# 7.6: Stem-and-Leaf Plots

**Basic**Created by: CK-12

**Practice**Stem-and-Leaf Plots

### Stem-and-Leaf Plots

In statistics, data is represented in tables, charts, and graphs. One disadvantage of representing data in these ways is that the actual data values are often not retained. One way to ensure that the data values are kept intact is to graph the values in a stem-and-leaf plot. A **stem-and-leaf plot** is a method of organizing the data that includes sorting the data and graphing it at the same time. This type of graph uses a stem as the leading part of a data value and a leaf as the remaining part of the value. The result is a graph that displays the sorted data in groups, or classes. A stem-and-leaf plot is used most when the number of data values is large, and it allows you to easily calculate the mode and the median of a data set.

#### Constructing a Stem-and-Leaf Plot

Construct a stem-and-leaf plot for the following data values:

\begin{align*}& 20 \quad 12 \quad 39 \quad 38 \quad 18 \quad 58 \quad 49 \quad 59 \quad 66 \quad 50\\ & 23 \quad 32 \quad 43 \quad 53 \quad 67 \quad 35 \quad 29 \quad 13 \quad 42 \quad 55\\ & 37 \quad 19 \quad 38 \quad 22 \quad 46 \quad 71 \quad 9 \quad \ 65 \quad 15 \quad 38\end{align*}

The stem-and-leaf plot can be constructed as follows. We will first create the stem-and-leaf plot, and then we will organize the values of the leaves.

#### Interpreting Stem-and-Leaf Plots

The following numbers represent the growth (in centimeters) of some plants after 25 days.

Construct a stem-and-leaf plot to represent the data, and list 3 facts that you know about the growth of the plants.

\begin{align*}& 18 \quad 10 \quad 37 \quad 36 \quad 61\\ & 39 \quad 41 \quad 49 \quad 50 \quad 52\\ & 57 \quad 53 \quad 51 \quad 57 \quad 39\\ & 48 \quad 56 \quad 33 \quad 36 \quad 19\\ & 30 \quad 41 \quad 51 \quad 38 \quad 60\end{align*}

Note that there are no leaves for the second stem. If there are no values in a class, do not enter a value in the leaf\begin{align*}-\end{align*}just leave it blank.

As for 3 facts that you know about the growth of the plants, answers will vary, but the following are some possible responses:

- From the stem-and-leaf plot, the growth of the plants ranged from a minimum of 10 cm to a maximum of 61 cm.
- The median of the data set is the value in the \begin{align*}13^{\text{th}}\end{align*} position, which is 41 cm.
- There was no growth recorded in the class of 20 cm, so there is no number in the leaf row.
- The data set is multimodal.

#### Finding the Mode and the Median

The following stem-and-leaf plot represents the ages of 23 people waiting in line at Tim Horton’s. What is the mode and the median of the ages? How many people are older than 32?

The mode is the value that appears the most often, and here it is 32. As for the median, since there are 23 data values, the median is the value that appears in the \begin{align*}12^{\text{th}}\end{align*} position. From the stem-and-leaf plot, it's clear that this value is 32, so the median of the data set is 32 as well. Finally, to find the number of people older than 32, count the number of all the digits after the number 2 in the row that has 3 as its stem. There are a total of 11 digits, so 11 people are older than 32.

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### Example

At a local veterinarian school, the number of animals treated each day over a period of 20 days was recorded.

#### Example 1

Construct a stem-and-leaf plot for the data set, which is as follows:

\begin{align*}& 28 \quad 34 \quad 23 \quad 35 \quad 16\\ & 17 \quad 47 \quad 05 \quad 60 \quad 26\\ & 39 \quad 35 \quad 47 \quad 35 \quad 38\\ & 35 \quad 55 \quad 47 \quad 54 \quad 48\end{align*}

What is the mode and the median of the data set?

**Step 1:** Create the stem-and-leaf plot.

Some people prefer to arrange the data in order before the stems and leaves are created. This will ensure that the values of the leaves are in order. However, this is not necessary and can take a great deal of time if the data set is large.

The leading digit of a data value is used as the stem, and the trailing digit is used as the leaf. The numbers in the stem column should be consecutive numbers that begin with the smallest class and continue to the largest class.

**Step 2:** Organize the values in each leaf row.

Now that the graph has been constructed, there is a great deal of information that can be learned from it.

The number of values in the leaf column should equal the number of data values that were given in the table. The value that appears the most often in the same leaf row is the trailing digit of the mode of the data set. The mode of this data set is 35. For 7 of the 20 days, the number of animals receiving treatment was between 34 and 39. The veterinarian school treated a minimum of 5 animals and a maximum of 60 animals on any one day. The median of the data can be quickly calculated by using the values in the leaf column to locate the value in the middle position. In this stem and leaf plot, the median is the mean of the sum of the numbers represented by the \begin{align*}10^{\text{th}}\end{align*} and the \begin{align*}11^{\text{th}}\end{align*} leaves: \begin{align*}\frac{35+35}{2}=\frac{70}{2}=35\end{align*}.

### Review

- The following stem-and-leaf plot shows the cholesterol levels of a random number of students. These values range from 2.3 to 8.9. What percentage of the students have levels between 5.0 and 7.1, inclusive?
- 6 %
- 20%
- 24%
- 28%

- Just like Presidents of the United States, Canadian Prime Ministers must be sworn into office. The following data represents the ages of 22 Canadian Prime Ministers when they were sworn into office. Construct a stem-and-leaf plot to represent the ages, and list 4 facts that you know from the graph. \begin{align*}& 52 \quad 74 \quad 60 \quad 39 \quad 65 \quad 46 \quad 55 \quad 66 \quad 54 \quad 51 \quad 70 \quad 47 \quad 69 \quad 47 \quad 57 \quad 46\\ & 48 \quad 66 \quad 61 \quad 59 \quad 46 \quad 45\end{align*}

Use the stem-and-leaf plot below to answer the following questions:

- What is the mode of the data set?
- What is the median of the data set?
- What is the minimum of the data set?
- What is the maximum of the data set?

Use the stem-and-leaf plot below to answer the following questions:

- What is the mode of the data set?
- What is the median of the data set?
- How many of the data values are greater than 40?
- What percentage of the data values are less than 40?

### Review (Answers)

To view the Review answers, open this PDF file and look for section 7.6.

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### Image Attributions

Here you'll learn how to construct stem-and-leaf plots and the importance of stem-and-leaf plots in statistics.

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