11.2.3 Mode

Introduction

The mode is another measure of central tendency. It is the value that appears most frequently in a dataset. Unlike the mean and median, the mode can be used for both numerical and categorical data. A dataset may have one mode, more than one mode, or no mode at all. This topic explains the definition, formula, and calculation of the mode for both ungrouped and grouped data.

Definition

Mode: The mode is the value in a dataset that appears maximum number of times.

- A dataset may have one mode (unimodal), more than one mode (bimodal or multimodal), or no mode at all (amodal) i.e., all values occur the same number of times.

Symbol: It may be denoted by x (read as x-hat or x cap).

Formula (ungrouped data):

Mode (x) = The most frequent or repeated value occurring in a given dataset

Main Idea

The mode is the most frequently occurring value in a dataset.

• It is the value that appears maximum number of times.
• A dataset can have no mode, one mode, or multiple modes.
• It is useful for categorical data.

Core Concept

Formula for Mode

Where:

• l = lower boundary of modal group
• f_m = maximum frequency of a group
• f₁\) = previous frequency of \(f_m
• f₂\) = next frequency of \(f_m
• h = class interval of modal group

Detailed Explanation

1. Mode for Ungrouped Data

Definition: The mode is the value in a dataset that appears maximum number of times.

Formula:

Mode (x) = The most frequent or repeated value occurring in a given dataset

Simple explanation: The mode is the number that appears most often in the data.

Types of mode:

• Unimodal: One mode.
• Bimodal: Two modes.
• Multimodal: More than two modes.
• Amodal: No mode (all values occur the same number of times).

2. Mode for Grouped Data

Formula:

Mode (x) = l + (f_m - f₁) × h(f_m - f₁) + (f_m - f₂)

Where:

• l = lower boundary of modal group
• f_m = maximum frequency of a group
• f₁\) = previous frequency of \(f_m
• f₂\) = next frequency of \(f_m
• h = class interval of modal group

Simple explanation: For grouped data, the mode is calculated using the modal group (the group with the highest frequency).

Solved Examples

Example 1: Mode (Ungrouped Data, Word)

Dataset: Find the mode from the word "BIOLOGY".

Solution:

Given: The word "BIOLOGY"

Required: Mode

Formula:

Mode (x) = The most repeated value of the data

Calculation:

Letters in "BIOLOGY": B, I, O, L, O, G, Y

The letter "O" appears twice. All other letters appear once.

Final Answer: Mode = O

Example 2: Mode (Ungrouped Data, Unimodal)

Dataset: Find mode from the following data: 3, 5, 6, 6, 9, 10, 9, 6, 6

Solution:

Given: 3, 5, 6, 6, 9, 10, 9, 6, 6

Required: Mode

Formula:

Mode (x) = The most repeated value of the data

Calculation:

• 3 appears 1 time
• 5 appears 1 time
• 6 appears 4 times
• 9 appears 2 times
• 10 appears 1 time

The number 6 appears most frequently (4 times).

Final Answer: Mode = 6

Example 3: Mode (Ungrouped Data, Bimodal)

Dataset: Find the mode from the following data: 3, 5, 5, 6, 9, 9, 10, 12

Solution:

Given: 3, 5, 5, 6, 9, 9, 10, 12

Required: Mode

Formula:

Mode (x) = The most repeated value of the data

Calculation:

• 3 appears 1 time
• 5 appears 2 times
• 6 appears 1 time
• 9 appears 2 times
• 10 appears 1 time
• 12 appears 1 time

Both 5 and 9 appear most frequently (2 times each).

Final Answer: Mode = 5 and 9 (bimodal)

Example 4: Mode (Grouped Data)

Dataset: Find the mode from the following frequency distribution:

Solution:

Given:

• Modal group = 30–39 (highest frequency = 40)
• l = lower boundary of modal group = 29.5
• f_m = maximum frequency = 40
• f₁ = previous frequency = 25
• f₂ = next frequency = 20
• h = class interval = 10

Required: Mode

Formula:

Mode (x) = l + (f_m - f₁) × h(f_m - f₁) + (f_m - f₂)

Substitution:

Mode (x) = 29.5 + (40 - 25) × 10(40 - 25) + (40 - 20)

Calculation:

Mode (x) = 29.5 + 15 × 1015 + 20

= 29.5 + 15035

= 29.5 + 4.29

= 33.79

Final Answer: Mode = 33.79

Summary Table

Types of Mode

Intermediate Level

The mode is the value in a dataset that appears maximum number of times.

Formula for ungrouped data:

Mode (x) = The most repeated value of the data

Formula for grouped data:

Mode (x) = l + (f_m - f₁) × h(f_m - f₁) + (f_m - f₂)

Where:

• l = lower boundary of modal group
• f_m = maximum frequency
• f₁ = previous frequency
• f₂ = next frequency
• h = class interval

Key points:

• Mode can be unimodal, bimodal, or multimodal.
• Mode can be used for categorical data.
• Mode is not affected by extreme values.
• Symbol: x (x-hat).

Important Points

• - The mode is the value in a dataset that appears maximum number of times.
• - A dataset may have one mode (unimodal), more than one mode (bimodal or multimodal) or no mode at all (amodal) i.e. all values occur in same number of times.
• - It may be denoted by x (read as x-hat or x cap).
• - Mode = the most frequent or repeated value occurring in a given dataset.
• - Formula for calculating Mode for Group data is given below:
• Mode (x) = l + (f_m - f₁) × h(f_m - f₁) + (f_m - f₂)
• - Where, l = lowercase boundary of model group, f_m = maximum frequency of a group, f₁ = previous frequency of f_m, f₂ = next frequency of f_m, h = class interval of model group.

Remember

• - Mode = most frequently occurring value
• - Symbol = x (x-hat)
• - Ungrouped formula = Most repeated value
• - Grouped formula = l + (f_m - f₁) × h(f_m - f₁) + (f_m - f₂)
• - Types = unimodal, bimodal, multimodal, amodal
• - Not affected by extreme values
• - Useful for categorical data

Common Mistakes

• 1. Thinking there is always one mode.
• - A dataset can have no mode, one mode, or multiple modes.
• 2. Confusing mode with mean or median.
• - Mode = most frequent value.
• - Mean = average.
• - Median = middle value.
• 3. Forgetting the formula for grouped data.
• - Grouped data uses l + (f_m - f₁) × h(f_m - f₁) + (f_m - f₂).
• 4. Mixing up f_m, f₁, and f₂.
• - f_m = maximum frequency.
• - f₁ = previous frequency.
• - f₂ = next frequency.
• 5. Thinking the mode is always in the center of the data.
• - The mode can be anywhere in the dataset.
• 6. Believing the mode is not useful.
• - The mode is useful for categorical data and identifying the most frequent value.
• 7. Forgetting the symbol for mode.
• - Mode is denoted by x (x-hat).

Quick Revision

Types of Mode

60-Second Revision

• Mode = most frequently occurring value
• Symbol = x (x-hat)
• Ungrouped formula = Most repeated value
• Grouped formula = l + (f_m - f₁) × h(f_m - f₁) + (f_m - f₂)
• Types = unimodal, bimodal, multimodal, amodal
• Not affected by extreme values
• Useful for categorical data
• Warning: Do not think there is always one mode. A dataset can have no mode, one mode, or multiple modes. The mode is not necessarily in the center of the data.