Mode

In mathematics, the mode is the value that appears most frequently in a data set. A data set can have:
 

• One mode (unimodal)
   
• Two modes (bimodal)
   
• Three modes (trimodal)
   
• No mode, if no number repeats
   

The mode is a measure of central tendency because it indicates the most common or typical value in the data.

Examples:
Find the mode of the data set: 5, 7, 9, 9, 2, 6.

Answer: The number 9 appears most frequently.
Mode = 9
 

The mode is the value that occurs most frequently in a data set. The method for finding the mode depends on whether the data are grouped or ungrouped.
 

• Grouped data is organized into class intervals and is often represented in a graph or frequency table.
   
• Ungrouped data consists of individual values and can be shown in a simple list or table.

Based on this classification, we can use different methods to find the mode for ungrouped data and grouped data.

In statistics, the mode is the value that occurs most frequently in the data set, also called the modal value. It represents the number or value with the highest frequency in the set. Mode is one of the three main measures of central tendency, along with the mean and median.
 

• For ungrouped data, the mode is simply the value that appears most often in the list.
   
• For grouped data, the mode is the class interval with the highest frequency.

Let:

Mode =  \(L + \frac{f_m - f_1}{2 f_m - f_1 - f_2} \times h\)

Where:
 

• L = lower limit of the modal class
   
• h = size of the class interval
   
• fₘ = frequency of the modal class
   
• f₁ = frequency of the class just before the modal class
   
• f₂ = frequency of the class just after the modal class

For grouped data, individual values are not available, so we cannot directly identify the mode. Instead, we use the modal class, the class with the highest frequency, and estimate the mode from it.
 

Consider a histogram of the data:
 

• Let h be the size of the modal class.
   
• Let fₘ (or f₁) be the frequency of the modal class.
   
• Let f₀ be the frequency of the class preceding the modal class.
   
• Let f₂ be the frequency of the class succeeding the modal class.
   
• Let I₀ be the lower limit of the modal class.
   

We assume that the mode is located at a distance x from the lower limit, so:

Mode = I₀ + x

From the histogram and similar triangles:
 

\(\triangle AEB \sim \triangle DEC \ gives: \frac{AB}{CD} = \frac{BE}{DE} = \frac{f_1 - f_0}{f_1 - f_2} \)
 

Using\(\triangle BEF \sim \triangle BDC\)we get:

\(\frac{FE}{BC} = \frac{BE}{BD} = \frac{f_1 - f_0}{(f_1 - f_0) + (f_1 - f_2)} = \frac{f_1 - f_0}{2f_1 - f_0 - f_2}\)
 

So,
 

\(FE = \frac{f_1 - f_0}{2 f_1 - f_0 - f_2} \times BC\) \(= \frac{f_1 - f_0}{2 f_1 - f_0 - f_2} \times h\)
 

Hence,

\(x = \frac{f_1 - f_0}{2 f_1 - f_0 - f_2} \times h\)


Then the mode formula becomes:

Mode = \(I_0 + \frac{f_1 - f_0}{2 f_1 - f_0 - f_2} \times h\)

There are different types of modes depending on the number of modal solutions. We will now see the different types of modes:

Unimodal: If the dataset has only one mode, it is called unimodal.

For example: In set A= {2, 3, 3, 5, 6, 6, 6, 7}, the mode is 6.

Bimodal: If there are two modes present in the given data, it is called bimodal.

For example: In set B = {3, 3, 3, 5, 7, 7, 7, 4}, the modes are 3 and 7 because both numbers are repeated three times.

Trimodal: If there are exactly three modes in a dataset, it is called trimodal.

For example: In set C =  {3, 3, 3, 5, 5, 5, 6, 7, 7, 7, 4}, the modes are 3, 5, and 7 as each of these numbers is repeated three times.

Multimodal: A dataset can contain multiple modes and if it has more than three modes, it is known as multimodal.

For example, if set A= {1, 2, 2, 2, 3, 4, 4, 4, 5, 6, 6, 6, 8, 8, 8}, the data set has four modes: 2, 4, 6, 8, as these numbers are repeated the same number of times.

The Merits and Demerits of the Mode are discussed below:

The mode is the most frequently occurring value in a data set. It helps identify common trends, but has both advantage and disadvantage when analyzing and interpreting data. 
 

Step 1: Identify the class interval with the highest frequency. This class is called the modal class.

Step 2: Determine the size of the class interval (h). It is calculated as:

h = Upper limit−Lower limit of the class

Step 3: Calculate the mode using the formula:

Mode =  \(L + \frac{f_m - f_1}{2 f_m - f_1 - f_2} \times h\)
 

Where:
 

L = Lower limit of the modal class

fₘ = Frequency of the modal class

f₁ = Frequency of the class preceding the modal class

f₂ = Frequency of the class succeeding the modal class

h = Size of the class interval

To calculate the mode of ungrouped data,

1. List the values in the dataset in either increasing or decreasing order.
    
2. Count how many times each value occurs in the dataset.
    
3. Identify the number that appears the most times, which is the mode.
    
4. Based on frequency, the mode can be categorized as unimodal, bimodal, trimodal, or multimodal.
    

Formula: 

Mode = \(= x_i \quad \text{where} \quad f_i = \max(f_1, f_2, \dots, f_n) \)
 

Here, x_i represents the data values and f_i​ represents their respective frequencies.

Understanding the differences between mean, median, and mode enables solving the problems related to them easily. The key differences are listed below:

Mode is a complex and difficult topic to understand. Therefore, in this section, we will discuss some tips and tricks to master the mode.

• Use visual tools: Plot the data using bar graphs or histograms — the tallest bar instantly shows the mode visually, especially in large data sets.
   
• Watch out for multiple modes: Some data sets can have more than one mode (bimodal or multimodal). Always check if two or more values occur with the same highest frequency.
   
• Arrange data clearly: Before finding the mode, organize the data in ascending order or create a frequency table. This makes it easier to spot repeating values.
   
• Identify when to use mode: Use mode when dealing with categorical or non-numeric data, like favorite colors, most-sold items, or popular destinations.
   
• Understand what mode means: Mode is the value that appears most frequently in a data set. Always look for the number or category with the highest frequency.
   
• Use familiar data such as class attendance, favorite colors, or daily temperatures to make learning engaging and relatable.
   
• Use phrases like: “Mode is the number we see most often in the list” to help children grasp the concept quickly.
   
• Teach children first to check whether the data are grouped or ungrouped, as the method for finding the mode depends on this.
   
• Draw bar graphs or histograms to help children visually spot the tallest bar (the mode). This makes it easier for them to understand.

The mode is the most frequently used in real-life applications such as businesses, social media, etc. Let’s understand the top uses of mode in day-to-day life. 

Businesses: Businesses use mode quite often to understand which product is in demand. The mode techniques help in understanding the on-demand product, any increase in production rate, etc.

Fashion Retail Industry: In fashion stores, the retailers will stock up on the most sold sizes, colors, or styles of fashion clothing by using the mode to maximize their sales.

Social Media: Social media influencers use mode to decide what type of content they should post on social media by analyzing their likes, shares, and comments over their mode value. For example, a YouTuber may check the most frequent video topic viewers engage with.
 

Transportation: In many countries, the mode is used to analyze which time is the most frequently traveled, to understand the most crowded time of buses, trains, or any public transports.

Education: Teachers and schools use mode to find the most common test score or grade range among students. This helps in understanding the overall class performance and identifying learning trends.