Mode

Mode in statistics is defined as the value which repeatedly occurs in a given set. Mode or Modal Value is defined as the number with the highest frequency in a set.

Example: Mode of the given set of values {3, 3, 5, 8, 9} is 3 as 3 has the highest frequency.

For a given set we can have one mode, more than one mode, or no mode at all.

Definition of Mode

A mode is defined as the value that has a higher frequency in a given set of values. It is the value that appears the most number of times.

Example: In the given set of data: 2, 4, 5, 5, 6, 7, the mode of the data set is 5 since it has appeared in the set twice.

Mode has a lot of real-life applications, for example, a company sells 5 different types of products in a month which product has the highest number of units sold is considered as the mode of the data.

Bimodal, Trimodal & Multimodal

• When there exist two modes in the given data set, then it is called a bimodal.

Example: In Set A = {1,1,1,3,4,4,6,6,6} the mode is 1 and 6 because both 1 and 6 have the highest frequency in the given set.

• When there exist three modes in the given data set, then it is called trimodal.

Example: In Set A = {2,2,2,3,4,4,6,6,6,7,9,9,9} the mode is 2, 6, and 9 because 2, 6, and 9 have the highest frequency in the given set.

• When there are four or more modes in the given data set, then it is called multimodal.

Example: In Set A = {1,1,1,3,4,4,6,6,6,7,9,9,9,11,11,11} 1, 6, 9, and 11 are the mode because 1, 6, 9, and 11 have the highest frequency in the given set.

Mode Formula (Ungrouped Data)

The value with the highest frequency in the given set is considered the mode of the given data set. For ungrouped data first, we arrange them in either ascending or descending order then the number which appears the highest number of times is called the mode of the data. In this method, the mode is found simply by observing the data.

Example: 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.

Mode Formula (Grouped Data)

For determining the mode in case of data is grouped, simple observation does not help, we use a special formula to calculate the mode in case of grouped data is given. The formula for finding the mode when grouped data is given is discussed below in this article.

In the formula given above,

l is the lower limit of the modal class
h is the size of the class interval
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

How to Find the Mode?

The method of calculating mode is discussed below in this article.

For Ungrouped Data: 

The mode of ungrouped data is simply calculated by observing the data after it is arranged in either increasing or decreasing order.

Example: For the given set of values {2,3,2,3,4,2,6,2,4,7} the mode of the data set is 2 as it appeared the most (4 times)

For Grouped Data:

Use the following steps for finding the mode of grouped data

Step 1: Study the given table and write down all the values used in the formula,

Mode = l + [(f₁ – f₀) / (2f₁ – f₀ – f₂)]×h

Step 2: Put all the values in the above formula and calculate the mode.

Step 3: Simplify the above calculation to get your answer

Points To Remember:

Some important points about mode are discussed below:

• For any given data set, mean, median, and mode all three can have the same value sometimes.
• Mode can be easily calculated when the given set of values is arranged in ascending or descending order.
• For ungrouped data, the mode can be found by observation, whereas for grouped data mode is found using the mode formula.
• Mode is used to find Categorical Data.

Mean, Median, Mode Formula

The relationship between Mean, Median, and Mode is given by the formula discussed below,

Mode = 3 Median – 2 Mean

Read More

• Central Tendency
• Mean
• Statistics Formulas

Solved Examples on Mode

Example 1: 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 2: 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 3: For a class of 40 students marks obtained by them in maths out of 50 are given below in the table. Find the mode of data given.

Marks Obtained	Number of Students
20-30	7
30-40	23
40-50	10

Solution:

Maximum Class Frequency = 23 

Class Interval corresponding to maximum frequency = 30-40

Modal class is 30-40

Lower limit of the modal class (l) = 30

Size of the class interval (h) = 10

Frequency of the modal class (f₁) = 23

Frequency of the class preceding the modal class (f₀) = 7

Frequency of the class succeeding the modal class (f₂)= 10

Using these values in the formula 

Mode = l + [(f₁ – f₀) / (2f₁ – f₀ – f₂)]×h

          = 30 + [(23-7) / (2×23 – 7- 10)]×10

          = 45.86

Example 4: Find the mode in the given set of data: 15, 8, 26, 35, 15, 33, 20

Solution:

First arrange the given set of data in ascending order

8, 15, 15, 20, 26, 33, 35

Therefore, the mode of the data set is 15 since it has appeared in the set twice.

FAQs on Mode

Question 1: What is meant by mode?

Answer:

Mode, in mathematics, is defined as the number in grouped data with the highest frequency.

Question 2: What is meant by the mode and median of data?

Answer:

A number that has the highest frequency in a data set is called the Mode of given data. Median is defined as the middle value of the data set when it is arranged in ascending and descending order. 

Question 3: How the mode for a given set of values is found?

Answer:

For any given set of values, the number with the highest frequency is called the mode of the data. 

Example: In the given group of data find the mode 3, 4, 4, 4, 4, 5, 5, 6. 4 has the highest frequency hence, it is the mode of the data.

Question 4: What are the types of modes?

Answer:

The various types of modes are 

• Unimodal
• Bimodal
• Trimodal
• Multimodal

Question 5: What is the bimodal mode?

Answer:

If a given set has two modes then it is called a bimodal mode.

Question 6: What is meant by the range of data?

Answer:

The difference between the highest and the lowest value in data is called the range of the data.