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11.9: Stem-and-Leaf Plots

Difficulty Level: At Grade Created by: CK-12
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The city of Northtown tracked the speeds of cars along a street before and after placing a speed limit sign up. The city wants to create a display that illustrates the following data:

Speeds before: 19, 20, 24, 25, 25, 26, 30, 31, 32, 36, 38, 39, 40, 42, 42, 45, 45, 47, 48, 48, 52, 54, 55, 55, 56, 56, 58, 61, 62, 62, 63, 68, 71

Speeds after: 11, 18, 22, 23, 25, 28, 29, 30, 30, 32, 32, 32, 35, 39, 40, 43, 44, 45, 47, 47, 50, 51, 51, 53, 56, 62, 64, 65, 67, 73.

How can the city create a data display? 

In this concept, you will learn how to create and read stem-and-leaf plots.

Creating and Reading Stem-and-Leaf Plots

Data is a set of numerical or non-numerical information. Data can be analyzed in many different ways. In this concept you will analyze numerical data using stem-and-leaf plots.

Mean is a numerical value that measures the spread of a data set. The mean is one way to determine the central or typical value of the set. Mean is also called the average. 

Median is a numerical value that represents the middle term in a data set. When ordered sequentially, this value that is in the middle is the median

Mode is the numerical value that occurs most frequently in a data set.  

The range of a set of data is the difference between the largest and smallest values. The range identifies how far apart the values in the data set are. 

One way to display data is in a stem-and-leaf plot. A stem-and-leaf plot is one way to show the distribution of the data. The data values in a stem-and-leaf plot are split into a stem and leaf.   

Let's look at an example. 

Construct a stem-and-leaf plot for the data below.

Science test scores for third period (out of 100%): 97, 92, 77, 82, 96, 75, 68, 80, 79, 96

First, arrange the data in order from least to greatest.

68, 75, 77, 79, 80, 82, 85, 92, 96, 96

Then, determine how to separate the data values into a stem and leaves. In this case, the stem will be the tens unit and the leaves will be the ones unit.

Next, create the stem-and-leaf plot by placing the tens values in one column and the ones values in another column. For 68, the 6 from 60 will go in the stem and the 8 from the ones will go in the leaf. For 75, 77, and 79, the 7 from 70 will go in the stem and the 5, 7, and 9 from the ones will go in the leaf.  

Lastly, create a key for the stem-and-left plot. The key tells the reader what values are represented in the display. To do this, select one term in the plot and separate the digits by a vertical line, then use the equal sign to show what it is equivalent to. It will look like this, 6|8=68.

The answer is the stem-and-leaf plot will look like the one below.

Examples

Example 1

Earlier, you were given a problem about Northtown and their speed data.

The city of Northtown tracked the speed of cars driving on a road before and after posting a speed limit. The city wants to display the the following data:

Speeds before: 19, 20, 24, 25, 25, 26, 30, 31, 32, 36, 38, 39, 40, 42, 42, 45, 45, 47, 48, 48, 52, 54, 55, 55, 56, 56, 58, 61, 62, 62, 63, 68, 71

Speeds after: 11, 18, 22, 23, 25, 28, 29, 30, 30, 32, 32, 32, 35, 39, 40, 43, 44, 45, 47, 47, 50, 51, 51, 53, 56, 62, 64, 65, 67, 73.

How can the city create a data display? 

To do this, create two stem-and-leaf plots in one display. In this type of display, the stem will have leaves on both sides. To create the plot, start with determining the stem. In this case, the values in both data sets include numbers in the tens to seventies. Therefore, the stem will go from 1 to 7 to represent 10 to 70. 

Next, list the before speeds on the left side of the stem. These should be listed in order from right to left. 

Next, list the after speeds on the right side of the stem. These should be listed in order from left to right. 

Lastly, create a key for the stem-and-left plot. To do this, select terms from both sides of the plot and separate the digits by vertical lines, then use the equal sign to show what the stem-and-leaf is equivalent to.

The answer is the stem-and-leaf plot will look like the one below.

Example 2

Create a stem-and-leaf plot for the data set: 8, 12, 17, 18, 19, 22, 23, 31, 35, 40.

First, arrange the data from least to greatest. In this case, the data is already ordered from least to greatest.

Next, determine how to separate the data values into a stem and leaves. In this case, the stem will be the tens unit and the leaves will be the ones unit.

Next, create the stem-and-leaf plot by placing the tens values in one column and the ones values in another column. For 8, there is no 10 so the stem will be 0 and the 8 from the ones will go in the leaf. For 12, 17, 18, and 19, the 1 from 10 will go in the stem and the 2, 7, 8 and 9 from the ones will go in the leaf.  

Lastly, create a key for the stem-and-left plot. To do this, select one term in the plot and separate the digits by a vertical line, then use the equal sign to show what it is equivalent to. It will look like this, 1|2=12.

The answer is the stem-and-leaf plot will look like the one below.

Example 3

Create a stem-and-leaf plot for the data set: 14, 16, 17, 18, 18, 20, 22, 24, 29, 31, 33

First, arrange the data from least to greatest. In this case, the data is already ordered from least to greatest.

Next, determine how to separate the data values into a stem and leaves. In this case, the stem will be the tens unit and the leaves will be the ones unit.

Next, create the stem-and-leaf plot by placing the tens values in one column and the ones values in another column. For 14, the 1 from the 10 will go in the stem and the 4 from the ones will go in the leaf. For 33, the 3 from 30 will go in the stem and the 3 from the ones will go in the leaf.  

Lastly, create a key for the stem-and-left plot. To do this, select one term in the plot and separate the digits by a vertical line, then use the equal sign to show what it is equivalent to. It will look like this, 1|4=14.

The answer is the stem-and-leaf plot will look like the one below.

Example 4

The stem-and-leaf plot below represent the quiz scores for a history class. The maximum score of the quiz is 20. Calculate the mean, median, mode, and range of the scores.

First, calculate the mean. To do this, use the key to list the values in the stem-and-leaf plot.

0, 0, 0, 2, 3, 6, 7, 9, 9, 10, 12, 14, 14, 14, 15, 17, 18, 19, 19, 19, 20, 20, 20

Next, add all the values together. 

0 + 0 + 0 + 2 + 3 + 6 + 7 + 9 + 9 + 10 + 12 + 14 + 14 + 14 + 15 + 17 + 18 + 19 + 19 + 19 + 20 + 20 + 20 = 267

Then, divide the sum by the number of terms in the list. In this case there are 23 terms, so divide 267 by 23.

\begin{align*}267\div 23=11.6\end{align*}267÷23=11.6

This is the mean of the quiz scores.

Next, calculate the median. Since there are 23 terms, the median is the 12th term, 14. 

Next, find the mode. The mode is the number, or numbers, that occurs most often. There are three numbers that occur most often, 14, 19, and 20. So all these are the mode. 

Next, find the range. The range is the difference between the smallest and largest numbers. To do this, subtract 0 from 20, which is 20.

The answer is the mean is 11.6, the median is 14, the mode is 14, 19, and 20, and the range is 20. All values are quiz scores. 

Example 5

The stem-and-leaf plot below shows the weight (in pounds) of ten trout caught in a fishing competition. Calculate the mean, median, mode, and range of the data on the stem-and-leaf plot.

First, calculate the mean. To do this, use the key to list the values in the stem-and-leaf plot.

2.9, 3.1, 4.0, 4.5, 5.2, 6.2, 7.6, 8.3, 9.2, 9.2

Next, add all the values together.

\begin{align*}2.9+3.1+4+4.5+5.2+6.2+7.6+8.3+9.2+9.2=60.2\end{align*}2.9+3.1+4+4.5+5.2+6.2+7.6+8.3+9.2+9.2=60.2

Then, divide the sum by the number of terms in the list. In this case there are ten terms, so divide 60.2 by 10.

\begin{align*}60.2\div 10=6.02\end{align*}60.2÷10=6.02

This is the mean of the weight (in pounds).

Next, calculate the median. The median is the number in the middle. Since there are ten terms, the median is between the fifth and sixth terms, 5.2 and 6.2. To find the value in the middle, add the two numbers together and divide by 2.

\begin{align*}5.2+6.2=11.4\end{align*}5.2+6.2=11.4

\begin{align*}11.4\div 2=5.7\end{align*}11.4÷2=5.7

This is the median of the weight (in pounds).

Next, find the mode. The mode is the number that occurs most often. There is only one number that occurs more than once, 9.2, so this is the mode.

Next, find the range. The range is the difference between the smallest and largest numbers. To do this, subtract 2.9 from 9.2.

\begin{align*}9.2-2.9=6.3\end{align*}9.22.9=6.3

This is the range.  

The answer is the mean is 6.02, the median is 5.7, the mode is 9.2, and the range is 6.3. All values are weight (in pounds). 

Review

Create stem-and-leaf plots and answer the questions on each.

  1. Make a stem-and-leaf plot to display the data: 22, 25, 27, 29, 31, 34, 34, 39, 40, and 44.
  2. Make a stem-and-leaf plot to display the data: 88, 96, 72, 65, 89, 91, 90, 100, 101, and 86.

The data table below depicts the number of miles ten students commute to school each day.

\begin{align*}8 \quad 7 \quad 11 \quad 6 \quad 9 \quad 15 \quad 6 \quad 20 \quad 12 \quad 4\end{align*}87116915620124

  1. Create a stem-and-leaf plot to display the data.
  2. Use the stem-and-leaf plot to determine the mean.
  3. Use the stem-and-leaf plot to determine the median.
  4. Use the stem-and-leaf plot to determine the mode.
  5. Use the stem-and-leaf plot to determine the range of the data.

The data table below depicts the final score each basketball game for an entire season.

\begin{align*}27 \quad 36 \quad 31 \quad 29 \quad 25 \quad 39 \quad 21 \quad 26 \quad 34 \quad 40 \quad 38 \quad 29\end{align*}273631292539212634403829

  1. Use the data to create a stem-and-leaf plot.
  2. Use the data to determine the mean.
  3. Use the data to determine the median.
  4. Use the data to determine the mode.
  5. Use the data to determine the range.

The stem-and-leaf plot depicts the class sizes for two grade levels at Huntington Middle School. Use the information on the stem-and-leaf plot to answer the questions below.

\begin{align*}7^{th}\end{align*}7th and \begin{align*}8^{th}\end{align*}8th Grade Class Size

\begin{align*}7^{th}\end{align*}7th Grade: \begin{align*}8^{th}\end{align*}8th Grade:
Leaf Stem Leaf
9 8 1 6 7 7
1 2 4 2 3 4
0 3 2
Key: \begin{align*}4 \big | 2 = 24\end{align*}42=24 Key: \begin{align*}1 \big | 6 = 16\end{align*}16=16
  1. Identify the stems for this set of data.
  2. Which grade level has a greater mean class size?
  3. Determine the range in class size for \begin{align*}7^{th}\end{align*}7th grade and \begin{align*}8^{th}\end{align*}8th grade.
  4. Which grade level has a smaller median class size?

The data on the stem-and-leaf plots below compare the average daily temperature in Austin, Texas and Seattle, Washington for ten days in January.

Temperature in Two Cities (in Fahrenheit)
Temperature in Austin, Texas Temperature in Seattle, Washington
Leaf Stem Leaf
9 4 0 0 2 4 5 7
9 6 6 3 1 5 1 2 4 6
7 4 2 1 6
Key: \begin{align*}1 \big | 5 = 51\end{align*}15=51 Key: \begin{align*}4 \big | 0 = 40\end{align*}40=40
  1. What are the stems for the data?
  2. What is the coolest temperature in Austin, Texas?
  3. What is the coolest temperature in Seattle, Washington?
  4. Which city has the lowest mean temperature?
  5. What is the range of the temperatures in Austin?
  6. What is the range of the temperatures in Seattle?
  7. What are the modes in both cities?

Review (Answers)

To see the Review answers, open this PDF file and look for section 11.9.

Resources

 

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Vocabulary

bins

Bins are groups of data plotted on the x-axis.

Continuous

Continuity for a point exists when the left and right sided limits match the function evaluated at that point. For a function to be continuous, the function must be continuous at every single point in an unbroken domain.

Leaf

The leaves of a stem-and-leaf plot are the rightmost digits of each of the original data values.

Mean

The mean of a data set is the average of the data set. The mean is found by calculating the sum of the values in the data set and then dividing by the number of values in the data set.

Median

The median of a data set is the middle value of an organized data set.

Mode

The mode of a data set is the value or values with greatest frequency in the data set.

Range

The range of a data set is the difference between the smallest value and the greatest value in the data set.

Stem-and-leaf plot

A stem-and-leaf plot is a way of organizing data values from least to greatest using place value. Usually, the last digit of each data value becomes the "leaf" and the other digits become the "stem".

Truncate

To truncate is to cut off a decimal number at a certain point without rounding.

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