In this explainer, we will learn how to use grouped frequency tables to identify the modal class and estimate the mode using a histogram.

Let’s begin by recalling what a grouped frequency table is.

### Definition: Grouped Frequency Table

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

Grouped frequency tables allow for easier analysis of large data sets and those with a wide range of values. However, we need different approaches for performing statistical methods such as finding the averages. In this explainer, we will focus on how we can find the mode of a grouped frequency table. Let’s recap what we mean by the mode of a data set.

### Definition: Mode

The mode is the most commonly or frequently occurring value (or values). We sometimes call this the modal value.

Consider the following data set: 1, 1, 2, 3, 3, 5, 5, 6, 7, 7, 7, 7, 8. As this is an ordered list, we can see that the most commonly occurring value is 7. This is the mode of the set. If we presented this data in a frequency table as shown below, the value with the highest frequency is the mode.

Value | 1 | 2 | 3 | 4 | 5 | 6 | 8 | |
---|---|---|---|---|---|---|---|---|

Frequency | 2 | 1 | 2 | 0 | 2 | 1 | 1 |

The highest frequency in the table is 4, which means that the corresponding value of 7 is the mode.

However, we cannot find an exact value for the mode when data is presented in a grouped frequency table. Instead, we can find the modal class.

### Definition: Modal Class

The modal class of a grouped frequency distribution is the class (or classes) with the highest frequency.

Let’s see how we find the modal class in a grouped frequency table.

### Example 1: Identifying the Modal Class for a Grouped Data Set

For the given frequency distribution, what is the modal class?

Class | 0– | 10– | 20– | 30– | 40– | 50– |
---|---|---|---|---|---|---|

Frequency | 6 | 10 | 5 | 12 | 8 | 4 |

### Answer

We recall that the modal class of a grouped frequency distribution is the class (or classes) with the highest frequency.

We can observe that the highest frequency value in the given table is 12. This is the frequency of the class 30–, values that are greater than or equal to 30 and less than 40 (the lower boundary of the next class).

Therefore, the modal class is 30–.

A common error when finding the mode, or modal class, from a grouped or ungrouped frequency distribution is to give the answer as the value of the highest frequency rather than the labeling of the class itself. In the previous example, the highest frequency was 12. This means that 12 individual data points exist in the class of 30–, and this is higher than the number of data points in the other classes.

As previously mentioned, we cannot find the exact mode from a grouped frequency
distribution, but we will now consider how we can find a graphical *estimate*
for the mode in a grouped frequency distribution, by using a histogram. This
applies when the histogram has equal class widths. To do this, we apply the
following steps.

### How To: Estimating the Mode from a Histogram with Equal Class Widths

- Draw a histogram of the data and identify the modal class. The modal class is the class with the highest bar.
- Draw a straight line connecting the top-left corner of the tallest bar to the top-left corner of the bar representing the frequency of the following class.
- Draw a straight line connecting the top-right corner of the tallest bar to the top-right corner of the bar representing the frequency of the class immediately before.
- From the point of intersection of these lines, draw a vertical line down to the -axis. This value is the estimate for the mode.

An example of this is given below. Here, the mode of the frequency distribution has been estimated graphically as 27.

We will now see how we can use this method to find an estimate of the mode from a given histogram.

### Example 2: Finding an Estimation of the Mode from a Histogram

For the given histogram, which of the following is the best estimation of the mode?

- 12
- 40
- 50
- 30
- 44

### Answer

We recall that the modal class of a grouped frequency distribution is the class (or classes) with the highest frequency. When using a histogram with classes of equal width, the class with the highest frequency will be the class with the highest bar in the histogram. Therefore, we can identify that the modal class is that of 40–.

This notation of class 40– means that this class contains values that are 40 or greater but less than 50, the boundary of the next class. We could also write this class using the notation that for some variable : the class is .

In order to identify the best estimate for the mode, we can first recognize that the mode will lie within the modal class. This means that, in the given answer options, options A and D of estimates 12 and 30 could not be correct. We can now apply the steps needed to find an estimate for the mode graphically.

Having identified the modal class, we draw a straight line connecting the top-left corner of the tallest bar to the top-left corner of the rectangle representing the frequency of the following class.

Next, we draw a straight line connecting the top-right corner of the tallest bar to the top-right corner of the rectangle representing the frequency of the class immediately before.

Then, from the point of intersection of the lines, we draw a vertical line down to the -axis.

This point on the -axis represents an estimate for the mode. We observe that this line lies slightly to the left of the midpoint of 40 and 50. Hence, we can give the answer that an estimate for the mode must be the value given in option E, 44.

There are alternative ways in which data may be represented in a grouped frequency table. We have seen that 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 be values of greater than or equal to 10 (since is equivalent to ) but less than 20.

We will now see another example of estimating the mode, where the data is given in classes using inequality notation.

### Example 3: Estimating the Mode from a Histogram

The speeds, in kilometres per hour, of cars driving on a road were recorded in the table below and are represented in the histogram.

Speed (km/h) | |||||
---|---|---|---|---|---|

Frequency | 8 | 15 | 17 | 25 | 7 |

Which of the following is the best estimation of the modal speed of the cars?

### Answer

In this problem, we need to find a single value that is an estimate for the mode. However, in order to find an estimate for the mode of grouped data, we begin by determining the modal class. The modal class is the class, or classes, with the highest frequency. We can identify this either from the table or from the histogram. The highest frequency in the table is 25, and so the corresponding class is that of . From the histogram, the highest bar is that of . Therefore, the class represents the modal class. In context, this means that the most common speed of cars along the road is 80 km/h or greater but less than 90 km/h.

We can then use the histogram to find a graphical estimate for the modal speed. We apply the following steps.

On the bar representing the modal class (the tallest bar),

- draw a line from the left corner of the tallest bar to the left corner of the rectangle representing the frequency of the following class,
- draw a line from the right corner of the tallest bar to the right corner of the rectangle representing the frequency of the class immediately before,
- from the point of intersection of the lines, draw a vertical line to the -axis; this value is the estimate for the mode.

The histogram will now appear as shown below.

Hence, we can give the answer that an estimate for the mean is 83 km/h.

In the following questions, we will need to draw a histogram before using it to find an estimate for the mean. Remember that we use histograms to represent continuous data. The edges of the bars will lie on the lower and upper boundaries of each class. For example, a bar representing the class , for some variable , will have its vertical lines from 5 and 10 on the -axis. The top of the bar will have a horizontal line representing the frequency of that class.

### Example 4: Estimating the Mode for a Grouped Data Set

The table represents the time taken by some people to travel to work.

Time () | 4.5– | 9.5– | 14.5– | 19.5– |
---|---|---|---|---|

Frequency | 10 | 15 | 7 | 3 |

### Answer

The table gives the time taken for people to travel to work in the form of a grouped frequency table. The first class represents times of 4.5 minutes or more and less than 9.5 minutes. We can recall that the mode of a data set is the most common value.

We cannot extract the mode from a grouped data set; however, we can calculate an estimate for it. To do this, we utilize the modal class: the most commonly occurring class in the frequency distribution. Using the table, we can see that the class with the highest frequency is the class 9.5–.

We begin by drawing a histogram. In a histogram, we do not have bars with spaces between them; rather, we have a continuous -axis representing the variable. In this problem, this will be the time, in minutes. The rectangular bars representing the frequencies will have their vertical line segments on the upper and lower boundaries of each class. A histogram can be drawn as below.

We now draw two lines from each of the top corners of the modal class bar to the adjacent corners of the classes on each side.

A vertical line is then drawn from the point of intersection down to the -axis.

We can use the grid lines to help us identify the estimate for the mode. Thus, an estimate for the modal number of minutes can be given as 11.5 minutes.

We will now see a final example.

### Example 5: Estimating the Mode for a Grouped Data Set

Some seeds are planted, and the heights of the resulting plants, in centimetres, are measured after 6 weeks.

Height (cm) | 1– | 7– | 13– | 19– | 25– |
---|---|---|---|---|---|

Frequency | 15 | 20 | 45 | 47 | 23 |

Using the given table, calculate an estimated mode for the heights of the plants in centimetres.

### Answer

We recall that the mode of a data set is the most commonly occurring value, that is, the value with the highest frequency. Since we cannot calculate an exact mode from a grouped frequency table, we can only calculate an estimate for the mode. We do this by identifying the modal class and drawing a histogram.

We begin by drawing the histogram showing the height of the plants.

Next, we draw a straight line connecting the top-left corner of the tallest bar to the top-left corner of the rectangle representing the frequency of the following class. Then, we draw a straight line connecting the top-right corner of the tallest bar to the top-right corner of the rectangle representing the frequency of the class immediately before.

Finally, from the point of intersection of these lines, we draw a vertical line down to the -axis. This value is the estimate for the mode.

The line from the intersection point meets the -axis just before 20. In fact, we can use the minor gridlines to identify that this would be at the point 19.5. Hence, the answer for the estimated mode is 19.5 cm.

We can now summarize the key points.

### Key Points

- The mode is the most commonly or frequently occurring value (or values). We sometimes call this the modal value.
- The modal class of a grouped frequency distribution is the class (or classes) with the highest frequency.
- We cannot find an exact mode from a grouped frequency table, so instead we determine an estimate for the mode.
- To find an estimate for the mode graphically, by using a histogram, we
follow the steps below:
- Draw a histogram of the data and identify the modal class. When all the class widths are equal, the modal class is the class with the highest bar.
- Draw a straight line connecting the top-left corner of the tallest bar to the top-left corner of the rectangle representing the frequency of the following class.
- Draw a straight line connecting the top-right corner of the tallest bar to the top-right corner of the rectangle representing the frequency of the class immediately before.
- From the point of intersection of these lines, draw a vertical line down to the -axis. This value is the estimate for the mode.