How to Calculate the Mode of Grouped Data Step by Step
The concept of Mode of Grouped Data is essential in mathematics and statistics, especially for board exam preparation and understanding data trends. It helps students analyze and interpret real-world datasets effectively.
Understanding Mode of Grouped Data
Mode of Grouped Data refers to finding the most frequently occurring value—in other words, the value that appears with the highest frequency—in a grouped frequency distribution. Unlike ungrouped data, where the mode can be spotted directly, grouped data requires a formula-based approach. This concept is widely used in statistics, data analysis, and central tendency calculations.
Why is Mode of Grouped Data Important?
Learning about the Mode of Grouped Data is important for several reasons:
- It is a key measure of central tendency (along with mean and median).
- Featured in CBSE class 10 syllabus and competitive exams.
- Helps make sense of large datasets in surveys, reports, and business.
- Enables you to summarize information with one representative value.
- Builds foundational skills for advanced statistics and data science.
Grouped vs Ungrouped Data: What’s the Difference?
Ungrouped data lists individual values, making it easy to spot the mode directly. Grouped data organizes values into class intervals with corresponding frequencies. In grouped data, you must use a specific formula to accurately estimate the mode.
| Type | How to Find Mode | Example |
|---|---|---|
| Ungrouped Data | Identify the value with highest frequency. | For 2, 6, 4, 2, 5, 2: Mode = 2 |
| Grouped Data | Find modal class, then apply formula. | Class intervals with frequencies |
Key Definitions in Mode of Grouped Data
- Class Interval: Range of values grouped together (e.g., 0–10, 10–20).
- Frequency: Number of data points in a class interval.
- Modal Class: The class interval with the highest frequency.
- Mode: The value that appears most frequently, estimated for grouped data using a formula.
Mode of Grouped Data Formula
The standard formula to calculate the mode for grouped data is:
Mode = l + [(f1 - f0) / (2f1 - f0 - f2)] × h
Where:
- l = Lower limit of the modal class
- h = Class size (interval width)
- f1 = Frequency of modal class
- f0 = Frequency of class before modal class
- f2 = Frequency of class after modal class
Step-by-Step Calculation of Mode of Grouped Data
Let’s see the entire process in a full example, as you would solve it in class 10 board exams:
1. Prepare the frequency distribution table of the data.2. Identify the modal class—the class interval with the highest frequency (f1).
3. Note down:
4. Plug all these values into the mode formula.
5. Calculate stepwise as follows:
(a) Subtract f0 from f1 and from 2f1.
(b) Subtract f2 from result.
(c) Divide (f1 - f0) by (2f1 - f0 - f2).
(d) Multiply the outcome by h.
(e) Add this to l to get the mode.
6. Final answer: Write it clearly and highlight as your mode.
Mode of Grouped Data Table Example
Here’s a table to illustrate how to find the mode of grouped data:
| Class Interval | Frequency |
|---|---|
| 0 – 2 | 6 |
| 2 – 4 | 7 |
| 4 – 6 (Modal Class) | 8 |
| 6 – 8 | 2 |
| 8 – 10 | 1 |
In this table, 4 – 6 is the modal class as it has the highest frequency.
Worked Example – Solving a Problem
Let’s solve the above example step by step:
1. Modal class = 4 – 6, so:2. Use the mode formula:
3. Calculate numerator and denominator:
4. Final value:
So, the mode of this grouped data is 4.29.
Special Cases: Two Modal Classes and Unequal Intervals
If two or more class intervals share the highest frequency, both are called modal classes and the distribution is bimodal or multimodal. In such cases, mode is usually not calculated unless specified by your exam or teacher. For unequal class intervals, mode calculation is generally not expected at school level; consult your textbook or Data Management page for advanced steps.
Common Mistakes to Avoid
- Choosing the highest frequency but not checking for adjacent intervals.
- Plugging in the wrong frequencies (f0, f1, f2).
- Forgetting to use the lower boundary (l) of the modal class.
- Using incorrect class size (h).
Practice Problems
- Given frequencies for class intervals 0–10, 10–20, ..., which is the modal class and what is the mode?
- If a dataset's highest frequency occurs in two intervals, what type of distribution is it?
- Find the mode for the class intervals: 0–5 (3), 5–10 (5), 10–15 (8), 15–20 (6), 20–25 (2).
- Explain what happens if all class intervals have the same frequency.
Real-World Applications
The concept of Mode of Grouped Data is used in real-world scenarios like finding the most common marks range in school results, analyzing sales data, and understanding survey outcomes. Vedantu often uses practical worksheets and visual explanations to connect students to these applications.
We explored the idea of Mode of Grouped Data, how to calculate it step by step, and its importance in statistics and everyday applications. To master more statistics concepts, keep practicing on Vedantu and try out similar central tendency measures like mean and median.
Quick Revision Table
| Key Point | Details |
|---|---|
| Formula | Mode = l + [(f₁−f₀)/(2f₁−f₀−f₂)] × h |
| Modal Class | Class interval with highest frequency |
| Common Error | Wrong value for l, h, f₀, f₁, or f₂ |
| Used In | Board exams, business, surveys |
Explore Related Concepts
- Mean, Median, Mode – See all central tendency measures together.
- Central Tendency – Theory and in-depth explanation for exams.
- CBSE Class 10 Maths Important Topics – Syllabus mapping for easier study.
- Variance – Learn more about spread of data.
- Statistics – Applications of data analysis and handling.
FAQs on Mode of Grouped Data Explained with Formula & Examples
1. What is the mode of grouped data?
The mode of grouped data is the value that appears most frequently within a grouped frequency distribution. It is determined by identifying the modal class (the class interval with the highest frequency) and applying the mode formula to estimate the exact mode value within that class.
2. How do you find the mode in a grouped frequency distribution?
To find the mode of grouped data, follow these steps:
1. Identify the modal class (the class with the highest frequency).
2. Note the frequencies of the modal class (f₁), the class before it (f₀), and the class after it (f₂).
3. Note the lower limit (l) and class width (h) of the modal class.
4. Use the formula:
Mode = l + [(f₁ - f₀) / (2f₁ - f₀ - f₂)] × h
This formula helps approximate the mode value inside the modal class.
3. What is the formula for mode of grouped data for class 10?
The mode formula for grouped data as per the CBSE Class 10 syllabus is:
Mode = l + [(f₁ - f₀) / (2f₁ - f₀ - f₂)] × h
where:
l = lower limit of the modal class,
f₁ = frequency of the modal class,
f₀ = frequency of the class preceding the modal class,
f₂ = frequency of the class succeeding the modal class,
h = class interval size.
4. How do you handle two modal classes?
When a frequency distribution has two classes with the highest and equal frequencies, it is called bimodal distribution. To handle two modal classes:
• Identify both modal classes.
• Calculate the mode for each modal class using the usual mode formula separately.
• Report both mode values as the dataset's modes.
This situation requires recognizing that the data has multiple modes, which is common in multimodal distributions.
5. What is the difference between mode of grouped and ungrouped data?
The key differences are:
• Ungrouped data mode is the value that appears most frequently in a raw data set.
• Grouped data mode requires identifying the modal class from class intervals and estimating mode using a formula.
• For ungrouped data, mode is exact and directly seen;
for grouped data, mode is an estimate within an interval.
• Grouped data mode is used when data is organized in frequency distributions for larger datasets.
6. Why is choosing the correct modal class important in statistics exams?
Choosing the correct modal class is crucial because:
• The mode calculation depends entirely on the class with the highest frequency.
• An incorrect modal class leads to a wrong mode value.
• It helps demonstrate a clear understanding of data interpretation in frequency distributions.
• Correct identification ensures accuracy in board and competitive exams where precision matters.
7. Why do students often mix up mean, median, and mode when data is grouped?
Students may confuse mean, median, and mode for grouped data because:
• All three are measures of central tendency but calculated differently.
• Different formulas and steps apply for grouped data, causing confusion.
• Misunderstanding concepts like modal class versus class midpoints (for mean and median).
• Lack of clarity on when to apply each measure based on the data and question context.
8. What mistakes happen if class size (h) is not constant?
If the class size (h) is not constant:
• The standard mode formula becomes inaccurate.
• Unequal class intervals require specialized methods or adjustments.
• Students often use the formula assuming equal class width, leading to errors.
• Such mistakes can shift the mode estimate away from the true modal value.
• Proper handling involves calculating weighted class widths or using alternative statistical techniques.
9. Why is mode of grouped data formula rarely tested for unequal intervals?
The mode formula for grouped data assumes classes are of equal width. It is rarely tested for unequal intervals because:
• The formula is designed for uniform class intervals, which simplifies calculations.
• Unequal intervals require complex corrections, making it less suitable for standard exams.
• Most educational boards focus on uniform intervals to keep the syllabus straightforward.
• When unequal intervals occur, alternate methods or data grouping adjustments are preferred.
10. How can using a calculator improve mode accuracy for grouped datasets?
Using a calculator or digital tool for mode calculation helps by:
• Quickly identifying the modal class from large frequency data.
• Accurately performing formula calculations and decimal operations.
• Reducing human errors in arithmetic or transcription.
• Allowing visualization of frequency distributions through graphs.
• Helping students focus on understanding concepts rather than manual computation.
11. What is the mode of a dataset with all unique values?
If all values in a dataset are unique and occur only once, the dataset is said to have no mode because no value repeats more than others. In such cases:
• The mode is undefined or considered to be 'no mode'.
• For grouped data, it means no clear modal class exists.
• Students should then use other central tendency measures, like mean or median.
12. Can the mode be used for both qualitative and quantitative data?
Yes, the mode is the only measure of central tendency applicable to both qualitative (categorical) and quantitative data because:
• It identifies the most frequently occurring category or value.
• For grouped quantitative data, mode helps estimate the most common range.
• This versatility makes mode especially useful in descriptive statistics.
