Mode for Ungrouped Data in Statistics – Class 11 Notes
Written by Neeraj Anand | Published by ANAND TECHNICAL PUBLISHERS
Introduction
In statistics, the mode is the value that appears most frequently in a given dataset. It is one of the three measures of central tendency, along with mean and median. The mode helps in identifying the most common observation in a dataset, making it useful for analyzing real-world data distributions.
For ungrouped data, the mode can be determined simply by counting the occurrences of each value and identifying the one that appears the most.
Definition of Mode
The mode of a dataset is the value that appears with the highest frequency. A dataset may have:
- One mode (Unimodal) – If only one value occurs most frequently.
- Two modes (Bimodal) – If two values appear with the same highest frequency.
- More than two modes (Multimodal) – If multiple values have the highest frequency.
- No mode – If no value repeats in the dataset.
Steps to Find Mode for Ungrouped Data
To find the mode of a given set of numbers, follow these steps:
- List the Data – Write all the given observations.
- Count the Frequency – Find how many times each value appears.
- Identify the Mode – The number with the highest frequency is the mode.
| Type | Definition | Example Data Set | Modes |
|---|---|---|---|
| Unimodal | When there is only one and only one mode in a dataset. | Set X = {1, 2, 2, 3, 6, 7, 7, 7, 8, 9} | Only 7 |
| Bimodal | When there are two modes in the given data set. | Set A = {1, 1, 1, 3, 4, 4, 6, 6, 6} | 1 and 6 |
| Trimodal | When there are three modes in the given data set. | Set A = {2, 2, 2, 3, 4, 4, 6, 6, 6, 7, 9, 9, 9} | 2, 6, and 9 |
| Multimodal | When there are four or more modes in the given data set. | Set A = {1, 1, 1, 3, 4, 4, 6, 6, 6, 7, 9, 9, 9, 11, 11, 11} | 1, 6, 9, and 11 |
Note : A dataset without recurring values, however, lacks a mode.
Example 1: Finding Mode of Ungrouped Data
Consider the dataset: 2, 4, 5, 3, 2, 6, 4, 4, 2, 5, 4, 6, 4, 3
Step 1: Count the occurrences of each number
\begin{array}{|c|c|} \hline \text{Number} & \text{Frequency} \\ \hline 2 & 3 \\ 3 & 2 \\ 4 & 5 \\ 5 & 2 \\ 6 & 2 \\ \hline \end{array}
Step 2: Identify the Mode
The number 4 appears 5 times, which is more frequent than any other number.
Answer:
Hence, Mode = 4
Example 2: Trimodal Data
Consider the dataset: 3,5,7,3,8,5,7,3,5,8,7,7,5,8,8
Step 1: Count the occurrences of each number
\begin{array}{|c|c|} \hline \text{Number} & \text{Frequency} \\ \hline 3 & 3 \\ 5 & 4 \\ 7 & 4 \\ 8 & 4 \\ \hline \end{array}
Step 2: Identify the Mode
Here, 5, 7, and 8 appear 4 times each, meaning the dataset is trimodal (having three modes).
Answer:
Hence, Modes=5,7,8
How to Calculate Mode of Ungrouped Data
To find the mode of the ungrouped dataset, we observe the most occurring value in the dataset. The values in the dataset must be rearranged either in increasing or decreasing order and their frequency should be noted.
The value which is appearing the most number of times has the highest frequency in the dataset and it is the Mode of the data.
Steps for Calculating Mode for Ungrouped Data
To calculate the mode of any given ungrouped data set, we use the following steps:
Step 1: Sort the data in ascending or descending order, whichever is more convenient.
Step 2: Determine the value that occurs most frequently in the data set. This value is the mode.
Step 3: If there are two or more values that occur with the same highest frequency, then the data set has multiple modes.
Solved Examples For Mode of Ungrouped Data
Example.1: Find the mode in the given set of data: 4, 6, 8, 16, 22, 24, 41, 24, 42, 24, 15, 13, 61, 24, 29.
Solution:
Arrange the given set of data in ascending order,
4, 7, 8, 13, 15, 16, 22, 24, 24, 24, 24, 29, 41, 42, 61.
The mode of the data set is 24 as it appeared in the given most.
Example.2 : Imagine a shoe store that tracks the sizes of shoes sold over a month. The sizes are recorded as:
6, 7, 8, 7, 9, 7, 8, 8, 7, 6, 7, 8, 8, 7, 8, 8, 9, 8, 7, 8, 6, 7, 7, 10, 8, 9, 7, 8, 8, 8, 7, 7, 7, 9, 8, 7, 7, 10, 7, 8, 8, 7, 8, 7, 8, 8, 8, 6, 7, 9, 8, 7, 6, 8, 8, 7, 7, 9, 8, 10, 7, 7, 7, 8, 8, 7, 7, 6, 8, 8, 9, 7, 7, 8, 10
Solution :
- Size 6: 6 times
- Size 7: 26 times
- Size 8: 27 times
- Size 9: 8 times
- Size 10: 4 times
Here, the most frequently sold shoe size is 8, which occurs 27 times. Therefore, the mode of this data set is 8.
Example 3: Find the mode in the given set of data: 3, 6, 7, 15, 21, 23, 40, 23, 41, 23, 14, 12, 60, 23, 28
Solution:
First arrange the given set of data in ascending order:
3, 6, 7, 12, 14, 15, 21, 23, 23, 23, 23, 28, 40, 41, 60
Therefore, the mode of the data set is 23 since it has appeared in the set four times.
Example 4: Find the mode in the given set of data: 1, 3, 3, 3, 6, 6, 6, 4, 4, 10
Solution:
First arrange the given set of data in ascending order:
1, 3, 3, 3, 4, 4, 6, 6, 6, 10
Therefore, the mode of the data set is 3 and 6, because both 3 and 6 is repeated three times in the given set.
Example 5: Find the mode of the following marks obtained by 25 students in a mathematics test out of 50.
34, 46, 45, 39, 43, 22, 27, 37, 46, 35, 34, 39, 40, 30, 30, 41, 37, 46, 39, 29, 34, 39, 35, 43, 30
Solution:
The ascending order of the data:
22, 27, 29, 30, 30, 30, 34, 34, 34, 35, 35, 37, 37, 39, 39, 39, 39, 40, 41, 43, 43, 45, 46, 46, 46
The most frequently occurred value is 39.
Hence, the mode of given marks is 39.
Alternatively, let us form the table with observations and their frequencies to get the mode.
The mode of the given data can be obtained by making the frequency table and choosing the highest frequency. Such as:
| Observation | 22 | 27 | 29 | 30 | 34 | 35 | 37 | 39 | 40 | 41 | 43 | 45 | 46 |
| Frequency | 1 | 1 | 1 | 3 | 3 | 2 | 2 | 4 | 1 | 1 | 2 | 1 | 3 |
Here, the highest frequency is 4.
Therefore, the mode is 39.
Key Points to Remember
- If no value repeats, there is no mode.
- If multiple values have the same highest frequency, the dataset is bimodal or multimodal.
- The mode is useful in categorical data analysis where numerical averages (mean/median) do not make sense.
Applications of Mode in Real Life
- Fashion Industry: Identifying the most popular clothing size.
- E-commerce: Determining the most sold product.
- Education: Analyzing the most common score in a test.
- Medical Studies: Finding the most frequent health condition in a study.
Frequently Asked Questions – FAQs
What is Mode Definition in Statistics?
Mode refers to the value that appears most frequently in a dataset. It is one of the measures of central tendency, along with the mean and the median.
Can there be Two Modes in a Given set of Data?
Yes, there can be two modes or any greater number of modes for any given data sets as there can be same number of observations repeating the maximum number of times. If the data set has more than one mode, dataset is called multimodal data.
Can the Mode be used with Continuous Data?
Yes, mode can be used for the continuous set of data, but as continuous data has very less chances of any value to repeat it is not an optimum measure for continuous data.
Is it possible for Data to have No Mode?
Yes, it is possible for data to have no mode i.e., when each observation only comes in the dataset exactly once then the dataset is said to have no mode.
What is the mode of 2, 5, 7, 5, 2, 5, 5, 3?
The number which is repeated most number of times is called mode.
Here the number 5 is repeated most number of times so mode is 5.
What is the definition of mode?
The number which is repeated most number of times is called the mode.
What is the mode of 20, 21, 22, 22, 23, 22, 25?
Here the mode is 22 which is repeated most number of times.
The mode of the data 12, 10, 11, 12, 12, 15, 16, 17 is?
Here the mode is 12 which is repeated the most number of times.
Mode of the data 35, 38, 32, 35, 38, 39 is?
Here the mode is 35 which is repeated most number of times.
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