Mode Formula

Introduction to Mode  

In statistics, the mode is one of the three main measures of central tendency, alongside mean and median. It represents the value that shows up most often in a dataset. The mode formula helps calculate this frequent value accurately.  

 

Table of Contents

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

 

What is Mode in Statistics?  

What is mode in statistics? The mode is the value that occurs most frequently in a dataset. Unlike mean or median, which take all data points into account, mode only focuses on the most repeated value.  

Example:  

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

If no value repeats, the dataset has no mode.  

"What is mode?" is a common question asked when analyzing categorical and numerical data.  

 

Types of Mode in Data

In statistics, data can be described based on how many modes, or most frequent values, it has. They are classified as follows:

1. Unimodal Data

• Definition: A dataset is called unimodal when only one value appears most often.

• This is the most common type of distribution.

• It has a single peak when displayed on a graph.

• The mode formula is straightforward in unimodal data since there is only one most frequent value.

 

Example:  

Data: 2, 4, 4, 5, 6, 7  

Here, the number 4 appears twice, while all other numbers appear only once.  

Mode = 4  

This is a unimodal dataset.

 

2. Bimodal Data

• Definition: A dataset is called bimodal when two different values occur with the same highest frequency.

• This type of data has two peaks in its frequency distribution.

• Bimodal distributions can show two different trends or groups in the data.

 

Example:  

Data: 1, 2, 2, 3, 4, 4, 5  

Here, both 2 and 4 appear twice.  

Modes = 2 and 4  

This dataset is bimodal because it has two modes.

 

3. Trimodal and Multimodal Data

• Trimodal: A dataset is trimodal if it has three values that appear with the same highest frequency.

• Multimodal: If a dataset has more than two modes, such as three, four, or more values with equal and highest frequencies, it is called multimodal.

• These datasets show multiple peaks and are often more complicated to analyze.

• The mode formula is not typically used the same way for multimodal data, since the result is a list of modes instead of a single calculated value.

 

Example (Trimodal):  

Data: 3, 3, 5, 5, 7, 7, 8  

Here, 3, 5, and 7 all appear twice.  

Modes = 3, 5, 7  

This is a trimodal dataset.

 

Example (Multimodal):  

Data: 2, 2, 4, 4, 6, 6, 8, 8  

Here, 2, 4, 6, and 8 all appear twice.  

Modes = 2, 4, 6, 8  

This is a multimodal dataset.

 

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 varies based on whether the data is grouped or ungrouped. The general idea is to find the value with the highest frequency.  

 

Mode Formula for Ungrouped Data  

The mode formula for ungrouped data is simple:  

Mode = Value with the maximum frequency  

This formula is best when the dataset is small and values are easy to see.  

 

Mode Formula for Grouped Data  

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

Mode =  L + ((f1 - f0) / (2f1 - f0 - f2)) × 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.  

 

Derivation of Mode Formula  

The mode formula for grouped data is:  

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  

 

Step-by-Step Derivation:  

1. Identify the modal class:  

   The modal class is the class interval that has the highest frequency (f1).  

2. Draw a histogram:  

   Draw bars for each class interval. The tallest bar is the modal class.  

   The class before it has frequency f0.  

   The class after it has frequency f2.  

3. Use interpolation:  

   Assume frequency increases linearly from f0 to f1 and decreases linearly from f1 to f2.  

4. Let the mode be M and lie within the modal class.  

The mode is closer to the class with higher frequency.  

5. Using similar triangles (from histogram visualization), we estimate the distance from L to the mode:  

 

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

So,  

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

 

Solved Example

Question:
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

 

Question 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: Plug values into the formula:

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

 

Question 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

 

Question 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

 

Question 5:
Find the mode of the following data:

 

 

Solution:
Step 1: Modal class = 30 - 40 (frequency = 17)
L = 30, f1 = 17, f0 = 13, f2 = 11, h = 10

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.  

 

Related Links

Mean, Median, Mode - Understand the differences and applications of mean, median, and mode in statistics, with clear definitions and solved examples.

Mean - Learn how to calculate the mean (average) of a data set, with step-by-step explanations and practical examples to strengthen your understanding of statistics.

Mode - Discover how to find the mode of a data set, understand its significance in statistics, and explore examples that demonstrate real-world usage.

 

Frequently Asked Questions on Mode  

1. How to calculate using mode?  

Ans: 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?  

Ans: 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?  

Ans: 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?  

Ans: 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.  

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