Mode in Mathematics: Definition, Formula, Types, and Examples

The mode in mathematics is the value that appears most frequently in a dataset. It helps you quickly identify the most common number or category in a set of data.

For example, in a class survey, if 10 students prefer Mathematics, 8 prefer Science, and 5 prefer English, the mode is Mathematics because it is chosen the most.

The mode is especially useful for understanding frequent choices or patterns in data. A formula is used only when the data is grouped into class intervals; otherwise, the mode can be found by simply identifying the most frequent value.

Table of Contents

• What is Mode?
• Types of Mode in Mathematics
• Mode Formula
• How to Find Mode in Ungrouped Data
• How to Find Mode in Grouped Data
• Real-Life Applications of Mode
• Solved Examples on Mode
• Fun Facts on Mode
• Common Misconceptions on Mode
• Practice Questions on Mode
• Conclusion
• Frequently Asked Questions on Mode
  
  

What is Mode?

Mode is the number that appears most often in a set of numbers. It shows what is most common in a dataset.

Example: In 3, 5, 5, 6, 7, the mode is 5 because it appears twice.

There can be more than one mode, or no mode if all numbers appear only once. The mode is useful for finding the most common value, like the most popular shoe size in a class.

Types of Mode in Mathematics

There are different types of mode depending on the number of values that appear with the highest frequency:

• Unimodal: A dataset with only one mode, where one value occurs most frequently.
  Example - Unimodal: 2, 4, 4, 6
  Mode = 4
• Bimodal: A dataset with two modes, where two values appear with the same highest frequency.
  Example - Bimodal: 2, 3, 3, 5, 5
  Modes = 3, 5
• Trimodal: A dataset with three modes.
  Example - Trimodal: 1, 2, 2, 3, 3, 4, 4
  Modes = 2, 3, 4
• Multimodal: A dataset with more than three modes.
  Example - Multimodal: 1, 1, 2, 2, 3, 3, 4, 4
  Modes = 1, 2, 3, 4

These types help you understand how values are distributed in data.

Mode Formula

The mode formula for grouped data is:

Mode = L + [(fₘ - f₁) / (2fₘ - f₁ - f₂)] × 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 preceding the modal class
• f2 = Frequency of the class succeeding the modal class

This formula is used only for grouped data (data in class intervals). Mode Formula

For ungrouped data, the mode is simply the most frequent value in the set. We can find it by arranging the data in ascending or descending order and selecting the value that appears most often.

How to Find Mode in Ungrouped Data

For ungrouped data, follow these steps:

1. Arrange the data in ascending or descending order.
2. Count the frequency of each number.
3. The number that appears most frequently is the mode.

Example:

Data: 3, 5, 6, 8, 8, 8, 9, 10

The number 8 appears most frequently, so the mode is 8.

How to Find Mode in Grouped Data

To find the mode for grouped data, follow these steps:

1. Identify the modal class, which is the class interval with the highest frequency.
2. Find the class width (h), which is the difference between the upper and lower limits of the modal class.
3. Substitute the values in the mode formula and calculate the result.

Formula:

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

Example:

Class Interval	Frequency
0 - 5	5
5 - 10	3
10 - 15	7
15 - 20	2

The highest frequency is 7, so the modal class is 10 - 15. Here, the lower limit (L) is 10, the frequency of the modal class (fₘ) is 7, the frequency of the preceding class (f₁) is 3, and the frequency of the succeeding class (f₂) is 2. The class width (h) is 5.

Substituting the values in the formula:

Mode = 10 + [(7 - 3) / (2×7 - 3 - 2)] × 5
Mode = 10 + [4 / (14 - 5)] × 5
Mode = 10 + (4/9) × 5
Mode = 10 + 2.22
Mode = 12.22

Read more:

• Arithmetic Mean
• Median
• Empirical Relation between Mean, Median and Mode
• Mode for Class 7
  
  

Solved Examples on Mode

Example 1: Find the mode for the following data: 5, 8, 8, 10, 15, 20, 20, 20, 25.

Solution: The number 20 appears most frequently, so the mode is 20.

Example 2: For the class intervals below, find the mode.

Class Interval	Frequency
0 - 10	8
10 - 20	12
20 - 30	20
30 - 40	5

The modal class is 20 - 30 (frequency = 20). Using the mode formula:

Mode = 20 + [10 × (20 - 12)] / [(20 - 12) + (20 - 5)]
= 20 + (10 × 8) / (8 + 15)
= 20 + 80 / 23
≈ 23.48

So, the mode is approximately 23.48.

Example 3: Find the mode of the given data set: 3, 3, 6, 9, 15, 15, 15, 27, 27, 37, 48.

Solution: In the following list of numbers, 3, 3, 6, 9, 15, 15, 15, 27, 27, 37, 48 - 15 is the mode since it appears more times than any other number.

Example 4: Find the mode of 4, 4, 4, 9, 15, 15, 15, 27, 37, 48.

Solution: A data set can have more than one mode if more than one value occurs with equal frequency. Here both 4 and 15 are modes of the set.

Example 5: Find the mode of the following data set: 2, 5, 7, 9, 11.

Solution: Each number appears only once. Since no number repeats, there is no mode for this data set.

Learn more about Mean, Median and Mode
Practice questions with Mean, Median and Mode Questions

Fun Facts on Mode

1. The mode works for both numbers and categories
The mode can be used for numbers as well as non-numerical data like names or choices.
Example: Numbers - 2, 4, 4, 6
Mode = 4
Colours - Red, Blue, Blue, Green
Mode = Blue

2. A dataset can have more than one mode
Sometimes, more than one value appears the same highest number of times.
Example: 1, 2, 2, 3, 3
Modes = 2 and 3

3. The mode is useful in real life
It helps find the most common or popular choice in different situations.
Example 1 - Shop sales:
Shirts sold: S, M, M, L
Mode = M (most sold size)

Example 2 - Class survey:
Favourite subject: Maths, Science, Maths
Mode = Maths

Common Misconceptions on Mode

1. The mode is always one value
Correct - A dataset can have more than one mode, or even no mode.
Example: 2, 3, 3, 5, 5
Modes = 3 and 5 (bimodal)
In this: 1, 2, 3, 4 - No mode (no number repeats)

2. The mode must always be a number
Correct - The mode can also be a category.
Example: Favourite colours: Red, Blue, Blue, Green
Mode = Blue

3. The mode and mean are the same
Correct - The mean is the average, while the mode is the most frequent value.
Example: Data: 1, 2, 2, 10
Mean = 3.75, Mode = 2 (different values)

Practice Questions on Mode

1. Find the mode of: 6, 8, 8, 10, 12, 14, 8.
2. Find the mode(s) of: 5, 7, 7, 9, 9, 11, 13.
3. Find the mode of: 2, 4, 6, 8, 10, 12.
4. In a class, students chose their favourite fruit: Apple, Banana, Apple, Mango, Banana, Apple. Which fruit is the mode?
5. The mode of the data set is 15. 10, 15, 20, 15, __, 25. What could the missing number be if 15 must remain the mode?
6. The marks scored by students are: 40 (2 times), 50 (5 times), 60 (3 times), 70 (5 times). Find the mode(s).
7. The number of hours students studied in a week: 2, 3, 4, 3, 5, 3, 6. Find the mode.
8. Find the mode: 12, 15, 18, 15, 21, 24, 15, 18, 18.
9. If the mode of a data set is 9, which number is most likely to appear more often: 9, 6, or 12?
10. T-shirts sold in a shop: Red-10, Blue-15, Green-15, Yellow-8. Which colour(s) represent the mode?
    
    

Conclusion

The mode in maths is a vital tool for understanding data sets and finding the most frequently occurring values. Whether dealing with ungrouped data or grouped data, knowing how to find the mode is an essential skill in both academic and practical applications. It helps identify trends, make informed decisions, and analyze data effectively.

Frequently Asked Questions On Mode

1. What is the mode?

Answer. The mode is the value that appears most frequently in a data set.

2. How is the mode useful?

Answer. The mode helps identify the most common occurrence in a dataset, making it useful for analyzing frequency-based data.

3. Can a dataset have more than one mode?

Answer. Yes, a dataset can have multiple modes (bimodal, trimodal, or multimodal).

4. Can there be no mode?

Answer. Yes, if all values in the dataset appear with the same frequency or if no value repeats, there is no mode.

5. How do I calculate the mode for grouped data?

Answer. Use the formula:

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

Learn how to identify the mode in a data set and why it's important in statistics. Explore more exciting math concepts with Orchids The International School!