Kalena’s cheerleading squad is raising money for prom by selling candy at school football games. After a month of sales, the squad is running low on candy and decides to review the sales so far to help them decide what to order when they restock.

The table below describes the sales that the girls have recorded. How could the girls create a visual representation of the data so they can more easily present their findings to the purchasing committee? Ideally, they would like to order more of the item that is proving the most profitable, and so they want to present the data to the committee in a way that highlights this. We will review this question after the lesson.

Item |
# of Sales |

Popsicles $1 ea | 850 |

Chocolate Bars $0.85 ea | 1300 |

Bag of Lemon Drops $1.25 ea | 340 |

Ice Cream Bars $1 ea | 670 |

### Creating Pie Charts

Pie charts are an excellent way to compare multiple values that make up parts of a whole. Each piece of the pie is called a ** sector,** and each sector encompasses an angle that is proportional to the frequency of the data it represents. The formula relating the angle of a sector to frequency is:

\begin{align*}Sector \ Angle=\frac{Frequency \ of \ Data \ in \ the \ Sector}{Total \ Frequency \ of \ Data \ in \ Sample} \times 360^\circ\end{align*}

In order to construct an accurate pie chart, you will need to calculate the sector angles for each of the categories or intervals in your sample, verifying that they total \begin{align*}360^\circ\end{align*}.

Once you know the angles for each sector construct a circle and mark the sectors within it with lines running from the center of the circle to the edge (radii). Make sure that the angle between the lines is equal to the calculated sector angle for each category.

Finally you need to either label the sectors directly or create a key similar to the one in the concept question above so your audience can easily identify which sector corresponds to each category in your sample.

**Constructing a Pie Chart **

1. In Karen’s school, there were 480 students in 1997, 540 students in 2000, 710 students in 2003, and 900 students in 2006. Construct a pie chart to represent the relative numbers of students each year.

First, calculate the total number of students over all four categories (years):

\begin{align*}\text{Total number of students} = 480 + 540 + 710 + 900 = 2,630\end{align*}

Now calculate the angle of each sector with \begin{align*}Sector \ Angle=\frac{Frequency \ of \ Data \ in \ the \ Sector}{Total \ Frequency \ of \ Data \ in \ Sample} \times 360^\circ\end{align*}

- 1997: \begin{align*}sector \ angle=\frac{480}{2630} \times 360^\circ \rightarrow .183 \times 360^\circ \rightarrow \approx 66^\circ\end{align*}
- 2000: \begin{align*}sector \ angle = \frac{540}{2630} \times 360^\circ \rightarrow .205 \times 360^\circ \rightarrow \approx 74^\circ\end{align*}
- 2003: \begin{align*}sector \ angle=\frac{710}{2630} \times 360^\circ \rightarrow .270 \times 360^\circ \rightarrow \approx 97^\circ\end{align*}
- 2006: \begin{align*}sector \ angle=\frac{900}{2630} \times 360^\circ \rightarrow .342 \times 360^\circ \rightarrow \approx 123^\circ\end{align*}

Verify that your degree measures total \begin{align*}360^\circ\end{align*}:

\begin{align*}66^\circ+74^\circ+97^\circ+123^\circ=360^\circ\end{align*}

Finally, construct your circle and draw the internal angles equal to the calculated sector angles, and color-code and/or directly label each sector:

2. Create a pie chart to display the data from the table below:

Ford | 57 |

Chevrolet | 49 |

Dodge | 36 |

Toyota | 27 |

Nissan | 16 |

BMW | 5 |

Mercedes | 3 |

All Others | 17 |

First total the number of cars in the entire population:

\begin{align*}57+49+36+27+16+5+3+17=210\end{align*}

Now find the degree measure of each sector using:

\begin{align*}Sector \ Angle=\frac{Frequency \ of \ Data \ in \ the \ Sector}{Total \ Frequency \ of \ Data \ in \ Sample} \times 360^\circ\end{align*}

- Ford: \begin{align*}Angle=\frac{57}{210} \times 360^\circ \approx 98^\circ\end{align*}
- Chevrolet: \begin{align*}Angle=\frac{49}{210} \times 360^\circ=84^\circ\end{align*}
- Dodge: \begin{align*}Angle=\frac{36}{210} \times 360^\circ \approx 62^\circ\end{align*}
- Toyota: \begin{align*}Angle=\frac{27}{210} \times 360^\circ \approx 46^\circ\end{align*}
- Nissan: \begin{align*}Angle=\frac{16}{210} \times 360^\circ \approx 27^\circ\end{align*}
- BMW: \begin{align*}Angle=\frac{5}{210} \times 360^\circ \approx 9^\circ\end{align*}
- Mercedes: \begin{align*}Angle=\frac{3}{210} \times 360^\circ \approx 5^\circ\end{align*}
- All Others: \begin{align*}Angle=\frac{17}{210} \approx 360^\circ \approx 29^\circ\end{align*}

Verify that the total is \begin{align*}360^\circ\end{align*}: \begin{align*}98^\circ+84^\circ+62^\circ+46^\circ+27^\circ+9^\circ+5^\circ+29^\circ=360^\circ\end{align*}

Construct a circle with sectors representing each degree measure and label directly or create a key:

**Using Technology **

Use a spreadsheet or compass and protractor to create two related pie charts of the data in the table below. If using a modern spreadsheet, create a 3-D graph. Highlight the data from the soccer participants in both graphs.

Sport |
Count of Middle School Participants |
Count of High School Participants |

Football | 186 | 279 |

Volleyball | 28 | 57 |

Soccer | 66 | 92 |

Track | 82 | 124 |

By now you are familiar with creating charts using a pencil and paper, so let’s walk through creating a chart in a modern spreadsheet.

If you do not have spreadsheet software on your computer, you can download the *free* ‘Open Office Calc’ spreadsheet software from from Open Office's website. The process described here is essentially the same in any modern spreadsheet software such as Open Office Calc, Microsoft Excel or Numbers.

First, highlight and copy the two columns of data under ‘Sport’ and ‘Count of Middle School Participants’ from the table in the question, include the column headers.

Now open a blank spreadsheet in your software and paste the two columns of data.

Highlight the data in the spreadsheet (including the headers), and either choose ‘Charts : Pie’ from the toolbar or click the icon that looks like a bar graph and choose ‘Pie.’

The data will be immediately converted to a pie chart for you! Now pick a 3-D style (since that is what was specified in the question), and press ‘enter’ to get a chart like the one below.

To highlight the soccer data, simply select the sector representing the soccer participants and drag it away from the center of the circle just slightly. Finally, select the entire chart in the spreadsheet and copy/paste it into your answer document or print it out to turn it in.

To create the chart for the high school data, just copy the data from the single ‘Count of High School Participants’ category and paste it right overtop of the Middle School column in your spreadsheet and repeat the steps above to convert the revised table into a chart.

The two final products should look something like these:

**Earlier Problem Revisited**

Create a pie chart that compares the income from each product for the cheerleading squad. Use the data in the table of candy sales.

In the example problems, the first step was to total the number of items sold. However, in this problem, we need to compare the *dollar value* of the items rather than just the *number* of items. That means we need to first evaluate the dollar value of each product sale, and then calculate the angle of each slice based on a comparison of the dollar value of each product with the total income from sales. Just to keep things neat, let’s add another column to the original table called “dollar value”, and another row at the bottom for the total.

Item |
# of Sales |
$ Value |

Popsicles $1 ea | 850 | $850.00 |

Chocolate Bars $0.85 ea | 1300 | $1,105.00 |

Bag of Lemon Drops $1.25 ea | 340 | $425.00 |

Ice Cream Bars $1 ea | 670 | $670.00 |

TOTAL | 3160 | $3,050.00 |

Now we can use the formula to calculate the angle of each slice:

- Popsicles: \begin{align*}Angle=\frac{\$850}{\$3,050} \times 360^\circ \approx 100^\circ\end{align*}
- Chocolate Bars: \begin{align*}Angle=\frac{\$1,105}{\$3,050} \times 360^\circ \approx 130^\circ\end{align*}
- Lemon Drops: \begin{align*}Angle=\frac{\$425}{\$3,050} \times 360^\circ \approx 50^\circ\end{align*}
- Ice Cream Bars: \begin{align*}Angle = \frac{\$670}{\$3,050} \times 360^\circ \approx 79^\circ\end{align*}

Finally, we construct our circle and mark the divisions of the sectors based on the angles we have calculated, label the sectors, and label the graph.

### Examples

#### Example 1

Larry Bird was a well know basketball player. He played for the Boston Celtics. Use the following information to create a pie chart.

Season: |
# of Points Scored |

1979-1980 | 1745 |

1980-1981 | 1741 |

1981-1982 | 1761 |

1982-1983 | 1867 |

1983-1984 | 1908 |

1984-1985 | 2295 |

1985-1986 | 2115 |

1986-1987 | 2076 |

1987-1988 | 2275 |

1988-1989 | 116 |

1989-1990 | 1820 |

1990-1991 | 1164 |

1991-1992 | 908 |

#### Example 2

What percent of his career points came from the 1988 season?

First we must find the total number of baskets Larry made in the portion of his career represented by the chart. The answer is 21,792. This number represents 100 percent of our chart. To find what percentage of those shots were made in 1988, we write an algebraic word problem that looks this:

What percent of 21,792 is 2275? As an equation: \begin{align*}\boxed{?} \times 21,792 = 2275 \rightarrow \boxed{?}=\frac{2275}{21792} \rightarrow \boxed{?} =10.4\%\end{align*}

Pecent of career shots made in 1988 is 10.4%

#### Example 3

What percent of his career points came from the the 1980 season?

Calculate this the same way: \begin{align*}x \% \times 21792=1741 \rightarrow \frac{1741}{21792}=.08 \rightarrow x=8\%\end{align*}

Percent of shots made in his career in 1980 is 8%

#### Example 4

The following table shows the grades achieved by 30 pupils in their end of year exam.

Grade |
A | B | C | D | F |

Frequency |
8 | 10 | 7 | 3 |
3 |

Based on the number of pupils, how many degrees of a circle graph are allotted per student? What percentage of the pie chart is allotted to "C" grades?

There are 31 total grades, 7 of which are “C’s”. To find percentage, just divide: \begin{align*}\frac{7}{31}=.225\end{align*}, and then multiply by 100 to get percentage. **“C” grades make up 22.5% of the chart**.

#### Example 5

Create a pie chart for the information from the previous example.

Your pie chart should look something like the image below, with sectors measuring:

- Grade A: \begin{align*}\frac{8}{31}=.26=26\%\end{align*} Convert to degrees: \begin{align*}.26 \times 360^\circ=94^\circ\end{align*}
- Grade B: \begin{align*}\frac{10}{31}=.32=32\%\end{align*} Convert to degrees: \begin{align*}.32 \times 360^\circ=115^\circ\end{align*}
- Grade C: \begin{align*}\frac{7}{31}=.225=23\%\end{align*} Convert to degrees: \begin{align*}.225 \times 360^\circ=81^\circ\end{align*}
- Grade D: \begin{align*}\frac{3}{31}=.097=10\%\end{align*} Convert to degrees: \begin{align*}.097 \times 360^\circ=35^\circ\end{align*}
- Grade F: \begin{align*}\frac{3}{31}=.097=10\%\end{align*} Convert to degrees: \begin{align*}.097 \times 360^\circ=35^\circ\end{align*}

### Review

1. What is each part of a pie chart called?

2. What part of the circle represents the whole relationship?

3. Based on the data in the table below, which candy sold the most, and which the least?

Candy Bar |
Number sold at the school store |
Percentage of Total Sales |
Number of Degrees on a Pie Chart |

Heath | 151 | ||

M&M | 191 | ||

Snickers | 61 | ||

Skittles | 107 | ||

Almond Joy | 91 |

4. Fill in the chart above with percentages of each candy type sold.

5. Fill in the chart with degrees of a circle required to represent this amount on a pie chart.

6. Create a Pie Chart to represent this data.

7. Based on the data in the table below, which stock has the greatest potential for making money for the investor who owns the stock?

Company |
Shares Owned |
Percentage of Total Portfolio |
Number of Degrees on a Pie Chart |

Hostess | 8 | ||

Pepsi | 11 | ||

Dell | 5 | ||

Conoco | 7 | ||

Ford Motor | 19 |

8. Fill in the chart with relative percentages of each type of stock.

9. Fill in the chart with degrees of a circle required to represent this amount on a pie chart.

10. Create a Pie Chart to represent this data.

Use the data presented in questions 11-15 to create pie charts for each:

11. Julie runs for one hour per day, reads for two and sleeps nine. She spends about two hours eating, at least one hour on the phone with friends. She hangs out with her family on average 4 hours a day, and spends at least five hours a day studying.

12. Mrs. Garcia makes $1200.00 a month. She puts 10% in savings, spends 20% on her car payment and insurance, and another 20% on groceries. She likes clothes, so 10% of her income goes towards her wardrobe. Of her remaining money she spends 30% on her mortgage, and the remainder on miscellaneous expenses. Determine exactly how much Mrs. Garcia spends in each category. What percent of her income goes to miscellaneous expenses?

13. Katie earned 500.00 doing odd jobs for people in her neighborhood. She spent \begin{align*}\frac{1}{16}\end{align*} of her money on the movies, she spent \begin{align*}\frac{3}{8}\end{align*} of her money going out with friends. She spent \begin{align*}\frac{1}{2}\end{align*} on clothes, and the remainder on books. How much money did Katie spend on movies? What percentage was spent on books?

14. The state of Colorado receives 28 inches of precipitation a year. The winter is when it gets most of it, but it does not see it until it melts and runs off in the spring. However, inches are counted when they accumulate, and so they represent precipitation for Colorado based on a 4 season year. Colorado receives \begin{align*}\frac{3}{5}\end{align*} of its precipitation in the winter, \begin{align*}\frac{3}{10}\end{align*} in the spring and the rest in the summer and fall months.

15. Students were preparing to go on a field trip, and their teacher let them choose the destination. There were 36 students in the class. 44.4% choose the Nature Preserve, 25% chose an Art Gallery. Half as many students wanted to go to the Symphony as the Nature Preserve, and the rest wanted to go to the Museum of Nature and Science. How many more students were there that wanted to go to the Nature Preserve than wanted to go to the Museum of Nature and Science?

### Review (Answers)

To view the Review answers, open this PDF file and look for section 4.14.