Mode Formula

The mode formula is an important concept in statistics that helps us find the value occurring most frequently in a dataset. Along with mean and median, mode is one of the three main measures of central tendency. It is especially useful when we want to identify the most common observation, such as the most popular product, the most repeated score in a test, or the most frequent response in a survey. Understanding the mode formula, its types, and applications makes it easier to analyze data in both academic and real-life situations.

 

Table of Contents

• What is Mode in Statistics?
• How to Find the Mode?
• Mode Formula
• Solved Examples
• Important Notes and Tips on Mode
• Conclusion
• FAQs On Mode Formula

 

What is Mode in Statistics?  

In statistics, the mode is the value that appears most often in a set of data. It is one of the three main measures of central tendency, along with mean and median. Unlike the mean (which gives the average) or the median (which gives the middle value), the mode simply shows the most common number or item.  

Example:  

In a dataset: 2, 4, 4, 4, 5, 6 - the mode is 4.  

If no value repeats, the dataset has no mode.  

 

How to Find the Mode?  

To find the mode:  

1. Organize the data.  

2. Identify the value or class with the highest frequency.  

3. Apply the mode formula:  

•  Use a simple frequency count for ungrouped data.  

•  Use the mathematical formula for grouped data.  

This answers the question of what mode is and how to calculate it.  

 

Mode Formula  

The mode formula is used to calculate the most frequently occurring value in a dataset. The formula changes depending on whether the data is ungrouped (individual values) or grouped (class intervals). The general idea is to find the value with the highest frequency.  

 

Mode Formula for Ungrouped Data  

When we work with small sets of data or single values that are not grouped into a class interval, we use the mode formula for ungrouped data. Here, the mode is just the number or value that appears most frequently in the list of data.

In simple words, if you are given a set of numbers, you just have to look at which number is repeated the most. That number is called the mode. Unlike mean and median, there is no calculation required here beyond counting frequencies.

The formula can be written as:

$\text{Mode} = \text{Value with the maximum frequency}$

 

Steps to find the mode in ungrouped data:

1. Write down all the numbers in the dataset clearly.

2. Count how many times each number occurs.

3. Identify the number that occurs the highest number of times.

4. That number is the mode.

Example:
Consider the dataset: 7, 8, 6, 9, 7, 6, 8, 7, 5, 6, 7

• The number 5 occurs 1 time

• The number 6 occurs 3 times

• The number 7 occurs 4 times

• The number 8 occurs 2 times

• The number 9 occurs 1 time

Here, 7 appears the most often (4 times).
So, the mode = 7.

Thus, the mode formula for ungrouped data makes it very simple to find the most common value in small sets of data.

 

Mode Formula for Grouped Data  

When data is large, it is often arranged in class intervals (like 0-10, 10-20, 20-30, etc.) to make it easier to study. In such cases, we cannot directly see which value appears most often. Instead, we use the mode formula for grouped data to estimate the most frequent value.

The mode formula for grouped data is used when data is presented in class intervals:  

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

Where:  

• L = Lower boundary of modal class  

• f1 = Frequency of modal class  

• f0 = Frequency of class before modal class  

• f2 = Frequency of class after modal class  

• h = Class width  

This mode formula for grouped data helps accurately calculate mode from frequency tables.  

Steps to find the mode in grouped data:

1. Prepare a frequency table of the data.

2. Find the modal class (the class interval with the highest frequency).

3. Identify the values of L, f₁, f₀, f₂, and h.

4. Substitute these values into the mode formula.

5. Solve the expression to get the mode.

Example:

The marks obtained by 50 students in a math test are arranged in the following frequency table. Find the mode.

Step 1: Identify the modal class
The modal class is the class with the highest frequency, which is 30 - 40 (frequency = 18).

Step 2: Identify values for the formula

• L = 30 (lower boundary of modal class)

• f₁ = 18 (frequency of modal class)

• f₀ = 12 (frequency of previous class)

• f₂ = 8 (frequency of next class)

• h = 10 (class width)

Step 3: Apply the mode formula

Mode = L + [(f₁ – f₀) / (2f₁ – f₀ – f₂)] × h
Mode = 30 + [(18 – 12) / (2×18 – 12 – 8)] × 10
Mode = 30 + (6 / (36 – 20)) × 10
Mode = 30 + (6 / 16) × 10
Mode = 30 + 60/16
Mode = 30 + 3.75
Mode = 33.75

 Answer:
The mode of the marks is approximately 33.75.

 

Solved Example

Example 1:
Find the mode of the following data:
7, 8, 6, 9, 7, 6, 8, 7, 5, 6, 7

Solution:
Step 1: Count the frequency of each value.

• 5 → 1 time

• 6 → 3 times

• 7 → 4 times

• 8 → 2 times

• 9 → 1 time

Step 2: Identify the value with the highest frequency.

• 7 occurs 4 times - highest frequency.

Answer:
Mode = 7

 

Example 2:
Find the mode for the following frequency distribution:

 

 

Solution:
Step 1: Identify the modal class (class with highest frequency).

• Modal Class = 30 - 40, frequency = 20
  
  

Step 2: Use the mode formula for grouped data:

Mode = L + [(f1 - f0) / (2f1 - f0 - f2)] × h

Where:
L = 30 (lower limit of modal class)
f1 = 20 (frequency of modal class)
f0 = 15 (frequency of previous class)
f2 = 12 (frequency of next class)
h = 10 (class width)

Step 3: Substituting values into the formula:

Mode = L + [(f1 - f0) / (2f1 - f0 - f2)] × h

Mode = 30 + [(20 - 15) / (2×20 - 15 - 12)] × 10

Mode = 30 + (5 / (40 - 27)) × 10

Mode = 30 + (5 / 13) × 10

Mode ≈ 30 + 3.85 Mode ≈ 33.85

Answer:
Mode ≈ 33.85

 

Example 3:
Find the mode of the data:
12, 15, 12, 17, 19, 15, 12, 18, 15, 15

Solution:
Step 1: Count frequency of each value:

• 12 → 3 times

• 15 → 4 times

• 17 → 1 time

• 18 → 1 time

• 19 → 1 time
  
  

Step 2: Highest frequency = 4 (for 15)

Answer:
Mode = 15

 

Example  4:
Find the mode of the data:
25, 27, 29, 27, 26, 25, 25, 27, 26, 27

Solution:
Frequency count:

• 25 → 3 times

• 26 → 2 times

• 27 → 4 times

• 29 → 1 time

Highest frequency = 4 (for 27)

Answer:
Mode = 27

 

Example  5:
Find the mode of the following data:

 

 

Solution:
Step 1: Identify the modal class (class with highest frequency).

Modal Class = 30 - 40, frequency = 17

Step 2: Use the mode formula for grouped data:

$\text{Mode} = L + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h$

Where:
L = 30 (lower limit of modal class)
f₁ = 17 (frequency of modal class)
f₀ = 13 (frequency of previous class)
f₂ = 11 (frequency of next class)
h = 10 (class width)

Step 3: Substituting values into the formula:

Use formula:

Mode = L + [(f₁ - f₀) / (2f₁ - f₀ - f₂)] × h
Mode = 30 + [(17 - 13) / (2 × 17 - 13 - 11)] × 10
Mode = 30 + (4 / (34 - 24)) × 10
Mode = 30 + (4 / 10) × 10
Mode = 30 + 4
Mode = 34

Answer:
Mode = 34

 

Important Notes and Tips on Mode  

• The mode formula is helpful for dealing with frequencies and repeated values.  

• "What is mode?" is a common question in exams; always remember it is the most frequent value.  

• Mode in statistics works best for categorical data, but it can also apply to numerical data.  

• Use the mode formula for grouped data when working with class intervals.  

• Use the mode formula for ungrouped data for small datasets or direct frequency counts.  

• In multimodal data, simply list all the modes.  

• If no value repeats, then mode is undefined or no mode.  

• The mode formula with examples is crucial to learn for school and competitive exams.  

 

Conclusion  

The mode formula is a valuable statistical tool to spot the most frequent value in a dataset. Whether it's mode for ungrouped data or mode for grouped data, understanding the mode formula with examples is beneficial for school, competitive exams, and real-world data analysis. Knowing what mode is and how to use it across datasets is a key part of learning mode in statistics.  

 

Frequently Asked Questions on Mode Formula

1. How to calculate using the mode?  

Answer: To calculate the mode, list the numbers in the dataset and find the number that shows up most often. If more than one value is most frequent, the data is multimodal.  

 

2. What does the mode formula do?  

Answer: The mode formula identifies the most common value in a dataset. For grouped data, the mode can be estimated with the formula:  

Mode = L + [(f₁ - f₀) / (2f₁ - f₀ - f₂)] × h  

where:  

• L = lower boundary of the modal class  

• f₁ = frequency of the modal class  

• f₀ = frequency of the class before the modal class  

• f₂ = frequency of the class after the modal class  

• h = class width  

 

3. What is the mode of the dataset 1, 4, 4, 5, 5, 9, 9, 9?  

Answer: The mode of the dataset is 9 since it appears three times, which is more than any other number.  

 

4. How to calculate mode quickly?  

Answer: To find mode quickly:  

• Arrange the data in order (if not already done).  

• Count how many times each value appears.  

• The value with the highest frequency is the mode.  
  

 

Learn more math concepts like Mode Formula at Orchids The International School!