Mode Formula in Statistics: Definition, Formula, Examples & Questions

The mode formula is used to find the value that occurs most frequently in a dataset. It is one of the important measures of central tendency, along with mean and median.

In simple terms, mode tells us which value appears the most in a set of data.

For ungrouped data, we find the mode by directly observing the value that repeats the most number of times. For grouped data, we use a mathematical formula based on class intervals to estimate the mode.

Table of Contents

• What is Mode?
• Mode Formula in Statistics
• Mode Formula for Ungrouped Data
• Mode Formula for Grouped Data
• Mode Formula with Example
• Mode Formula Class 6
• Mode Formula Class 8
• Mode Formula Class 9
• Mode Formula Class 10
• Mean Median Mode Formula Class 10
• Types of Mode
• Applications of Mode
• Solved Examples on Mode Formula
• Practice Questions on Mode Formula
  
  

What is Mode?

The mode is the value that appears most frequently in a dataset. It focuses only on the repeated value.

Example: Dataset: 2, 4, 4, 5, 6 - Since 4 appears most often, the mode is 4.

If no value repeats, the dataset has no mode.

Mode Formula in Statistics

The mode formula helps find the value or group that appears most often in a dataset.

Mode Formula for Grouped Data

Where:

• L = lower limit of modal class
• f₁ = frequency of modal class
• f₀ = frequency of preceding class
• f₂ = frequency of succeeding class
• h = class width
  
  

Mode Formula for Ungrouped Data

Mode Formula for Ungrouped Data

When data is not arranged into class intervals, the mode is simply the value that appears most frequently.

Steps to Find Mode in Ungrouped Data:

1. Write all observations clearly.
2. Count the frequency of each value.
3. Identify the value with the highest frequency.
4. That value is the mode.

Example: Dataset: 7, 8, 6, 9, 7, 6, 8, 7, 5, 6, 7

• 5 → 1 time
• 6 → 3 times
• 7 → 4 times
• 8 → 2 times
• 9 → 1 time

Since 7 appears most frequently: Mode = 7

Mode Formula for Grouped Data

When data is arranged into class intervals, the class interval with the highest frequency is called the modal class.

Steps:

1. Prepare the frequency distribution table.
2. Identify the modal class.
3. Find the values of L, f₁, f₀, f₂, and h.
4. Substitute values into the formula.

Example:

Class Interval	Frequency
0 – 10	3
10 – 20	7
20 – 30	12
30 – 40	18
40 – 50	8
50 – 60	2

Solution: Modal Class = 30–40, L = 30, f₁ = 18, f₀ = 12, f₂ = 8, h = 10

Mode = 30 + [(18 − 12) / (36 − 12 − 8)] × 10 = 30 + (6/16) × 10 = 30 + 3.75 = 33.75

Mode Formula with Example

Ungrouped Example: Dataset: 4, 5, 6, 6, 7, 6, 8 - 6 appears 3 times (highest). Mode = 6

Mode Formula - Class 6

Find mode in the dataset: 2, 4, 4, 5, 6 - 4 appears most often. Mode = 4

Mode Formula - Class 8

Find the mode of: 2, 4, 5, 4, 6, 4, 7

2 - once, 4 - 3 times, 5 - once, 6 - once, 7 - once

4 appears 3 times (highest frequency).

Mode = 4

Mode Formula - Class 9

Example:

Class Interval	Frequency
0 – 10	5
10 – 20	9
20 – 30	12
30 – 40	7
40 – 50	3

Modal class = 20–30 (frequency 12). L = 20, f₁ = 12, f₀ = 9, f₂ = 7, h = 10

Mode = 20 + [(12 − 9) / (24 − 9 − 7)] × 10 = 20 + (3/8) × 10 = 20 + 3.75 = 23.75

Mode Formula - Class 10

Example:

Class Interval	Frequency
0 – 10	3
10 – 20	5
20 – 30	9
30 – 40	12
40 – 50	7

Modal class = 30–40. L = 30, f₁ = 12, f₀ = 9, f₂ = 7, h = 10

Mode = 30 + [(12 − 9) / (24 − 9 − 7)] × 10 = 30 + (3/8) × 10 = 30 + 3.75 = 33.75

Mean Median Mode Formula - Class 10

The empirical relation between the three measures of central tendency:

Example: If Mean = 18 and Median = 20:

Mode = 3 × 20 − 2 × 18 = 60 − 36 = 24

Types of Mode

Type	Meaning
Unimodal	One mode
Bimodal	Two modes
Multimodal	More than two modes
No Mode	No repeated value

Applications of Mode

Mode is widely used in:

• Business analysis
• Market research
• Survey reports
• Classroom performance analysis
• Inventory planning
• Customer preference studies
  
  

Solved Examples on Mode Formula

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

5 - 4 times, 6 - 1, 7 - 2, 8 - 1, 9 - 1

Mode = 5

Example 2: Find the mode of: 14, 12, 11, 14, 13, 14, 15, 12, 14

11 - 1, 12 - 2, 13 - 1, 14 - 4, 15 - 1

Mode = 14

Example 3: Find the mode for grouped data:

Class Interval	Frequency
0 – 10	4
10 – 20	9
20 – 30	15
30 – 40	11
40 – 50	6

Modal class = 20 - 30. L = 20, f₁ = 15, f₀ = 9, f₂ = 11, h = 10

Mode = 20 + [(15 − 9) / (30 − 9 − 11)] × 10 = 20 + (6/10) × 10 = 26

Example 4: Find the mode of: 21, 18, 21, 19, 20, 21, 18, 20, 21

• 18 - 2, 19 - 1, 20 - 2, 21 - 4

Mode = 21

Example 5: Find the mode for grouped data:

Class Interval	Frequency
0 – 5	3
5 – 10	7
10 – 15	12
15 – 20	9
20 – 25	4

Modal class = 10 - 15. L = 10, f₁ = 12, f₀ = 7, f₂ = 9, h = 5

Mode = 10 + [(12 − 7) / (24 − 7 − 9)] × 5 = 10 + (5/8) × 5 = 10 + 3.125 ≈ 13.13

Practice Questions on Mode Formula

1. Find the mode of: 6, 8, 9, 6, 5, 6, 7, 8
2. Find the mode of: 12, 15, 17, 15, 14, 15, 18, 19
3. Find the mode of: 22, 24, 26, 24, 22, 24, 27, 28
4. Find the mode of: 31, 35, 36, 35, 38, 35, 40
5. Find the mode of: 5, 7, 7, 8, 9, 10, 7, 6

6. Find the mode for grouped data:

   Class Interval	Frequency
   0 – 10	5
   10 – 20	11
   20 – 30	16
   30 – 40	10

7. Find the mode for grouped data:

   Class Interval	Frequency
   0 – 5	2
   5 – 10	8
   10 – 15	14
   15 – 20	9

8. Find the mode of: 18, 20, 22, 18, 21, 18, 24
9. Find the mode of: 11, 13, 15, 13, 17, 13, 19, 20
10. If Mean = 25 and Median = 27, find Mode using the empirical relation.

Answers:

1. Mode = 6
2. Mode = 15
3. Mode = 24
4. Mode = 35
5. Mode = 7
6. Mode ≈ 24.54
7. Mode ≈ 12.73
8. Mode = 18
9. Mode = 13
10. Mode = 31