Mode of Ungrouped Data: Mode can be defined as the observation with the highest frequency. It is referred to as a significant phenomenon associated with measures of central tendency. For example, consider an institution providing \(15\) different courses for pupils. The course with the highest number of admissions from the pupils will be regarded as the mode of the given data. Thus, mode tells us about any item in the data set having a maximum frequency.

In this article, we will learn how to find the mode of ungrouped data with the help of formulas and solved examples. In order to find a mode of ungrouped data, the dataset has to be set in ascending or descending fashion. Now utilising the formula to calculate the mode of ungrouped data, we can deduce the observation with the highest frequency in the given dataset, which will be the mode of the particular ungrouped data. 

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What is Ungrouped Data?

The data obtained from direct observation is called raw or ungrouped data.
The marks of ten students in a yearly test are an example of raw data.

Consider the marks obtained by \(10\) students in a science  test as given below:

The data is in the form of ungrouped data. Can we find the highest and the lowest marks by observing the given format?

Is it not time-consuming? So, to save time, we will organise the data in ascending and descending order.

Now, we can see that the lowest mark is \(30\) and the highest mark is \(98\).

Practice Exam Questions

Tabular Presentation Using Ungrouped Data

Two attributes are needed in the tabular representation of ungrouped data, such as observations and frequency.

Observation: Each numerical figure in data is known as observation.

Frequency: The number of times a particular observation occurs is called its frequency.

For example, consider the marks of \(15\) students in a class.

The above data are ungrouped data without frequency.
To make the frequency table of ungrouped data, first, we need to arrange the given data in ascending or descending order. Let us arrange them in ascending order.

We know that each numerical figure is known as an observation, and the number of times a particular observation occurs is called its frequency.

Here, \(20\) marks were obtained by \(2\) students. So, the frequency of \(20\) marks is \(2\).
Similarly, the frequency for marks \(22\) is \(1\) and so on.
The below table is known as the frequency distribution table.

Marks of students \(\left(x_{i}\right)\)	Frequency \(\left(f_{i}\right)\)
\(20\)	\(2\)
\(22\)	\(1\)
\(30\)	\(3\)
\(33\)	\(1\)
\(40\)	\(1\)
\(45\)	\(1\)
\(55\)	\(2\)
\(60\)	\(1\)
\(66\)	\(1\)
\(85\)	\(2\)

Central Tendency of Ungrouped Data

The measure of central tendency is a value that defines a set of data by recognising the central position within that data set. The mean, median, and mode are the measures of central tendency. The mode is the value that occurs most frequently in a data set. Let us learn in detail about the mode of ungrouped data.

Mode

A mode is that observation that occurs most often, that is, the value of the observation having a maximum frequency. 

Steps to Calculate Mode

1. Find the set of observations.
2. Determine the frequency distribution. That means checking how many times each observation occurs.
3. Obtain the value that occurs the highest number of times, i.e., find the value which has the maximum frequency.
4. The value we get from the above step is the mode.

Properties of Mode

1. The mode can be computed graphically.
2. The mode can be used for open-ended distribution and qualitative data.

Uses of Mode

Mode is used to determine the actual size or item from a set of objects. For Example: in the banking sector, Information technology sectors, ready-made manufacturing shoe companies etc.

Suppose a shoe manufacturer manufactures a series of watches. We use mode to know which of those is in high demand and sold widely. The same way when a garment company designs garments. It can be known which item is in high demand using mode. Here, the model is the most appropriate measure because it shows the highest number of repetitions in the series.

Mode of Ungrouped Data

We need to determine the observation that appears maximum times for ungrouped data.

For example, the number of students in \(10\) different classes is as follows:
\(3, 6, 4, 5, 0, 2, 1, 3, 2, 3\)

The given data set is ungrouped data without a frequency table.
Here, \(3\) is the number of students in the maximum number of classes. Hence, the mode of the data is \(3\).

Let us take an example to know how to identify the mode of ungrouped data using a frequency distribution table. Let us say a clothing company manufactured Jeans-pants with the sizes as stated in the frequency distribution table:

Size of Jeans-Pants	\(28\)	\(30\)	\(32\)	\(34\)	\(36\)	\(38\)	\(40\)
Number of Jeans-Pants	\(10\)	\(8\)	\(15\)	\(20\)	\(12\)	\(5\)	\(6\)

We can observe that size \(34\) has a maximum frequency that is \(20\). Hence, the model for the size of the jeans-pants is \(34\).

Solved Examples – Mode of Ungrouped Data

Q.1. The following is the number of female employees in different private bank branches.
\(2, 4, 6, 1, 3, 5, 3, 6, 1, 4, 4, 2, 1, 3, 6, 4, 2, 5\)
Make a frequency distribution.
Ans: We know that each numerical component is known as observation in data, and the number of times an observation appears is called its frequency.
There is only \(1\) employee in \(3\) branches. Thus, the frequency of \(1\) is \(3\).
Similarly, the frequency for \(2\) is \(3\), and so on.

Number of employees \(\left(x_{i}\right)\)	Branches \(\left(f_{i}\right)\)
\(1\)	\(3\)
\(2\)	\(3\)
\(3\)	\(3\)
\(4\)	\(4\)
\(5\)	\(2\)
\(6\)	\(3\)

Q.2. The fruits taken by a Mr Das From \(10\) different sections of fruit stalls are as follows:
\(2,6,9,5,9,2,4,3,9,3\)
Find the mode of the data.
Ans: A mode is that observation that occurs most often, that is, the value of the observation having a maximum frequency.
Mode \(=\) Observation with the maximum frequency
Hence, \(9\) is the number of fruits taken by Mr Das in the maximum number from the fruit stall. Therefore, the mode of the data is \(9\).

Q.3. Find the mode from the following table:

Size of T-shirts	\(30\)	\(32\)	\(34\)	\(36\)	\(38\)	\(40\)	\(42\)
Number of T-shirts	\(15\)	\(30\)	\(10\)	\(20\)	\(24\)	\(12\)	\(8\)

Ans: Here, the frequency table is given.
We know that each numerical figure in data is known as observation, and the number of times a particular observation appears is called its frequency.
So,

Observation	\(30\)	\(32\)	\(34\)	\(36\)	\(38\)	\(40\)	\(42\)
Frequency	\(15\)	\(30\)	\(10\)	\(20\)	\(24\)	\(12\)	\(8\)

Mode \(=\)Observation with the maximum frequency
We can observe that size \(38\) has a maximum frequency that is \(24\).
Hence, the mode for the size of the T-shirts is \(38\).

Q.4. Find the mode of the following data.
\(1, 2, 3, 1, 2, 8, 9, 0,4,6\).
Ans: Consider the given data \(1, 2, 3, 1, 2, 8, 9, 0, 4, 6\), here \(1, 2\) appears twice. This data has \(2\) modes, which is called bimodal data. Therefore, the mode of the given data set is \(1, 2\).

Q.5. Find the mode of the given data.
\(5, 7, 3, 5, 7, 8, 9, 0,3\)
Ans: Consider the given data \(5, 7, 3, 5, 7, 8, 9, 0, 3\). Here \(5, 7, 3\) appears twice. This data has \(3\) modes called trimodal data. Therefore, the mode of the given data set is \(3, 5, 7\).

Summary

We have talked about the mode that provides us with an idea about which observations in a data set occur most frequently in this article. We have discussed the definition of ungrouped data, the method to find the mode of the ungrouped data, uses and properties of the mode.

FAQs Related to Mode of Ungrouped Data

Frequently asked questions related to the mode of ungrouped data are listed as follows:

Q.1. How do you find the mode of ungrouped data?
Ans: A mode is that observations that occur most often, that is, the value of the observation having a maximum frequency.
For ungrouped data, we need to find the observation which occurs maximum times.
Mode \(=\) Observation with the maximum frequency

Q.2. What if there are \(2\) modes in ungrouped data?
Ans: If there are \(2\) modes in ungrouped data, they are called bimodal data.
Consider the given data \(1, 2, 3, 1, 2, 8, 9, 0, 4, 6\), here \(1, 2\) appears twice. Hence, the mode of the given data set is \(1, 2\).

Q.3. What is an ungrouped data example?
Ans: The data which are obtained from direct observation is called raw data or ungrouped data.
For example, the number of a family on \(10\) different floors of a building is as follows:
\(3, 6, 4, 5, 0, 2, 1, 3, 2, 3\)
The given data set is ungrouped data.

Q.4. What is the mode of data?
Ans: A mode is that observation that occurs most often, that is, the value of the observation having a maximum frequency.

Q.5. How do you find the mode of a set of data?
Ans: We can consider two types of data: ungrouped data and grouped data.
For ungrouped data, we need to find the observation which appears maximum times.
Mode = Observation with the maximum frequency.
More than one observation may have the maximum frequency in a frequency distribution. In such situations, the data is said to be multimodal.

We hope this detailed article on the mode of ungrouped data helped you in your studies. If you have any doubts or queries regarding this topic, feel to ask us in the comment. Happy learning!