For his science class, Alex must present on the levels of air pollution in his city. He has collected readings, on 30 separate occasions, of the amount of carbon monoxide (CO) in the air in parts per million (PPM) below. He has organized his data in a table, but how can he create a data display that will clearly illustrate the air pollution?

In this concept, you will learn how to create and read frequency tables.

### Creating and Reading Frequency Tables

**Data** is a set of numerical or non-numerical information. Data can be analyzed in many different ways. In this concept you will analyze numerical data using frequency tables.

A **frequency table** shows the frequency, or amount of occurrences, of a specific category or group of data. Said another way, a frequency table shows the number of times that a group of data occurred.

Groups of data are referred to as **bins**. Bins can be many different sizes, but they never overlap. The size of the bins is determined by the range of the data.

The **range** of a set of data is the difference between the largest and smallest values. The range identifies how far apart the values in the data set are.

A frequency table has two columns, one for the bins (or categories) and the other for the number of occurrences.

Let's look at an example.

Twenty people were asked how many hours they sleep each night. The results of the survey are listed below. Create a frequency table to display the data.

7, 8, 6, 9, 10, 12, 5, 7, 8, 9, 10, 11, 12, 7, 6, 7, 8, 10, 11, 9

First, create a two column table. The left hand column will be the number of hours and the right will be the frequency. Since the range of the data is relatively small, each bin will be one hour.

Bins: Number of Hours Slept |
Frequency |

5 | |

6 | |

7 | |

8 | |

9 | |

10 | |

11 | |

12 |

Next, calculate the frequency of each bin. To do this, count up how many data points are 5 and place this value in the right column of the table. Then count up how many data points are 6 and place this value in the table. Continue until the entire table is complete.

The answer is the table should look like the one below.

Bins: Number of Hours Slept |
Frequency |

5 | 1 |

6 | 2 |

7 | 4 |

8 | 3 |

9 | 3 |

10 | 3 |

11 | 2 |

12 | 2 |

Let's look at another example.

The data set below represents the amount of time (in minutes) 20 middle school students spent on the computer each day. Create a frequency table to display the data.

10, 32, 8, 55, 5, 0, 30, 20, 25, 45, 40, 60, 45, 15, 5, 56, 47, 12, 15, 20

First, create a two column table. The left hand column will be the number of hours and the right will be the frequency.

Next, determine the size of the bins. The range of this data is from 0 to 60 minutes. Given this relatively larger range of data, we will create bins of size 10.

Bins: Minutes Spent on Computer |
Frequency |

0-10 | |

11-20 | |

21-30 | |

31-40 | |

41-50 | |

51-60 |

Next, calculate the frequency of each bin. To do this, count up how many data points are between 0 and 10, and place this value in the right column of the table. Then count up how many data points are between 11 and 20, and place this value in the table. Continue until the entire table is complete.

The table should look like the one below.

Bins: Minutes Spent on Computer |
Frequency |

0-10 | 5 |

11-20 | 5 |

21-30 | 2 |

31-40 | 2 |

41-50 | 3 |

51-60 | 3 |

### Examples

#### Example 1

Earlier, you were given a problem about Alex and his data on air pollution.

Alex will present data on the amount of air pollution in his city. He organized his data into the table below. Alex's table is hard to read and does not clearly indicate the amounts of pollution. How can Alex create a table that will better represent the amount of pollution?

Date |
CO Level (PPM) |

2 May | 4 |

17 April | 12 |

12 May | 15 |

3 May | 20 |

6 May | 20 |

8 May | 22 |

18 April | 25 |

23 April | 25 |

9 May |
27 |

20 April | 30 |

25 April | 30 |

26 April | 30 |

27 April | 34 |

4 May | 34 |

28 April | 36 |

15 May | 37 |

30 April | 38 |

1 May | 40 |

16 April | 42 |

22 April | 43 |

5 May | 45 |

21 April | 52 |

7 May | 55 |

24 April | 58 |

10 May | 58 |

11 May | 60 |

19 April | 63 |

13 May | 63 |

14 May | 71 |

29 April | 74 |

Alex's data is not well organized and does not clearly indicate the amount of pollution that is present in the city. A frequency table would be a better way to display his data.

To create a frequency table, first create a two column table. The left hand column of the table will be the amount of carbon monoxide (CO levels) in parts per million (PPM) and the right column will be the frequency.

Next, determine the size of the bins. The range of this data is from 4 to 74 PPM. Given this large range of data, we will create bins of size 10, starting at 0.

Bins: CO Level (PPM) |
Frequency |

0-10 | |

11-20 | |

21-30 | |

31-40 | |

41-50 | |

51-60 | |

61-70 | |

71-80 |

Next, calculate the frequency of each bin. To do this, count up how many data points are between 0-10, and place this value in the right column of the table. Then count up how many data points are between 11-20, and place this value in the table. Continue until the entire table is complete.

The answer is the table should look like the one below. The frequency table below is a better visual representation of the data table that Alex started with. In reading the table, one can more easily see the frequency of the amount of pollution in Alex's city.

Bins: CO Level (PPM) |
Frequency |

0-10 | 1 |

11-20 | 4 |

21-30 | 7 |

31-40 | 6 |

41-50 | 3 |

51-60 | 5 |

61-70 | 2 |

71-80 | 2 |

#### Example 2

The data below shows the height (in meters) a ball bounced after being dropped from different heights. Create a frequency table to display the data. Then state one conclusion about the data.

\begin{align*}6 \quad 9 \quad 4 \quad 12 \quad 11 \quad 5 \quad 7 \quad 9 \quad 13 \quad 5 \quad 6 \quad 10 \quad 14 \quad 7 \quad 8\end{align*}

First, create a two column table. The left hand column will be the height (in meters) and the right will be the frequency.

Next, determine the size of the bins. The range of this data is from 4 to 14 minutes. Given this relatively medium sized range of data, we will create bins of size 2, starting at 4.

Bins: Height (meters) |
Frequency |

4-5 | |

6-7 | |

8-9 | |

10-11 | |

12-13 | |

14-15 |

Next, calculate the frequency of each bin. To do this, count up how many data points are between 4-5, and place this value in the right column of the table. Then count up how many data points are between 6-7, and place this value in the table. Continue until the entire table is complete.

Next, analyze the data by looking at the frequencies of the heights and state one conclusion.

The first answer is the table should look like the one below. The second answer is one conclusion that can be stated about the data is: when dropped the ball most frequently reached a height of 6 or 7 meters.

Bins: Height (meters) |
Frequency |

4-5 | 3 |

6-7 | 4 |

8-9 | 3 |

10-11 | 2 |

12-13 | 2 |

14-15 | 1 |

#### Example 3

Kelsey asked her classmates how long they went on vacation for over the summer. Below is the data Kelsey collected.

- 1 classmate was gone for 8 days
- 18 classmates were gone for 9 days
- 16 classmates were gone for 10 days
- 6 classmates were gone for 11 days
- 9 classmates were gone for 12 days
- 4 classmates were gone for 13 days

Create a frequency table the represents Kelsey's data. Then state two conclusions about the data.

First, create a two column table. The left hand column will be the vacation days and the right will be the frequency.

Next, determine the size of the bins. The range of this data is relatively small. Given this, we will create bins of size 1, starting at 8.

Next, calculate the frequency of each bin. In this case, Kelsey has already done this as represented in her list. Use her values to fill in the entire table.

Next, analyze the data by looking at the frequencies of the vacation days and state two conclusions.

The first answer is the table should look like the one below. The second answer is two conclusions are: the most frequent vacation length among classmates was 9 days; and the least frequent vacation length among classmates was 1 day.

Bins: Vacation Days |
Frequency |

8 | 1 |

9 | 18 |

10 | 16 |

11 | 6 |

12 | 9 |

13 | 4 |

#### Example 4

The frequency table below lists the scores of a class on two separate exams. Compare and contrast the test scores and explain which test was more challenging.

Bins: Test Score (%) |
Frequency Test 1 |
Frequency Test 2 |

0-50 | 0 | 5 |

51-60 | 2 | 3 |

61-70 | 7 | 4 |

71-80 | 10 | 7 |

81-90 | 5 | 8 |

91-100 | 6 | 3 |

First, analyze the data by comparing the frequencies of scores between test 1 and test 2. Then, determine which test was more challenging.

The answer is test 2 was more challenging, because (a) there was a greater distribution of scores across all bins than test 1 and (b) there were a greater number of students who scored 70 or less.

### Review

Use each set of data to answer the following questions.

There are 175 7th grade students. Here is a list of the number of days that 7th students at Marrimack Middle School bought lunch.

0 Days = 15 students

1 Day = 13 students

2 Days = 30 students

3 Days = 21 students

4 Days = 35 students

5 Days = 60 students

- How many students are in the \begin{align*}7^{th}\end{align*} grade?
- What is the most popular number of days?
- What is the least popular number of days?
- Create a frequency table to show the data.
- Were any students left out of the count?
- How do you know?
- Create a frequency table to display the data below.

2, 5, 3, 1, 6, 5, 7, 8, 3, 1

- Were there any numbers not represented?
- What is the most popular number or numbers?

The following frequency table shows data regarding the number of people who attended different movies in one week. Use the following frequency table to answer each question.

# of People at the movies per week |
Frequency |
---|---|

20 | 4 |

50 | 3 |

85 | 3 |

90 | 5 |

120 | 2 |

- If we were to create a list of this data, is the following list correct or incorrect?

20, 20, 20, 20, 50, 50, 50, 90, 90, 90, 85, 85, 85, 120, 120

- Would you consider the list in number 1 to be organized or unorganized data?
- How many showings had 90 people or more in attendance?
- How many showings had less than 50 people in attendance?
- How many showings had less than 70 people in attendance?
- True or false. This data also tells you which showings had the most people in attendance.
- True or false. There were two showings that had 78 people in attendance.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 11.11.