Suppose that your physical education teacher asked each of the members of your gym class to do as many pull-ups as possible. If you recorded the number of pull-ups completed by each person, would you be able to represent this data with a stem-and-leaf plot? How about with a histogram?

### Graphing Data

Understanding data is a very important mathematical ability. You must know how to use data and interpret the results to make informed decisions about politics, food, and income. This section will show two ways to graph data:

- as a stem-and-leaf plot and
- as a histogram

#### Stem-and-Leaf Plots

A **stem-and-leaf plot** is an organization of numerical data into categories based on place value. The stem-and-leaf plot is a graph that is similar to a histogram but it displays more information. For a stem-and-leaf plot, each number will be divided into two parts using place value.

The **stem** is the left-hand column and will contain the digits in the largest place. The right-hand column will be the **leaf** and it will contain the digits in the smallest place.

#### Let's represent the data below in a stem-and-leaf plot:

25 |
60 | 120 | 64 | 65 | 28 | 110 | 60 |

70 | 34 | 35 | 70 | 58 | 100 | 55 | 95 |

55 | 95 | 93 | 50 | 75 | 35 | 40 | 75 |

90 | 40 | 50 | 80 | 85 | 50 | 80 | 47 |

50 | 80 | 90 | 42 | 49 | 84 | 35 | 70 |

The stems will be arranged vertically in ascending order (smallest to largest) and each leaf will be written to the right of its stem horizontally in order from least to greatest.

Stem |
Leaf |
---|---|

2 | 5, 8 |

3 | 4, 5, 5, 5 |

4 | 0, 0, 2, 7, 9 |

5 | 0, 0, 0, 0, 5, 5, 8 |

6 | 0, 0, 4, 5 |

7 | 0, 0, 0, 5, 5 |

8 | 0, 0, 0, 4, 5 |

9 | 0, 0, 3, 5, 5 |

10 | 0 |

11 | 0 |

12 | 0 |

This stem-and-leaf plot can be interpreted very easily. By looking at stem 6, you see that 4 males spent 60 ‘some dollars’ on prom night. By counting the number of leaves, you know that 40 males responded to the question concerning how much money they spent on prom night. The smallest and largest data values are known by looking and the first and last stem-and-leaf. The stem-and-leaf is a ‘quick look’ chart that can quickly provide information from the data. This also serves as an easy method for sorting numbers manually.

#### Histograms

Suppose you took a survey of 20 algebra students, asking their number of siblings. You would probably get a variety of answers. Some students would have no siblings while others would have several. The results may look like this.

1, 4, 2, 1, 0, 2, 1, 0, 1, 2, 1, 0, 0, 2, 2, 3, 1, 1, 3, 6

We could organize this many ways. The first way might just be to create an ordered list, relisting all numbers in order, starting with the smallest.

0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 4, 6

Another way to list the results is in a table.

Number of Siblings |
Number of Matching Students |
---|---|

0 | 4 |

1 | 7 |

2 | 5 |

3 | 2 |

4 | 1 |

5 | 0 |

6 | 1 |

A table showing the number of times a particular category appears in a data set is a **frequency distribution**.

You could also make a visual representation of the data by making categories for the number of siblings on the \begin{align*}x-\end{align*}axis and stacking representations of each student above the category marker. We could use crosses, stick-men or even photographs of the students to show how many students are in each category.

This graph is called a histogram.

A **histogram** is a bar chart that describes a frequency distribution.

The horizontal axis of the histogram is separated into equal intervals. The vertical bars represent how many items are in each interval.

#### Let's interpret what the following histogram is showing:

Jim collected data at a local fair. The above histogram relates the number of children of particular ages who visited the face-painting booth.

- You can find the sum of the heights of each bar to determine how many children visited the face painting booth.

\begin{align*}1+3+4+5+6+7+3+1=30 \ children\end{align*}

- The tallest bar is at age seven. There were more seven-year olds than any other age group.
- There is no bar at one or ten. This means zero ten-year-olds and zero one-year-olds had their faces painted.

#### Creating Histograms Using a Graphing Calculator

Drawing a histogram is quite similar to drawing a scatter plot. Instead of graphing two lists, \begin{align*}L_1\end{align*} and \begin{align*}L_2\end{align*}, you will graph only one list, \begin{align*}L_1\end{align*}.

The following un-ordered data represents the ages of passengers on a train carriage. Let's make a histogram of the data on a graphing calculator.

35, 42, 38, 57, 2, 24, 27, 36, 45, 60, 38, 40, 40, 44, 1, 44, 48, 84, 38, 20, 4,

2, 48, 58, 3, 20, 6, 40, 22, 26, 17, 18, 40, 51, 62, 31, 27, 48, 35, 27, 37, 58, 21.

Begin by entering the data into **[LIST 1]** of the **[STAT]** menu.

Choose **\begin{align*}[2^{nd}]\end{align*}** and \begin{align*}[Y=]\end{align*} to enter into the **[STATPLOT]** menu.

Press **[WINDOW]** and ensure that **Xmin** and **Xmax** allow for all data points to be shown. The **Xscl** value determines the bin width.

Press **[GRAPH]** to display the histogram.

### Examples

#### Example 1

Earlier, you were asked if you could represent the number of pull-ups completed by each person in your class by a stem-and-leaf plot or a histogram.

Yes, you could represent this data with a stem-and-leaf plot or a histogram. Both are great ways to represent large amounts of data.

#### Example 2

Studies (and logic) show that the more homework you do the better your grade in a course will be. In a study conducted at a local school, students in grade 10 were asked to check off what box represented the average amount of time they spent on homework each night. The following results were recorded:

Time Spent on Homework (Hours) |
Tally |
Frequency (# of students) |
---|---|---|

\begin{align*}[0-0.5)\end{align*} | \begin{align*}\cancel{||||} \ \cancel{||||} \ ||\end{align*} | 12 |

\begin{align*}[0.5-1.0)\end{align*} | \begin{align*}\cancel{||||} \ \cancel{||||} \ \cancel{||||} \ \cancel{||||} \ |||\end{align*} | 23 |

\begin{align*}[1.0-1.5)\end{align*} | \begin{align*}\cancel{||||} \ \cancel{||||} \ \cancel{||||} \ \cancel{||||} \ \cancel{||||} \ \cancel{||||} \ ||||\end{align*} | 34 |

\begin{align*}[1.5-2.0)\end{align*} | \begin{align*}\cancel{||||} \ \cancel{||||} \ \cancel{||||} \ \cancel{||||} \ \cancel{||||} \ |\end{align*} | 26 |

\begin{align*}[2.0-2.5)\end{align*} | \begin{align*}\cancel{||||}\end{align*} | 5 |

\begin{align*}2.5^+\end{align*} | 0 |

Convert this data into a histogram.

### Review

- What is the
*stem*in a stem-and-leaf plot? What is a*leaf*? What is an advantage to using a stem-and-leaf plot? - Describe a
*histogram*. What is an advantage of using a histogram? - For each of the following examples, describe why you would likely use a histogram.
- Frequency of the favorite drinks for the first 100 people to enter the school dance
- Frequency of the average time it takes the people in your class to finish a math assignment
- Frequency of the average distance people park their cars away from the mall in order to walk a little more

- Prepare a histogram using the following scores from a recent science test. Offer four conclusions you can make about this histogram? Include mean, median, mode, range, etc.

Score (%) |
Tally |
Frequency |
---|---|---|

50-60 | \begin{align*}||||\end{align*} | 4 |

60-70 | \begin{align*}\cancel{||||} \ |\end{align*} | 6 |

70-80 | \begin{align*}\cancel{||||} \ \cancel{||||} \ |\end{align*} | 11 |

80-90 | \begin{align*}\cancel{||||} \ |||\end{align*} | 8 |

90-100 | \begin{align*}||||\end{align*} | 4 |

- A research firm has just developed a streak-free glass cleaner. The product is sold at a number of local chain stores and its sales are being closely monitored. At the end of one year, the sales of the product are released. The company is planning on starting up an advertisement campaign to promote the product. The data is found in the chart below. \begin{align*}&266 && 94 && 204 && 164 && 219 && 163\\
&87 && 248 && 137 && 193 && 144 && 89\\
&175 && 164 && 118 && 248 && 159 && 123\\
&220 && 141 && 122 && 143 && 250 && 168\\
&100 && 217 && 165 && 226 && 138 && 131\end{align*}
- Display the sales of the product before the ad campaign in a stem-and-leaf plot.
- How many chain stores were involved in selling the streak-free glass cleaner?
- In stem 1, what does the number 11 represent? What does the number 8 represent?
- What percentage of stores sold less than 175 bottles of streak-free glass cleaner?

- Using the following data, answer the questions that follow. Data: 607.4, 886.0, 822.2, 755.7, 900.6, 770.9, 780.8, 760.1, 936.9, 962.9, 859.9, 848.3, 898.7, 670.9, 946.7, 817.8, 868.1, 887.1, 881.3, 744.6, 984.9, 941.5, 851.8, 905.4, 810.6, 765.3, 881.9, 851.6, 815.7, 989.7, 723.4, 869.3, 951.0, 794.7, 807.6, 841.3, 741.5, 822.2, 966.2, 950.1 A. Create a stem-and-leaf plot. Round each data point to the nearest tens place. Use the hundreds digit as the stem and the tens place as the leaf.
- What is the mean of the data?
- What is the mode of the data?
- What is the median of the data?

B. Make a frequency table for the data. Use a bin width of 50. C. Plot the data as a histogram with a bin width of:

- 50
- 100

- The women from the senior citizen’s complex bowl every day of the month. Lizzie had never bowled before and was enjoying this newfound pastime. She decided to keep track of her best score of the day for the month of September. Here are the scores that she recorded: \begin{align*}77&& 80 && 82 && 68 && 65 && 59 && 61\\ 57 && 50 && 62 && 61 && 70 && 69 && 64\\ 67 && 70 && 62 && 65 && 65 && 73 && 76\\ 87 && 80 && 82 && 83 && 79 && 79 && 77\\ 80 && 71 && && && && &&\end{align*} In order for Lizzie to see how well she is doing, create a stem-and-leaf plot of her scores.
- It is your job to entertain your younger sibling every Saturday morning. You decide to take the youngster to the community pool to swim. Since swimming is a new thing to do, your little buddy isn’t too sure about the water and is a bit scared of the new adventure. You decide to keep a record of the length of time he stays in the water each morning. You recorded the following times (in minutes): 12, 13, 21, 27, 33, 34, 35, 37, 40, 40, 41. Create a stem-and-leaf plot to represent this data. List two conclusions you can make from this graph.
- The following stem-and-leaf plot shows data collected for the speed of 40 cars in a 35 mph limit zone in Culver City, California.
- Find the mean, median, and mode speed.
- Complete a frequency table, starting at 25 mph with a bin width of 5 mph.
- Use the table to construct a histogram with the intervals from your frequency table.

- The following histogram displays the results of a larger-scale survey of the number of siblings. Use it to find:
- The median of the data
- The mean of the data
- The mode of the data
- The number of people who have an odd number of siblings
- The percentage of the people surveyed who have four or more siblings

- Use the stem-and-leaf plot to devise a situation that follows the data.
- An oil company claims that its premium-grade gasoline contains an additive that significantly increases gas mileage. To prove their claim, the company selected 15 drivers and first filled each of their cars with 45 L of regular gasoline and asked them to record their mileage. The company then filled each of the cars with 45 L of premium gasoline and again asked them to record their mileage. The results below show the number of kilometers each car traveled.

640 | 570 | 660 | 580 | 610 |
---|---|---|---|---|

540 | 555 | 588 | 615 | 570 |

550 | 590 | 585 | 587 | 591 |

659 | 619 | 639 | 629 | 664 |
---|---|---|---|---|

635 | 709 | 637 | 633 | 618 |

694 | 638 | 689 | 589 | 500 |

Display each set of data to explain whether or not the claim made by the oil company is true or false.

**Mixed Review**

- How many ways can a nine-person soccer team line up for a picture if the goalie is to be in the center?
- Graph \begin{align*}8x+5y=40\end{align*} using its intercepts.
- Simplify \begin{align*}\left ( \frac{7y^{-3} z^2}{4y^4 z^{-6}} \right )^{-2}\end{align*}.
- Rewrite in standard form: \begin{align*}y=-3(x-1)^2+4\end{align*}.
- Graph \begin{align*}f(x)=\frac{-x+2}{2}\end{align*}.
- A ball is dropped from a height of 10 meters. When will it reach the ground?
- Solve the following system: \begin{align*}\begin{cases}3x+4y=9\\ 9x+12y=27 \end{cases}\end{align*}.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 11.10.