In this explainer, we will learn how to construct, read, and interpret frequency tables for a given quantitative data set.

When we are dealing with small data sets, we commonly have the data values from which we may choose to create a tally chart and then a frequency table. Let’s recall what we mean by frequency.

### Definition: Frequency

The frequency of a value is the number of times it occurs.

If we consider the data set 3, 7, 9, 5, 3, 6, 4, 10, 8, 6, 2, 3, 1, 2, 2, 7, we can order the data as 1, 2, 2, 2, 3, 3, 3, 4, 5, 6, 7, 7, 8, 9, 10. We can then easily construct a frequency table for the data as follows:

Value | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

Frequency | 1 | 3 | 3 | 1 | 1 | 1 | 2 | 1 | 1 | 1 |

In this data set, the range of numbers is comparatively small: the smallest data value is 1, and the largest is 10. However, if the data set has a large range, then a frequency table such as this, which records the frequency of every individual data value, would be very large.

Let’s consider that a large number of adults completing a survey were asked to record their age. Instead of creating a frequency table to record the ages , it would be useful to group the ages into a grouped frequency table, for example, ages 18–27 years, 28–37 years, 38–47 years, . This would still allow us to make statistical conclusions about age and any variable under study, but the data would be more manageable. We can define the concept of a grouped frequency table below.

### Definition: Grouped Frequency Table

A grouped frequency table is a frequency table with data organized into smaller groups often referred to as classes.

There are some important points to consider when creating a grouped frequency table.

### How To: Creating a Grouped Frequency Table

We must ensure that a grouped frequency table meets the following criteria:

- Classes should be exhaustive: there should be a class for every data value in the set.
- Classes should be mutually exclusive: there should be no overlapping data values between classes.
- Classes should be continuous: there should be no gaps between classes.
- Classes are usually of equal width.

There is a number of different ways in which we may see the groups presented in a grouped frequency table, as we can see below.

10–19 | 20–29 |

10– | 20– |

Discrete data, such as ages, could be recorded in groups of 10–19 years, 20–29 years, and so forth. In this way, we know that there are no overlapping values and no missing values. However, continuous data cannot be represented this way, as it would not be clear where values between 19 and 20 would be placed.

There are two alternative ways in which discrete and continuous data may be represented. A class given as “10–” indicates values that are 10 or greater. The lower boundary of the subsequent group gives us the upper boundary for the 10– group. If this is 20–, then 10– represents values that are 10 or greater, and less than 20.

This is equivalent to classes presented with inequality notation for a variable , for example, . This class interval would contain values of greater than or equal to 10 (since is equivalent to ) but less than 20.

We will now see some examples of grouped frequency tables. In the first example, we will recap how to read a tally chart and how to calculate the frequencies in a grouped frequency table.

### Example 1: Completing a Frequency Table from a Tally Chart

The tally chart shows the grades that students in a class received in a mathematics test. Using the information from the tally chart, complete the frequency table.

Sets | 5– | 10– | 15– | 20– | 25– |
---|---|---|---|---|---|

Tally |

Sets | l5– | 10– | 15– | 20– | 25– |
---|---|---|---|---|---|

Frequency |

### Answer

The charts here are a tally chart and a frequency table for the same data on the scores of students in a mathematics test. The scores are given in groups, where 5– would represent a score of 5–9 points. This is because the second group begins with a score of 10 and we do not have overlapping values in a grouped frequency table. The second group, 10–, would represent scores of 10–14, and so forth.

We need to take the information from the given tallies and assign a frequency to each group. To do this, we recall that each vertical line in a tally represents 1 individual and a group of 4 vertical lines crossed through with a diagonal line represents 5. Groupings of 5 in a tally chart allow for faster counting of the frequencies.

In the first set of 5–, the tally has a group of 4 with a diagonal line, representing 5 students, plus an additional 4 students. The frequency of this set would be .

Sets | 5– | 10– | 15– | 20– | 25– |
---|---|---|---|---|---|

Tally |

Sets | 5– | 10– | 15– | 20– | 25– |
---|---|---|---|---|---|

Frequency |

We can then compute the remaining frequencies from the tallies.

Sets | 5– | 10– | 15– | 20– | 25– |
---|---|---|---|---|---|

Tally |

Sets | 5– | 10– | 15– | 20– | 25– |
---|---|---|---|---|---|

Frequency |

The frequencies can be given as 9, 14, 12, 16, and 6.

In the previous example, we were given the tally chart data and calculated this as a frequency. In the next example, we will need to carry out the tallying. It is good practice to take the data values in order and assign a tally mark to the correct group, rather than attempting to count the number of data values in each group.

### Example 2: Creating a Grouped Frequency Table from a Data Set

Using the data given for number of absences, complete the frequency table.

7 | 10 | 7 | 8 | 5 | 1 | 2 | 7 |

6 | 2 | 4 | 3 | 2 | 8 | 5 | 6 |

5 | 8 | 9 | 8 | 2 | 1 | 10 | 9 |

10 | 9 | 10 | 3 | 9 | 7 | 7 | 8 |

Number of Absences | 1-2 | 3-4 | 5-6 | 7-8 | 9-10 |
---|---|---|---|---|---|

Frequency |

### Answer

In this problem, we are required to complete a grouped frequency table from
the given data on absences. Each number in the table represents the number of
days absent for one individual. If we were inputting this in a (nongrouped)
frequency table, we would tally the number of 1s, 2s, and so forth, and then
calculate the frequency. We do the same in the grouped frequency table, but we
count the number of values in each group: for example, the first group would be
the frequency of the number of 1s *and* 2s in the data set.

It can be useful to begin with a tally chart. We proceed through the data values, inserting a tally mark in the correct group. It can also be helpful to cross through the tallied data values as we proceed. We can see in the table below how the chart appears after the first 3 values have been recorded in the tally chart.

8 | 5 | 1 | 2 | 7 | |||

6 | 2 | 4 | 3 | 2 | 8 | 5 | 6 |

5 | 8 | 9 | 8 | 2 | 1 | 10 | 9 |

10 | 9 | 10 | 3 | 9 | 7 | 7 | 8 |

Number of Absences | 1-2 | 3-4 | 5-6 | 7-8 | 9-10 |
---|---|---|---|---|---|

Tally | |||||

Frequency |

We can then complete the tally chart, recalling that 5 in a tally chart is represented by 4 vertical lines crossed through with a diagonal line.

Number of Absences | 1-2 | 3-4 | 5-6 | 7-8 | 9-10 |
---|---|---|---|---|---|

Tally | |||||

Frequency |

We now calculate the frequency from the tally of each group.

Number of Absences | 1-2 | 3-4 | 5-6 | 7-8 | 9-10 |
---|---|---|---|---|---|

Tally | |||||

Frequency |

The frequencies of the groups can be listed as 6, 3, 5, 10, and 8.

In the next example, we will see how we can interpret information given in a grouped frequency table.

### Example 3: Interpreting a Grouped Frequency Table

The table shows the grades that a class of students received in a mathematics test. How many students scored less than 45?

Grades | 20–24 | 25–29 | 30–34 | 35–39 | 40–44 | 45–49 | 50–54 | 55– | Total |
---|---|---|---|---|---|---|---|---|---|

Number of Students | 7 | 8 | 2 | 5 | 6 | 5 | 9 | 2 | 44 |

### Answer

The given table shows us the grades that students received in a test. The grades are grouped into classes of 5 points. For example, the first entry in the table indicates that 7 students received a grade of 20–24 points, (i.e., 20, 21, 22, 23, or 24 points in the test). Although we cannot extract the exact grade that each student received solely from the grouped frequency table, it can give helpful insight into the spread of results.

We need to establish how many students scored less than 45. This will be the total of the students in the first 5 groups, since all of these are grades of less than 45. Note that we could not accurately calculate this for scores that are smaller than the first value of a class, for example, scores less than 47 points.

Grades | 45–49 | 50–54 | 55– | Total | |||||
---|---|---|---|---|---|---|---|---|---|

Number of Students | 5 | 9 | 2 | 44 |

To find the total students with a score less than 45 points, we add .

There were 28 students who scored less than 45 in the test.

In the next example, we will calculate the number of values that are more than a specific value in a grouped frequency table.

### Example 4: Interpreting a Grouped Frequency Table

The table shows the grades of 100 students in a recent science exam. How many students received a grade of 40 or more?

Grade | 0–9 | 10–19 | 20–29 | 30–39 | 40–49 | 50– | Total |
---|---|---|---|---|---|---|---|

Frequency | 6 | 15 | 12 | 26 | 23 | 18 | 100 |

### Answer

The table gives the number of students (the frequencies) who attained particular grades in a science exam. We can interpret the table by recognizing that the first entry in the table indicates that 6 students got a grade of 0–9, the second entry indicates 15 students got a grade from 10–19, and so forth.

When considering the number of students achieving grades of 40 or more, it is a common error just to consider the 23 students in the group with grades 40–49. However, we should also note that any higher classes will also indicate grades that are greater than 40. Therefore, we have one more group, the group 50–, to add.

Hence, the number of students scoring 40 or more is found by adding .

There were 41 students who received a grade of 40 or more in the examination.

In the final example, we will construct a grouped frequency table, taking into account the information that we need to obtain from it.

### Example 5: Interpreting a Data Set by Constructing a Grouped Frequency Table

The table shows the number of days taken off work by 40 workers in a year. By constructing a frequency table or otherwise, calculate the number of workers who took 20 or more days off work.

16 | 18 | 5 | 20 | 15 | 26 | 12 | 17 | 8 | 5 |

5 | 12 | 26 | 30 | 23 | 22 | 21 | 30 | 25 | 21 |

27 | 15 | 12 | 28 | 11 | 12 | 26 | 12 | 13 | 23 |

6 | 29 | 28 | 5 | 20 | 12 | 10 | 24 | 23 | 14 |

### Answer

The data values in the table represent the number of days of absence, for example, 1 individual had 16 days of absence, another had 18 days of absence, and so forth. If we consider the range of data in the table, the smallest value is 5 and the highest value is 30. If we were to create a frequency table to represent every individual data value from 5–30, the table would be very large.

Therefore, the preferred option would be to have a grouped frequency table, with classes that group data values together. There are two ways to approach this problem. The first way would be to group the data into classes of 10 days of absence and create a tally chart. We must also be particularly careful to note the data value that we are asked to consider: 20 days or more. We do not want the value of 20 to appear inside of the group boundaries. Hence, one of the groups must begin with 20 days of absence.

When creating a grouped frequency table, we must ensure, firstly, that classes are exhaustive: there should be a class for every data value in the set. Secondly, the classes should be mutually exclusive: there should be no overlapping data values between classes. Thirdly, classes should be continuous: there should be no gaps between classes. In this scenario, classes for absences of 0–9 days, 10–19 days, 20–29 days, and 30–39 days would fit the requirements. We can create a tally column to help us log the data values.

Days Absent | Tally |
---|---|

0–9 | |

10–19 | |

20–29 | |

30–39 |

We take each data value in turn and place it into the appropriate group in the tally chart.

Days Absent | Tally |
---|---|

0–9 | |

10–19 | |

20–29 | |

30–39 |

Next, we can use the tallies to assign a frequency to each group in the table.

Days Absent | Tally | Frequency |
---|---|---|

0–9 | ||

10–19 | ||

20–29 | ||

30–39 |

Note that we can confirm that the total frequency is , and we were given that this is the data for 40 workers.

In order to calculate the number of workers who have taken 20 or more days off work, we need to find the total frequencies of classes with 20 or more days absent. This will be the two classes 20–29 days and 30–39 days.

Thus, we have

This approach in creating a grouped frequency table is useful if we have to answer several problems involving the data. However, an alternative method of solving this problem more simply would be to create a grouped frequency table that just has two groups: absences up to 19 days and absences of 20 days or more.

Days Absent | Tally | Frequency |
---|---|---|

0–9 | ||

20– |

Thus, once the tallies have been created, the frequency of the group 20 days or more would give us the required answer. Either method would demonstrate that the number of workers who took 20 or more days off work is 19.

We can now summarize the key points.

### Key Points

- A grouped frequency table is a frequency table with data organized into smaller groups or classes.
- Grouped frequency tables allow for for easier analysis of large data sets and those with a wide range of values.
- We must ensure that a grouped frequency table meets the following
criteria:
- Classes should be exhaustive: there should be a class for every data value in the set.
- Classes should be mutually exclusive: there should be no overlapping data values between classes.
- Classes should be continuous: there should be no gaps between classes.
- Classes are usually of equal width.

- There are different ways in which class groups may be defined. Discrete data can be given class notation such as 10–19, 20–29, and so forth. Discrete and continuous data can be represented by or by groups such as for a variable .