Calculate Mode (Modal Value)
Online calculator to compute the mode of a distribution in a data set
Mode Calculator
The Mode (Modal Value)
The mode (also called modal value) is the value with the highest frequency in a data set. It is particularly suitable for categorical and discrete data.
Mode Concept
The mode is the value that occurs most frequently.
In frequency distributions, it is the peak.
■ Highest Frequency (Mode) ■ Other Values
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What is the Mode (Modal Value)?
The mode is an important measure of location in descriptive statistics:
- Definition: Value with the highest frequency in a data set
- Designation: Also called modal value or density mean
- Determination: Count the frequencies of all values
- Advantage: Robust against outliers, suitable for categorical data
- Application: Discrete data, frequency distributions
- Special Feature: Can have multiple modes (multimodal)
Types of Distributions
Different distribution types are distinguished based on the number of modes:
Unimodal
One Mode: The distribution has exactly one peak.
This is the most common case.
Example: 1, 2, 2, 2, 3, 4 → Mode = 2
Bimodal
Two Modes: The distribution has two peaks of equal height.
Example: 1, 1, 2, 3, 3 → Modes = 1 and 3
Multimodal
Multiple Modes: The distribution has more than two peaks.
Example: 1, 1, 2, 2, 3, 3 → Modes = 1, 2, and 3
Applications of the Mode
The mode is particularly important for:
Business and Marketing
- Best-selling product
- Most popular size (clothing, shoes)
- Most frequent purchase quantity
- Typical customer preference
Surveys and Social Sciences
- Most frequent opinion in surveys
- Typical response (e.g., school grades)
- Modal value in Likert scales
- Categorical data analysis
Determination of Mode
Mode (Modal Value)
Value x with maximum frequency f(x)
Determination
1. Count the frequency of each value
2. Identify the value with the highest frequency
3. In case of ties: Multiple modes (multimodal)
For Grouped Data
L = lower class limit, h = class width
Properties
- Does not always exist (in uniform distribution)
- Can occur multiple times
- Only meaningful for discrete or categorical data
Symbol Explanations
| Mode | Modal value, most frequent value |
| \(f(x)\) | Frequency of value x |
| \(L\) | Lower class limit |
| \(h\) | Class width |
| \(f_0, f_1, f_2\) | Frequencies of adjacent classes |
Example Calculations for the Mode
Example 1: Unimodal Distribution
Calculate: Mode of the data set
1. Count Frequencies
| Value 2: | 1x |
| Value 3: | 1x |
| Value 4: | 3x |
| Value 5: | 3x |
| Value 7: | 1x |
| Value 8: | 1x |
| Value 9: | 1x |
2. Find Maximum
The highest frequency is 3.
Two values have this frequency: 4 and 5
3. Determine Mode
Bimodal Distribution!
Modes: 4 and 5
Calculator shows: 4 (smallest value)
Example 2: Clear Mode
Clear most frequent value
Frequency Table
| Value 1: | 1x |
| Value 2: | 3x ← Maximum |
| Value 3: | 1x |
| Value 4: | 2x |
| Value 5: | 1x |
Result
Unimodal Distribution
Mode: 2
The value 2 occurs 3 times and is thus the most frequent value.
Example 3: Categorical Data (School Grades)
Most frequent grade of a test
Grade Distribution
| Grade 1: | 1x (10%) |
| Grade 2: | 5x (50%) ← Mode |
| Grade 3: | 3x (30%) |
| Grade 4: | 1x (10%) |
Interpretation
Mode: Grade 2
The most frequent grade is 2 (50% of students).
The mode is perfect for categorical data like school grades,
as it represents the "typical" value, while the arithmetic mean
(2.3) lies between the grades.
Mathematical Foundations of the Mode
The mode (modal value) is a fundamental measure of location in descriptive statistics with special properties and applications.
Properties of the Mode
The mode has characteristic mathematical properties:
- Frequency-Based: Determined by counting, not by calculation
- Robust: Insensitive to outliers
- Existence: Does not always exist (e.g., in uniform distribution)
- Uniqueness: Can occur multiple times (multimodal)
- Scale Level: Can be used for nominal, ordinal, and metric data
Comparison with Other Measures of Location
Mode
Most frequent value
For all scale levels
Robust against outliers
Median
Middle value
From ordinal scale level
Robust against outliers
Arithmetic Mean
Average value
Only for metric data
Sensitive to outliers
When to Use Mode?
Mode is Suitable For
- Categorical Data: Colors, brands, categories
- Discrete Data: Shoe sizes, number of children
- Ordinal Data: School grades, ratings
- Typical Value: "What is most common?"
- Robust Analysis: With outliers
Limitations
- Continuous Data: Usually no clear mode
- Uniform Distribution: Does not exist
- Multimodality: Interpretation difficult
- Small Samples: Not very meaningful
- No Mathematical Property: Not further computable
Modal Class for Grouped Data
For grouped data (frequency distribution with classes) we speak of the modal class:
- Modal Class: The class with the highest frequency
- Interpolation: Exact mode can be interpolated
- Formula: Mode = L + [(f₁-f₀) / (2f₁-f₀-f₂)] · h
Where L is the lower class limit of the modal class, h is the class width, f₁ is the frequency of the modal class, f₀ is the frequency of the previous and f₂ of the following class.
Relationship to Distribution Shape
Symmetric Distribution
In a symmetric distribution (e.g., normal distribution):
Mode = Median = Arithmetic Mean
All three measures of location coincide.
Skewed Distribution
In a skewed distribution, the measures differ:
Right Skew: Mode < Median < Mean
Left Skew: Mode > Median > Mean
Summary
The mode is the only measure of location that can be used for nominal data. It is robust against outliers and shows the "typical" or "most common" value of a distribution. Its main strengths lie in analyzing categorical and discrete data (e.g., school grades, product preferences, clothing sizes). For continuous data, it is less meaningful, as values rarely repeat. A multimodal distribution (multiple modes) can indicate a heterogeneous population or multiple underlying groups.
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