A zoologist takes the weights of male and female chimpanzees at 1 year of age. She finds the following data, in pounds, and places it in order from smallest to largest.

Females: 14, 17, 19, 19, 20, 21, 23, 24

Males: 18, 22, 24, 25, 26, 28, 31, 32, 34

How can the zoologist represent this data in order to compare the males and the females in the same data display?

In this concept, you will learn to use stem-and-leaf plots.

### Stem and Leaf Plot

Measures of central tendency are an important method for interpreting a set of data. For a visual representation of data, plots or graphs can be used. Each type of visual tool has advantages and the best type of plot or graph depends on the situation. Indeed, sometimes it is a matter of preference as many different graphs could be used to illustrate the same data.

One common type of data visualization is a stem-and-leaf plot. Consider a stem and its leaves. The stem is a strong base from which the leaves sprout. This is the idea of a stem-and-leaf plot. They visually present groups of data and tendencies quickly while at the same time showing every single piece of data.

Stem-and-leaf plots are organized according to the largest base ten value and the smaller base ten values.

Let’s look at an example.

An accountant must consider the cost of healthcare for the employees at a company. The healthcare costs are based on age of the employees. She finds that the ages of the employees are as follows: 32, 19, 37, 22, 25, 46, 58, 35, 41, 45, 35, 27, 29, 42, 53, 70, 56, 34, 29, 30, 21, 24, 27, and 45.

This data is difficult to understand in an unorganized list. Measures of central tendency could be calculated but they would not help to determine healthcare costs. A stem-and-leaf plot will give her a better idea of numbers of employees per age group.

To create a stem-and-leaf plot, the data must first be put in order from smallest to largest:

19, 21, 22, 24, 25, 27, 27, 29, 29, 30, 32, 34, 35, 35, 37, 41, 42, 45, 45, 46, 53, 56, 58, 70.

Now, stem values must be chosen. That means to choose values that would be the first digit(s) of appropriate groupings. In this case, because our youngest employee is 19 and the oldest is 70, the tens place can be used as the stem.

Construct the stem vertically, as shown below. Then, place each piece of data, the leaves, in the plot, next to its stem. The leaves must be placed in order, only separated by a column. The stem, the tens place, is not repeated.

\begin{align*}\begin{array}{c|c c c c c c c c} \text{Stem} & \text{Leaves}\\
1&9\\
2&1,&2,&4,&5,&7,&7,&9,&9\\
3&0,&2,&4,&5,&5,&7\\
4&1,&2,&5,&5,&6\\
5&3,&6,&8\\
6 \\
7&0
\end{array}\end{align*}

The stem-and-leaf plot is complete.

Now, look at the plot. What trends are visible?

The bulk of the employees are in their 20’s and 30’s. By counting, the median can be quickly located. The importance of lining up the numbers in columns is also visible. The number of data items in each row is readily available.

If the situation had used numbers in the hundreds, then the hundreds would have become the largest stem. If it had been in the thousands, then the thousands would have been the largest stem.

If you have two data sets with similar values, you can plot them all on one stem-and-leaf plot by drawing leaves on either side of the stem - this is called a **double stem-and-leaf plot.**

### Examples

#### Example 1

Earlier, you were given a problem about the scientists who collected weights of chimpanzees and got the following data.

Females: 14, 17, 19, 19, 20, 21, 23, 24

Males: 18, 22, 24, 25, 26, 28, 31, 32, 34

You need to draw a double stem-and-leaf plot. The female data on the left begins with smallest data closest to the stem and increases as you go left.

\begin{align*}\begin{array}{c|c|c c c c c c} \text{Females} & \text{Stem} & \text{Males}\\
9,9,7,4 & 1 & 8 & \\
4,3,1,0& 2 & 2, & 4, & 5, & 6, & 8 \\
& 3 &1, & 2, & 4 \\
\end{array}\end{align*}

Some inferences to be made from this double stem-and-leaf plot:

You can see from the stem-and-leaf plot that the tendency is for the males to weigh more than the females after a year.

You can also see that there are more males in this group than females.

#### Example 2

Create a stem-and-leaf plot of the mass of geodes found at a volcanic site. Scientists measured 24 geodes in kilograms and got the following data:

0.8, 0.9, 1.1, 1.1, 1.2, 1.5, 1.5, 1.6, 1.7, 1.7, 1.7, 1.9, 2.0, 2.3, 4.1, 4.8, 5.3, 5.5, 5,7, 6.8, 7.0, 7.1, 7.5, 9.6

Build a stem-and-leaf plot to organize the data.

\begin{align*}\begin{array}{c|c c c c c c c c c} \text{Stem} & \text{Leaves}\\ 0&8,&9\\ 1&1,&1,&2,&5,&5,&6,&7,&7,&9\\ 2&0,&3 \\ 3\\ 4&1,&8\\ 5&3,&7\\ 6&8 \\ 7&0,&1,&5 \\ 8 \\ 9&6 \end{array}\end{align*}

The stem-and-leaf plot is complete.

Use this data set to answer the following questions.

22, 23, 24, 25, 33, 34, 40, 51, 52, 52, 60, 61, 62

#### Example 3

Draw a stem-and-leaf plot.

\begin{align*}\begin{array}{c|c c c c c c c} \text{Stem} & \text{Leaves}\\ 2&2,&3,&4,&5 \\ 3&3,&4\\ 4&0\\ 5&1,&2,&2\\ 6&0,&1,&2 \\ \end{array}\end{align*}

The stem-and-leaf plot is complete.

#### Example 4

Consider the stem-and-leaf plot from Example 1. Which stem would have the most leaves?

The most leaves are in the 20’s.

#### Example 5

Consider the stem-and-leaf plot from Example 1. Which stem would have the least leaves?

The least leaves are in the 40’s.

### Review

Use each situation to answer the following questions.

Students spent the following total minutes on homework last Thursday evening:

45, 45, 40, 43, 36, 50, 60, 55, 55, 45, 60, 63, 90, 75, 80

1. Make a stem-and-leaf plot that represents the data.

2. Which stem has the greatest number of values?

3. Which stem has the least number of values?

4. What can you interpret from the plot?

5. Explain the intervals that you chose.

6. Why is it necessary to show intervals for which there was no data?

A hybrid car and a gasoline-only car filled up on the same days of the month. The drivers recorded the gasoline costs for the two cars.

Hybrid:

$17, $24, $19, $21, $10, $12, $15, $20, $6, $16

Gasoline-Only:

$34, $27, $15, $31, $29, $27, $24, $14, $35, $28

7. Create a double stem-and-leaf plot to represent this data.

8. Which stem has the greatest number of values for a hybrid?

9. Which stem has the greatest number of values for a gasoline-only car?

10. What can you conclude from your stem-and-leaf plot?

Kelly earned the following amounts of money babysitting:

$30.00, $10.00, $15.00, $20.00, $18.00, $22.00, $35.00, $40.00 and $58.00.

11. Make a stem-and-leaf plot that represents the data.

12. Which stem has the greatest number of values?

13. Which stem has the least number of values?

14. What can you interpret from the plot?

15. What was the median amount of money that she made?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 10.3.