Complete Guide to Mode Formulas in Statistics

What is Mode?

The mode is the value that appears most frequently in a dataset. It is one of the three measures of central tendency, alongside mean and median. Unlike mean and median, a dataset can have no mode, one mode (unimodal), or multiple modes (bimodal or multimodal).

Complete List of Mode Formulas

1. Mode for Ungrouped Data

Example: In the dataset {2, 3, 4, 3, 5, 3, 6}, the mode is 3 (appears 3 times).

2. Mode for Grouped Data (Class Intervals)

Method 1: Modal Class Formula

Method 2: Alternative Grouping Method Formula

Where:

• Δ₁ = f₁ – f₀ (difference between modal class and preceding class)
• Δ₂ = f₁ – f₂ (difference between modal class and succeeding class)

3. Empirical Relationship Formulas

Karl Pearson’s Formula

Mode-Median-Mean Relationship

4. Mode Formulas for Special Cases

Continuous Distribution Mode

Step-by-Step Calculation Guide

For Ungrouped Data:

1. Count frequency of each value
2. Identify the value with highest frequency
3. That value is the mode

For Grouped Data:

1. Identify modal class (class with highest frequency)
2. Note the values: L, f₁, f₀, f₂, h
3. Apply formula:Mode = L+ f₁−f₀/2f₁−f₀−f₂ × h
4. Calculate the result

Worked Examples

Example 1: Ungrouped Data

Dataset: 12, 15, 18, 15, 20, 15, 22, 25, 15

Solution:

• Frequency count: 12(1), 15(4), 18(1), 20(1), 22(1), 25(1)
• Mode = 15 (highest frequency = 4)

Example 2: Grouped Data

Solution:

• Modal class=30-40 (highest frequency = 18)
• L = 30, f₁ = 18, f₀ = 12, f₂ = 15, h = 10
• Mode = 30 + 18−12/2(18)−12−15 × 10
• Mode = 30+ 6/36−27 × 10 = 30 + 6/9 × 10
• Mode = 30 + 6.67 = 36.67

Important Points to Remember

Main Characteristics:

• Mode is not affected by extreme values (unlike mean)
• A dataset can have multiple modes
• Mode can be used for all types of data (numerical and categorical)
• For open-ended classes, mode can still be calculated if modal class is not open-ended

Common Mistakes to Avoid:

1. Confusing modal class with mode – Modal class is an interval, mode is a specific value
2. Wrong identification of f₀ and f₂ – Always check frequencies of adjacent classes
3. Incorrect class width calculation – Ensure h is calculated properly
4. Forgetting to add L – The final answer must include the lower boundary

When to Use Mode:

• Qualitative data analysis
• Finding most common value
• Skewed distributions (mode is less affected by skewness)
• Business applications (most popular product, size, etc.)

Formula Summary

This comprehensive guide covers all essential mode formulas that students encounter from elementary to advanced statistics courses, ensuring complete understanding and practical application capability.

FAQs on Mode Formula

Q: What is the Mode Formula for Grouped Data in Statistics?

The mode formula for grouped data is:

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

Where:

• L = Lower boundary of the modal class (class with highest frequency)
• 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 (size of the class interval)

Step-by-step calculation:

1. Identify the modal class (highest frequency)
2. Find L (lower limit), f₁, f₀, f₂, and h
3. Substitute values into the formula
4. Calculate the final result

Example: If modal class is 30-40 with frequency 18, previous class frequency is 12, next class frequency is 15, and class width is 10:

Mode = 30 + [(18-12)/(2×18-12-15)] × 10 = 30 + (6/9) × 10 = 36.67

Q: How Do You Find Mode for Ungrouped Data?

For ungrouped data, finding the mode is straightforward:

Method:

1. List all values in the dataset
2. Count the frequency of each value
3. Identify the value that appears most frequently
4. That value is the mode

Example 1 (Single Mode): Dataset: 5, 7, 8, 7, 10, 7, 12

• Value 7 appears 3 times (highest frequency)
• Mode = 7

Example 2 (Bimodal): Dataset: 3, 5, 5, 6, 8, 8, 9

• Values 5 and 8 both appear twice
• Modes = 5 and 8 (bimodal dataset)

Example 3 (No Mode): Dataset: 2, 4, 6, 8, 10

• All values appear once
• No mode exists

Q: What is the Difference Between Mode, Mean, and Median Formulas?

All three are measures of central tendency but calculated differently:

Differences:

1. Calculation Method:

• Mean: Arithmetic calculation (addition and division)
• Median: Positional value (requires sorting)
• Mode: Frequency count (observation)

2. Effect of Outliers:

• Mean: Heavily affected by outliers
• Median: Minimally affected
• Mode: Not affected at all

3. Uniqueness:

• Mean: Always unique
• Median: Always unique (or average of two middle values)
• Mode: Can have multiple values or no mode

Example Dataset: 2, 3, 3, 4, 5, 5, 5, 6, 100

• Mean = 137/9 = 15.22 (affected by outlier 100)
• Median = 5 (middle value)
• Mode = 5 (appears 3 times)

Relationship in Symmetric Distribution: Mode = Median = Mean

Q: What is the Mode Formula for Class 10 CBSE?

For CBSE Class 10 Mathematics, the mode formula taught is:

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

Class 10 Exam Format: Students must:

1. Identify the modal class from frequency distribution
2. Extract all required values
3. Apply the formula correctly
4. Show complete working with proper steps

Important CBSE Marking Guidelines:

• Writing formula correctly: 1 mark
• Identifying modal class: 1 mark
• Substitution: 1 mark
• Final calculation: 1 mark

Class 10 Standard Question Format:

Given frequency distribution:

Solution Steps:

1. Modal class=20-30 (frequency = 36)
2. L = 20, f₁ = 36, f₀ = 16, f₂ = 34, h = 10
3. Mode = 20 + [(36-16)/(72-16-34)] × 10
4. Mode = 20 + (20/22) × 10 = 20 + 9.09 = 29.09

Additional Formulas for Class 10:

• Empirical formula: Mode = 3 Median – 2 Mean (for estimation)
• Used when direct calculation is complex

Q: Can a Dataset Have More Than One Mode?

A dataset can have multiple modes or no mode at all.

Types of Modal Distributions:

1. Unimodal (One Mode):

• Dataset: 2, 3, 4, 4, 4, 5, 6
• Mode = 4 (appears 3 times)

2. Bimodal (Two Modes):

• Dataset: 1, 2, 2, 2, 3, 4, 5, 5, 5, 6
• Modes = 2 and 5 (both appear 3 times)
• Common in datasets with two distinct groups

3. Multimodal (Multiple Modes):

• Dataset: 1, 1, 1, 2, 3, 3, 3, 4, 5, 5, 5
• Modes = 1, 3, and 5 (all appear 3 times)

4. No Mode:

• Dataset: 10, 20, 30, 40, 50
• No mode (all values appear with equal frequency)

For Grouped Data:

• Only one modal class exists (class with highest frequency)
• But the calculated mode value is unique

Real-World Examples:

Bimodal Distribution:

• Heights in a mixed gender group (two peaks: male and female averages)
• Test scores with two distinct student groups (high performers and low performers)
• Age distribution in areas with both young families and retirees

Practical Significance:

• Multiple modes indicate distinct subgroups in data
• Important for market research, demographics, and quality control
• Helps identify diverse patterns that mean and median might miss

Statistical Reporting: When reporting multiple modes, list all of them:

• “The data is bimodal with modes at 25 and 45”
• Never report just one mode when multiple exist

Q: How to Calculate Mode Using Karl Pearson’s Empirical Formula?

Karl Pearson’s Empirical Formula provides an approximate mode when:

• Direct calculation is difficult
• Data is moderately skewed
• Quick estimation is needed

The Formula:

Mode = 3 × Median – 2 × Mean

Alternative Form:Mode = Mean – 3(Mean – Median)

When to Use This Method:

• For moderately skewed distributions
• When you already know mean and median
• For quick approximations
• When grouped data calculation is complex

Step-by-Step Calculation:

Example: Consider a dataset where:

• Mean = 45
• Median = 42

Solution: Mode = 3 × Median – 2 × Mean Mode = 3 × 42 – 2 × 45 Mode = 126 – 90 Mode = 36

Verification of Formula Accuracy:

The empirical formula works best when:

• Distribution is moderately skewed (not heavily skewed)
• Data follows an approximately normal pattern
• Skewness is consistent

Relationship in Different Distributions:

Practical Example with Complete Calculation:

Dataset: 12, 15, 18, 20, 22, 25, 25, 28, 30, 55

Step 1: Calculate Mean Mean = (12+15+18+20+22+25+25+28+30+55)/10 = 250/10 = 25

Step 2: Calculate Median Arrange in order (already arranged) Median = (22+25)/2 = 23.5

Step 3: Apply Empirical Formula Mode = 3(23.5) – 2(25) Mode = 70.5 – 50 Mode = 20.5

Verification: The actual mode is 25 (appears twice), and our empirical estimate of 20.5 is reasonably close, confirming the formula’s utility for approximation.

Important Note:

• This is an approximation method, not exact
• For precise values, use the standard mode formula
• Most accurate for bell-shaped distributions