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# Stem-and-Leaf Plots

## Displaying and comparing actual data using place value.

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Stem-and-Leaf Plots

Have you ever examined the route of a race? Well, the students are now looking at the route of the Iditarod. Take a look.

During the third day of class on the Iditarod, the students began to examine the routes on the map. The race begins in Anchorage Alaska and travels 1150 miles of rough terrain to the city of Nome Alaska where it finishes.

While racing, the teams face all kinds of wild weather. There can be blizzards, ice and unbelievable winds not to mention that the surrounds can be woods or frozen tundra.

“Wow, can you imagine being out there and not seeing anyone?” Sam said to his friend Juan.

“I think they do see people at the checkpoints, isn’t that right Mr. Hawkins?” Juan asked as Mr. Hawkins walked around the room.

“Yes Juan. The mushers all have to check in at the checkpoints. It helps the race officials to keep track of everyone as well as check on the dogs, refuel, any precaution can be taken there.”

“How many are there?”

“Well, there are 24, and you can figure out the distances and organize them in a data display. That is the next task that you are all going to work on,” Mr. Hawkins said taking out a giant ruler. “Then I want you to figure out the median distance of the checkpoints.”

Juan took out a piece of paper and wrote down the distances that are between each check point. Here are his notes.

20, 29. 52, 34, 45, 30, 48, 75, 54, 18, 25, 59, 112, 52, 52, 42, 90, 42, 48, 48, 28, 18, 55, 22

Juan wants to create a stem-and-leaf plot to show the data. He knows that once he arranges the data in a stem-and-leaf plot, that it can help him to find the median distance between two checkpoints.

Do you have an idea how Juan can do this? Have you ever created a stem-and-leaf plot? Take the time to learn about them during this Concept. At the end of the Concept, you will see how Juan created his data display.

### Guidance

In this Concept, you will learn another way to display data. This way of organizing and displaying data helps us to see values according to their size and we can order them accordingly. Think about the base ten number system of 10,100,1000 etc. We can organize numbers by ones, tens, hundreds and thousands. In fact, you think about numbers in this way when we work with decimals. Now we are going to look at how to look at the stem of a number so that we can organize it for a visual display of data.

A stem-and-leaf plot organizes data in order. In a stem-and-leaf plot each data value is split into a stem and a leaf.

The leaf is the last digit to the right. The stem is the remaining digits to the left. For the number 243, the stem is 24 and the leaf is 3.

Yes it does. Identifying the leaf will help you to know what the stem is and this will assist you in creating an accurate stem-and-leaf plot.

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

Step 1: Arrange the data in order from least to greatest.

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

Step 2: Separate each number into a stem and a leaf.

Stem Leaf
6 8
7 5 7 9
8 0 2
9 2 6 6 7

Step 3: Create a key and give the stem-and-leaf plot a title.

Science Test Scores: 3rd\begin{align*}3^{rd}\end{align*} Period

Stem Leaf
6 8
7 5 7 9
8 0 2
9 2 6 6 7
Key: 68=68\begin{align*}6 \big | 8 = 68\end{align*}

If we were to analyze this data, you could see that most of the values cluster around the stem of 9. There are more numbers in this category.

Here is another situation where a stem-and-leaf plot is helpful.

Three friends began a babysitting service over the summer. The amount of money they made for each appointment is listed on the data table below. Use the information on the data table to create a stem-and-leaf plot.

June 26, 2006 $17.00 June 27, 2006$12.00
July 5, 2005 $22.00 July 9, 2005$23.00
July 15, 2006 $18.00 July 22, 2006$31.00
August 1, 2006 $40.00 August 5, 2006$35.00
August 13, 2006 $19.00 August 20, 2006$8.00

Step 1: Arrange the data values in order from least to greatest.

Since the data values are to the nearest whole number, arrange the data without the decimal points.

8, 12, 17, 18, 19, 22, 23, 31, 35, 40

Step 2: Separate each value into a stem and a leaf.

Stem Leaf
0 8
1 2 7 8 9
2 2 3
3 1 5
4 0

Step 3: Create a key and give the stem-and-leaf plot a title.

Babysitting Fund
Stem Leaf
0 8
1 2 7 8 9
2 2 3
3 1 5
4 0
Key: 08=8\begin{align*}0 \big | 8 = 8\end{align*}

Now that you know how to create a stem-and-leaf plot, let’s look at how we can use it to analyze data and draw conclusions. First, let’s review some of the vocabulary words that we used in the first lesson of this chapter.

The mean is sometimes also called the average of a set of data. To find the mean, add the data values and then divide the sum by the number of data values.

The median is the data value in the middle when the data is ordered from least to greatest. Since the data is ordered from least to greatest on a stem-and-leaf plot, find the data value in the middle of the stem-and-leaf plot.

The mode is the data value that occurs most often. On a stem-and-leaf plot, the mode is the most repeated leaf.

The range is the difference between the highest and the lowest data value.

Data from a stem-and-leaf plot can be used to determine the mean, median, mode, and range for a set of data. Let’s look at how we can do this.

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

Weight of Trout Caught
Stem Leaf
2 9
3 1
4 0 5
5 2
6 2
7 6
8 3
9 2 2
Key: 29=2.9\begin{align*}2 \big | 9 = 2.9\end{align*}

Step 1: Using the key, combine the stem with each of its leaves. The values are in order from least to greatest on the stem-and-leaf plot. Therefore, keep them in order as you list the data values.

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

Step 2: Recall that to determine the mean you add the data values and then divide the sum by the number of data values.

2.9+3.1+4.0+4.5+5.2+6.2+7.6+8.3+9.2+9.260.2÷10Mean=60.2=6.2=6.2 pounds

Step 3: The data is already arranged in order from least to greatest. Therefore, to determine the median, identify the number in the middle of the data set. In this case, two data values share the middle position. To find the median, find the mean of these two data values.

2.9, 3.1, 4.0, 4.5, 5.2, 6.2, 7.6, 8.3, 9.2, 9.25.2+6.2=11.411.4÷2=5.7Median=5.7 pounds

Step 4: Recall that the mode is the data value that occurs most. Looking at the stem-and-leaf plot, you can see that the data value that the number 9.2 appears twice. Therefore, the mode is 9.2.

Mode = 9.2 pounds

Step 5: Recall that the range is the difference of the greatest and least values. On the stem-and-leaf plot, the greatest value is the last value; the smallest value is the first value.

9.22.9Range=6.3=6.3

Mean = 6.2 pounds

Median = 5.7 pounds

Mode = 9.2 pounds

Range = 6.3 pounds

#### Example A

What is the stem of the number 25?

Solution: 2

#### Example B

What is the leaf of the number 56?

Solution: 6

#### Example C

What is the stem of the number 89?

Solution: 8

Here is the original problem once again. Reread it and then create your own stem-and-leaf plot. Compare your work with Juan’s.

During the third day of class on the Iditarod, the students began to examine the routes on the map. The race begins in Anchorage Alaska and travels 1150 miles of rough terrain to the city of Nome Alaska where it finishes.

While racing, the teams face all kinds of wild weather. There can be blizzards, ice and unbelievable winds not to mention that the surrounds can be woods or frozen tundra.

“Wow, can you imagine being out there and not seeing anyone?” Sam said to his friend Juan.

“I think they do see people at the checkpoints, isn’t that right Mr. Hawkins?” Juan asked as Mr. Hawkins walked around the room.

“Yes Juan. The mushers all have to check in at the checkpoints. It helps the race officials to keep track of everyone as well as check on the dogs, refuel, any precaution can be taken there.”

“How many are there?”

“Well, there are 24, and you can figure out the distances and organize them in a data display. That is the next task that you are all going to work on,” Mr. Hawkins said taking out a giant ruler. “Then I want you to figure out the median distance of the checkpoints.”

Juan took out a piece of paper and wrote down the distances that are between each check point. Here are his notes.

20, 29. 52, 34, 45, 30, 48, 75, 54, 18, 25, 59, 112, 52, 52, 42, 90, 42, 48, 48, 28, 18, 55, 22

Juan wants to create a stem-and-leaf plot to show the data. He knows that once he arranges the data in a stem-and-leaf plot, that it can help him to find the median distance between two checkpoints.

Now Juan is ready to create his stem-and-leaf plot. He begins by organizing his data in order from least to greatest.

18, 18, 20, 22, 25, 28, 29, 30, 34, 42, 42, 45, 48, 48, 48, 52, 52, 52, 54, 55, 59, 75, 90, 112

Next, he can organize the data in stems and leaves.

Stem Leaves
1 8 8
2 0 2 5 8 9
3 0 4
4 2 2 5 8 8 8
5 2 2 2 4 5 9
6
7 5
8
9 5
10
11 2

Juan can see that the median distance centers in the 40 – 50’s mile zone. He makes these notes.

42, 42, 45, 48, 48, 48, 52, 52, 52, 54, 55, 59

The median distance is between 48 and 52 miles. Since there isn’t a check point with a distance of 50 miles, it is accurate to say that the median is both 48 and 52 miles.

### Vocabulary

Stem-and-Leaf Plot
a visual way to organize data which divides numbers up into their stems and their leaves. You are able to easily count the number of values in each grouping.
Leaf
the last digit to the right in the number.
Stem
the rest of the digits to the left of the leaf.
Mean
the average of a set of numbers.
Median
the middle value in a set of numbers.
Mode
the value that occurs the most times in a set of numbers.
Range
the difference between the highest value in a set of numbers and the lowest value in a set of numbers.

### Guided Practice

Here is one for you to try on your own.

Make a stem-and-leaf plot to represent this data.

14, 16, 17, 18, 18, 20, 22, 24, 29, 31, 33

We have three stems: 1, 2, 3.

Here is the stem-and-leaf plot.

Annual DVD Sales (in millions)
Stem Leaf
1 4 6 7 8 8
2 0 2 4 9
3 1 3
Key: 20=20\begin{align*}2 \big | 0 = 20\end{align*}

### Practice

Directions: 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.

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

3. Create a stem-and-leaf plot to display the data.

4. Use the stem-and-leaf plot to determine the mean.

5. Use the stem-and-leaf plot to determine the median.

6. Use the stem-and-leaf plot to determine the mode.

7. 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.

273631292539212634403829\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*}

8. Use the data to create a stem-and-leaf plot.

9. Use the data to determine the mean.

10. Use the data to determine the median.

11. Use the data to determine the mode.

12. 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.

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

7th\begin{align*}7^{th}\end{align*} Grade: 8th\begin{align*}8^{th}\end{align*} Grade:
Leaf Stem Leaf
9 8 1 6 7 7
1 2 4 2 3 4
0 3 2
Key: 42=24\begin{align*}4 \big | 2 = 24\end{align*} Key: \begin{align*}1 \big | 6 = 16\end{align*}

13. Identify the stems for this set of data.

14. Which grade level has a greater mean class size?

15. Determine the range in class size for \begin{align*}7^{th}\end{align*} grade and \begin{align*}8^{th}\end{align*} grade.

16. 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*} Key: \begin{align*}4 \big | 0 = 40\end{align*}

17. What are the stems for the data?

18. What is the coolest temperature in Austin Texas?

19. What is the coolest temperature in Seattle Washington?

20. Which city has the lowest mean temperature?

21. What is the range of the temperatures in Austin?

22. What is the range of the temperatures in Seattle?

23. What are the modes in both cities?

### Vocabulary Language: English

bins

bins

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

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

Leaf

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

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

Median

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

Mode

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

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

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

Truncate

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