### Two-Sided Stem-and-Leaf Plots

As you have learned in a previous section, stem-and-leaf plots are an excellent tool for organizing data. Remember that stem-and-leaf plots are a visual representation of grouped discrete data, but they can also be referred to as a modal representation. This is because by looking at a stem-and-leaf plot, we can determine the mode by quick visual inspection. The previous section was about single-sided stem-and-leaf plots. In this section, you will learn about **two-sided stem-and-leaf plots**, which are also often called back-to-back stem-and-leaf plots.

#### Interpreting Stem-and-Leaf Plots

The girls and boys in one of BDF High School's AP English classes are having a contest. They want to see which group can read the most number of books. Mrs. Stubbard, their English teacher, says that the class will tally the number of books each group has read, and the highest mode will be the winner. The following data was collected for the first semester of AP English:

\begin{align*}& \text{Girls} \qquad 11 \quad 12 \quad 12 \quad 17 \quad 18 \quad 23 \quad 23 \quad 23 \quad 24 \quad 33 \quad 34 \quad 35 \quad 44 \quad 45 \quad 47 \quad 50 \quad 51 \quad 51\\ & \text{Boys} \qquad 15 \quad 18 \quad 22 \quad 22 \quad 23 \quad 26 \quad 34 \quad 35 \quad 35 \quad 35 \quad 40 \quad 40 \quad 42 \quad 47 \quad 49 \quad 50 \quad 50 \quad 51\end{align*}

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

b. Determine the mode for each group.

The mode for the girls is 23 books. It is the number in the girls column that appears most often. The mode for the boys is 35 books. It is the number in the boys column that appears most often.

c. Help Mrs. Stubbard decide which group won the contest.

Mrs. Stubbard should decide that the boys group has won the contest.

#### Constructing a Two-Sided Stem-and-Leaf Plots

1. Mrs. Cameron teaches AP Statistics at GHI High School. She recently wrote down the class marks for her current grade 12 class and compared it to the previous grade 12 class. The data can be found below. Construct a two-sided stem-and-leaf plot for the data and compare the distributions.

\begin{align*}\text{2010 class} \qquad & 70 \quad 70 \quad 70 \quad 71 \quad 72 \quad 74 \quad 74 \quad 74 \quad 74 \quad 75 \quad 76 \quad 76 \quad 77 \quad 78 \quad 79 \quad 80 \quad 81\\ & 82 \quad 82 \quad 82 \quad 83 \quad 84 \quad 85 \quad 85 \quad 86 \quad 87 \quad 93 \quad 98 \quad 100\\ \text{2009 class} \qquad & 76 \quad 76 \quad 76 \quad 76 \quad 77 \quad 78 \quad 78 \quad 78 \quad 79 \quad 80 \quad 80 \quad 82 \quad 82 \quad 83 \quad 83 \quad 83 \quad 85 \\ & 85 \quad 88 \quad 91 \quad 95\end{align*}

There is a wide variation in the marks for both years in Mrs. Cameron’s AP Statistics Class. In 2009, her class had marks anywhere from 76 to 95. In 2010, the class marks ranged from 70 to 100. The mode for the 2009 class was 76, but for the 2010 class, it was 74. It would seem that the 2009 class had, indeed, done slightly better than Mrs. Cameron’s current class.

2. The following data was collected in a survey done by Connor and Scott for their statistics project. The data represents the ages of people who entered into a new hardware store within its first half hour of opening on its opening weekend. The M's in the data represent males, and the F's represent females.

\begin{align*}&12M \quad \ 18F \quad 15F \quad \ 15M \quad \ 10M \quad 21F \quad 25M \quad 21M\\ & 26F \quad \ 29F \quad \ 29F \quad \ 31M \quad 33M \quad 35M \quad 35M \quad 35M\\ & 41F \quad \ 42F \quad \ 42M \quad 45M \quad 46F \quad \ 48F \quad \ 51M \quad 51M\\ & 55F \quad \ 56M \quad 58M \quad 59M \quad 60M \quad 60F \quad \ \ 61F \quad 65M\\ & 65M \quad 66M \quad 70M \quad 70M \quad 71M \quad 71M \quad \ 72M \quad 72F\end{align*}

Construct a back-to-back stem-and-leaf plot showing the ages of male customers and the ages of female customers. Compare the distributions.

For the male customers, the ages ranged from 10 to 72. The ages for the male customers were spread out throughout this range, with the mode being age 35. In other words, for the males found to be at the store in the first half hour of opening day, there was no real age category where a concentration of males could be found.

For the female customers, the ages ranged from 15 to 72, but they were concentrated between 21 and 48. The mode for the ages of the female customers was 29 years of age.

### Example

The boys and girls basketball teams at a high school had their heights measured at practice. The following data was recorded for their heights (in centimeters):

\begin{align*}& \text{Girls} \qquad 171 \quad 170 \quad 176 \quad 176 \quad 177 \quad 179 \quad 162 \quad 172 \quad 160 \quad 157 \quad 155\\ & \qquad \qquad 168 \quad 178 \quad 174 \quad 170 \quad 155 \quad 155 \quad 154 \quad 164 \quad 145 \quad 171 \quad 161\\ & \text{Boys} \qquad 168 \quad 170 \quad 162 \quad 153 \quad 176 \quad 167 \quad 158 \quad 180 \quad 181 \quad 176 \quad 172\\ & \qquad \qquad 168 \quad 167 \quad 165 \quad 159 \quad 185 \quad 184 \quad 173 \quad 177 \quad 167 \quad 169 \quad 177\end{align*}

#### Example 1

Construct a two-sided stem-and-leaf plot for the data. Determine the median and mode using the two-sided stem-and-leaf plot for each distribution. What can you conclude from the distributions?

The data suggests that there is a slightly wider variation in the heights for the group of girls than for the group of boys. For the girls, the heights ranged from 145 to 179 centimeters, whereas for the boys, the heights ranged from 153 to 185 centimeters. The median for the girls group is at 168.5 centimeters, and the mode is at 155 centimeters. For the group of boys, however, the median is at 169.5 centimeters, and the mode is at 167 centimeters. The boys seem to be taller than the girls.

### Review

The two-sided stem-and-leaf plot below shows the number of home runs hit by the members of 2 major league baseball teams. Use the two-sided stem-and-leaf plot to answer the following questions:

- What was the range for the number of home runs hit by the Mets? What was the range for the Phillies?
- What was the median for the number of home runs hit by the Mets? What was the median for the Phillies?
- What was the mode for the number of home runs hit by the Mets? What was the mode for the Phillies?
- Which team had more players hit 20 or more home runs?

30 girls and 35 boys participated in an intramural bowling league. The two-sided stem-and-leaf plot below shows the highest score of each of the participants. Use the two-sided stem-and-leaf plot to answer the following questions:

- What was the range for the highest scores for the girls? What was the range for the boys?
- What was the median for the highest scores for the girls? What was the median for the boys?
- What was the mode for the highest scores for the girls? What was the mode for the boys?
- Did a girl or a boy have the highest score in the intramural bowling league?

- Mr. Dugas, the senior high physical education teacher, is doing fitness testing this week in gym class. After each test, students are required to take their pulse rate and record it on the chart in the front of the gym. At the end of the week, Mr. Dugas looks at the data in order to analyze it. The data is shown below: \begin{align*}& \text{Girls} \qquad 70 \quad 88 \quad 80 \quad 76 \quad 76 \quad 77 \quad 89 \quad 72 \quad 72 \quad 76 \quad 72 \quad 75 \quad 77 \quad 80 \quad 76 \quad 68 \quad 68\\ & \qquad \qquad 82 \quad 78 \quad 60 \quad 64 \quad 64 \quad 65 \quad 81 \quad 84 \quad 84 \quad 79 \quad 78 \quad 70\\ &\text{Boys} \qquad 76 \quad 88 \quad 87 \quad 86 \quad 85 \quad 70 \quad 76 \quad 70 \quad 70 \quad 79 \quad 80 \quad 82 \quad 82 \quad 82 \quad 83 \quad 84 \quad 85\\ & \qquad \qquad 85 \quad 78 \quad 81 \quad 85\end{align*} Construct a two-sided stem-and-leaf plot for the data and compare the distributions.
- Starbucks prides itself on its low line-up times in order to be served. A new coffee house in town has also boasted that it will have your order in your hands and have you on your way quicker than the competition. The following data was collected for the line-up times (in minutes) for both coffee houses: \begin{align*}& \text{Starbucks} \qquad \qquad 20 \quad 26 \quad 26 \quad 27 \quad 19 \quad 12 \quad 12 \quad 16 \quad 12 \quad 15 \quad 17 \quad 20 \quad 8 \ \quad \ 8 \ \quad 18\\ & \text{Just Us Coffee} \qquad 17 \quad 16 \quad 15 \quad 10 \quad 16 \quad 10 \quad 10 \quad 29 \quad 20 \quad 22 \quad 22 \quad 12 \quad 13 \quad 24 \quad 15\end{align*} Construct a two-sided stem-and-leaf plot for the data. Determine the median and mode using the two-sided stem-and-leaf plot. What can you conclude from the distributions?

### Review (Answers)

To view the Review answers, open this PDF file and look for section 8.3.