 # Mode of Grouped Data

In statistics, we have learned that the mode is the most repeated value among the given set of data. Among the given set of observations, mode is the value which has the maximum frequency. Sometimes, it is possible to have more than one value which has the same maximum frequency. In such a case, we can say that the data is multimodal. In this article, we are going to learn how to find the mode of ungrouped and the grouped data with its formula and solved examples.

## Finding Mode of Ungrouped Data

Before learning about how to find the mode of grouped data, we will have a look at how to find the mode of ungrouped data.

For example, the wickets taken by a bowler in 10 cricket matches are 2, 6, 4, 5, 0, 2, 1, 3, 2, 3. Find the mode of the given data.

To find the mode of the ungrouped data, first construct the frequency distribution table.

 Number of Wickets 0 1 2 3 4 5 6 Number of Matches 1 1 3 2 1 1 1

So, it is clearly observed that, 2 is the number of wickets taken by the bowler in the maximum number of cricket matches i.e., 3. Therefore, the mode of the given data is 2.

Hence, we can easily determine the mode of the given data, if it is ungrouped.

## How to Find the Mode of Grouped Data?

In the case of grouped data, it is not possible to identify the mode of the data, by looking at the frequency of data. In this scenario, we can determine the mode value by locating the class with the maximum frequency called modal class. Inside a modal class, we can locate the mode value of the data by using the formula,

$$Mode = l + \left ( \frac{f_{1}-f_{0}}{2f_{1}-f_{0}-f_{2}} \right )\times h$$

Where,

f1 is the frequency of the modal class

f0 is the frequency of the class preceding the modal class

f2 is the frequency of the class succeeding the modal class

h is the size of the class intervals

l is the lower limit of the modal class

Now, let us understand how to find the mode of grouped data using this formula with the help of an example.

## Mode of Grouped Data Example

Question:

A survey has been conducted by a group of students on 20 households in a locality as shown in the following frequency distribution table. Find the mode for the given data.

 Size of Family 1-3 3-5 5-7 7-9 9-11 No. of Families 7 8 2 2 1

Solution:

From the given table, it is observed that the maximum class frequency is 8, and the corresponding class interval is 3-5.

Therefore, the modal class for the given data is 3-5.

The lower limit of modal class, l = 3

Class size, h = 2

Frequency of modal class, f1 = 8

Frequency of class proceeding to modal class, f0 = 7

Frequency of class succeeding to modal class, f2 = 2

We know that the formula to find the mode of the grouped data is:

$$Mode = l + \left ( \frac{f_{1}-f_{0}}{2f_{1}-f_{0}-f_{2}} \right )\times h$$

Now, substituting the values in the mode formula, we get,

$$Mode = 3 + \left ( \frac{8-7}{2(8)-7-2} \right )\times 2$$

Mode = 3 + (2/7)

Mode = (21+2)/7

Mode = 23/7

Mode = 3.286.

Therefore, the mode of the given grouped data is 3.286.

### Practice Problems

1. The information on the observed lifetimes (in hours) of 225 electrical components are given in the following frequency table. Find the modal lifetimes of the electrical components.

 The lifetime of electrical components (in hours) 0-20 20-40 40-60 60-80 80-100 100-200 Frequency 10 35 52 61 38 29
2. The following distribution table shows the number of runs scored by some top batsmen of the world in one-day international cricket matches. Find the mode of the given data.

 Runs scored by Top Batsmen Number of Batsmen 3000 – 4000 4 4000 – 5000 18 5000 – 6000 9 6000 – 7000 7 7000 – 8000 6 8000 – 9000 3 9000 – 10000 1 10000- 11000 1

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## Frequently Asked Questions on Mode of Grouped Data

### What is meant by the mode of the given data?

The mode is a value that occurs the maximum number of times in the given data set. In other words, the maximum number of repeated values in the given data set is called mode.

### What is meant by a multimodal dataset?

If a dataset has more than one value that has the same maximum frequency, then the dataset is called a multimodal dataset.

### Is it possible to find the mode of grouped data from the frequency table?

No, it is not possible to find the mode of grouped data by looking at the frequencies in the distribution table.

### What is the formula to find the mode of the grouped data?

The formula to find the mode of the grouped data is:
Mode = l + [(f1-f0)/(2f1-f0-f2)]×h.
Where, l = lower class limit of modal class, h = class size, f1 = frequency of modal class, f0 = frequency of class proceeding to modal class, f2 = frequency of class succeeding to modal class.

### What is a unimodal data set?

If a dataset has only one value which is often repeated, then the dataset is called a unimodal dataset.