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As you have learned in an earlier chapter, 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. In the last chapter, you learned about single-sided stem-and-leaf plots. In this lesson, you will learn about two-sided stem-and-leaf plots, which are also often called back-to-back stem-and-leaf plots.

Example 5

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:

& \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

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

b. Determine the mode for each group.

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

Solution:

a.

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

Example 6

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.

\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

Solution:

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.

Example 7

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.

&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

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

Solution:

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.

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