data: 52, 61, 68, 76, 76, 76, 79, 80, 80, 82, 85, 86, 89, 89, 90, 91, 91, 96, 96, 97, 97, 98, 99, 101, 102, 102, 105, 106, 108, 110, 111, 116
Step 2 — Decide on stems and leaves: Stems are the tens place and the leaves are the ones place (last digit) of any two or three-digit number in the data set.
Step 3 — List all possible stems vertically: Write the stems in a column from smallest to largest, even if some stems won't have any leaves. This shows where gaps exist in the data.
Step 4 — Add the leaves: Go through your sorted data and place each leaf next to its corresponding stem. Write leaves in order from smallest to largest.

$\begin{array}{c|lc}5&2\\6&18\\7&6669\\8&0025699\\9&011667789\\10&122568\\11&016\\\end{array}$

Step 5 — Add a key: Key: 5|2 = 52 or 10|1 = 101.

Example #6.3.5 🤔

The GPA’s (rounded to the nearest tenth) of a fundamental high school were noted:

2.8, 2.9, 2.0, 2.4, 3.4, 4.0, 3.8, 3.6, 3.8, 2.9, 2.0, 2.8, 2.7, 0.9, 4.0, 3.5, 3.5, 2.8, 2.4, 2.3, 3.5, 2.8, 2.9, 3.2, 3.5, 3.7, 4.0, 2.2

Draw a stem-and-leaf plot for the data.

✅ Solution:

Step 1 — Organize your data: 0.9, 2.0, 2.0, 2.2, 2.3, 2.4, 2.4, 2.7, 2.8, 2.8, 2.8, 2.8, 2.9, 2.9, 2.9, 3.2, 3.4, 3.5, 3.5, 3.5, 3.5, 3.6, 3.7, 3.8, 3.8, 4.0, 4.0, 4.0
Step 2 — Decide on stems and leaves: So, for the first time we have decimals. As long as the data doesn't have a long range of values, we can still construct a stem-and-leaf plot here. Stems, now, are the ones place and the leaves are the tenths place (last digit) of any number in the data set.
Step 3 — List all possible stems vertically: Write the stems in a column from smallest to largest, even if some stems won't have any leaves. This shows where gaps exist in the data.
Step 4 — Add the leaves: Go through your sorted data and place each leaf next to its corresponding stem. Write leaves in order from smallest to largest.

$\begin{array}{c|lc}0&9\\2&00234478888999\\3&2455556788\\4&000\\\end{array}$

Step 5 — Add a key: Key: 0|9 = 0.9. (This is why it is inportant to have the key here, because there are no decimal points in the stem-and-leaf plot.

Section 6.3: Stem-and-Leaf Plots [In-Class Exercises]

For Exercises 1 through 3, draw a stem-and-leaf plot for each data set given.

The following data represent the number of minutes 15 students spent studying for an exam:

45, 62, 58, 71, 67, 74, 53, 60, 68, 72, 59, 65, 55, 70, 64

The following data show the number of pages read by students last week:

102, 87, 95, 110, 124, 98, 105, 116, 89, 92, 101, 108, 117, 90, 96, 112, 99, 120

The following data represent sales (in dollars) at a café during their slowest hour:

145, 152, 138, 160, 149, 176, 155, 142, 158, 151, 147, 181, 132, 154, 152, 168, 118, 175

According to the California Department of Conservation, the table below shows the California's largest ever recorded earthquakes since 1800, ranked by date.

Magnitude	Date	Location	Damage
7.9	Jan. 9, 1857	Fort Tejon	Two killed; 220-mile surface scar
7.4	Mar. 26, 1872	Owens Valley	27 killed; three aftershocks of magnitude >6
7.8	April 18, 1906	San Francisco	Possibly 3,000 killed; 225,000 displaced
7.2	Jan. 22, 1923	Mendocino*	Damaged homes in several towns
7.1	Nov. 4, 1927	SW of Lompoc*	No major injuries; slight damage in 2 counties
6.4	March 10, 1933	SE of Long Beach	115 killed; led to new building codes for schools
7.0	May 18, 1940	El Centro	9 killed; $6 million in damage
7.3	July 21, 1952	Kern County	12 killed; 3 magnitude >6 aftershocks in 5 days
6.6	Feb. 9, 1971	San Fernando	65 killed; 2,000 injured; $505 million in damage
7.4	Nov. 8, 1980	W. of Eureka*	Injured 6; $2 million in damage
6.9	Oct. 17, 1989	Bay Area	63 killed; 3,753 hurt; up to $10 billion in damage
7.2	April 25, 1992	Cape Mendocino	356 injuries; $48.3 million in damage
7.3	June 28, 1992	Landers	One killed; 400 injured; $9.1 million in damage
6.7	Jan. 17, 1994	Northridge	57 killed; 9,000 hurt, up to $40 billion in damage
7.1	Oct. 16, 1999	Ludlow	Minimal damage due to remote location
7.1	July 5, 2019	Ridgecrest/Trona	Preceded by M6.4 quake; no fatalities

Draw a stem-and-leaf plot for the magnitude ratings.

Answers: 1. $\begin{array}{c|lc}4&5\\5&3589\\6&024578\\7&0124\\\end{array}$; 2. $\begin{array}{c|lc}8&79\\9&025689\\10&1258\\11&0267\\12&04\\\end{array}$; 3. $\begin{array}{c|lc}11&8\\13&28\\14&2579\\15&122458\\16&08\\17&56\\18&1\\\end{array}$; 4. $\begin{array}{c|lc}6&4679\\7&11122334489\\\end{array}$.

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