What is Grouping Method for Mode?

S3-SA3-0303

Grade Level:

Class 8

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

The Grouping Method for Mode is a special technique used to find the mode when the data is grouped into classes (like 0-10, 10-20) and it's not clear which class has the highest frequency. It helps us identify the 'modal class' more accurately by considering frequencies of neighbouring classes.

Simple Example

Quick Example

Imagine a shop selling different sizes of T-shirts. If 30 people bought size M, 32 bought size L, and 30 bought size XL, it's hard to say if M or L or XL is the most popular just by looking at the highest number. The Grouping Method helps us decide by looking at the sizes around them too.

Worked Example

Step-by-Step

Let's find the modal class for the following data:

Class Interval | Frequency
----------------|-----------
0-10 | 5
10-20 | 8
20-30 | 12
30-40 | 10
40-50 | 6

---
Step 1: Create a Grouping Table. This table will have 6 columns for frequencies.

Column 1: Original Frequencies (f)
Column 2: Sum of frequencies in twos (f1 + f2, f3 + f4, ...)
Column 3: Sum of frequencies in twos, starting from the second frequency (f2 + f3, f4 + f5, ...)
Column 4: Sum of frequencies in threes (f1 + f2 + f3, f4 + f5 + f6, ...)
Column 5: Sum of frequencies in threes, starting from the second frequency (f2 + f3 + f4, f5 + f6 + f7, ...)
Column 6: Sum of frequencies in threes, starting from the third frequency (f3 + f4 + f5, f6 + f7 + f8, ...)

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Step 2: Fill the Grouping Table:

Class Interval | f | Col 2 (f1+f2) | Col 3 (f2+f3) | Col 4 (f1+f2+f3) | Col 5 (f2+f3+f4) | Col 6 (f3+f4+f5)
---------------|---|---------------|---------------|------------------|------------------|------------------
0-10 | 5 | | | | |
10-20 | 8 | 13 (5+8) | | | |
20-30 | 12| 22 (12+10) | 20 (8+12) | 25 (5+8+12) | |
30-40 | 10| 16 (6) | 16 (12+10) | 28 (8+12+10) | 28 (12+10+6) |
40-50 | 6 | | 16 (10+6) | 28 (12+10+6) | 28 (12+10+6) | 28 (12+10+6)

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Step 3: Identify the highest frequency in each column and mark the corresponding class interval(s).

Col 1 (f): Max is 12 for 20-30
Col 2 (f1+f2): Max is 22 for 20-30 and 30-40 (sum of 12+10)
Col 3 (f2+f3): Max is 20 for 10-20 and 20-30 (sum of 8+12), and 16 for 20-30 and 30-40 (sum of 12+10) and 30-40 and 40-50 (sum of 10+6). In case of ties, consider all contributing classes.
Col 4 (f1+f2+f3): Max is 28 for 10-20, 20-30, 30-40 (sum of 8+12+10)
Col 5 (f2+f3+f4): Max is 28 for 20-30, 30-40, 40-50 (sum of 12+10+6)
Col 6 (f3+f4+f5): Max is 28 for 20-30, 30-40, 40-50 (sum of 12+10+6)

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Step 4: Create an Analysis Table. List the class intervals and mark a 'tick' for each time they appear as contributing to the highest frequency in any column.

Class Interval | Col 1 | Col 2 | Col 3 | Col 4 | Col 5 | Col 6 | Total Ticks
---------------|-------|-------|-------|-------|-------|-------|------------
0-10 | | | | | | |
10-20 | | | ✓ | ✓ | | | 2
20-30 | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | 6
30-40 | | ✓ | ✓ | ✓ | ✓ | ✓ | 5
40-50 | | | ✓ | | ✓ | ✓ | 3

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Step 5: The class interval with the highest total number of ticks is the modal class.

Here, 20-30 has 6 ticks, which is the highest.

Answer: The modal class is 20-30.

Why It Matters

This method is super useful in data analysis, helping us understand trends where data isn't straightforward. For example, in AI/ML, it can help identify the most common 'features' in a dataset. In economics, it helps find the most popular income group or price range, which is crucial for government policies or business strategies.

Common Mistakes

MISTAKE: Not creating all 6 frequency columns in the grouping table. | CORRECTION: Always make sure your grouping table has 6 columns for frequencies to cover all possible groupings of two and three frequencies.

MISTAKE: Incorrectly identifying the highest frequency in a column, especially when sums are tied. | CORRECTION: Carefully check each column and highlight the maximum sum. If sums are tied, mark all corresponding class intervals in the analysis table.

MISTAKE: Forgetting to mark ticks for ALL contributing classes in the analysis table. | CORRECTION: When a sum (e.g., f1+f2+f3) is the highest, put a tick against f1, f2, and f3's class intervals in the analysis table.

Practice Questions

Try It Yourself

QUESTION: A survey asked students about their daily screen time (in hours). Identify the modal class using the grouping method:

Screen Time (hours) | Number of Students
--------------------|--------------------
0-2 | 3
2-4 | 7
4-6 | 9
6-8 | 5
8-10 | 2

| ANSWER: The modal class is 4-6 hours.

QUESTION: Find the modal class for the following distribution of marks:

Marks | Number of Students
------|--------------------
0-10 | 6
10-20 | 10
20-30 | 15
30-40 | 12
40-50 | 8
50-60 | 4

| ANSWER: The modal class is 20-30.

QUESTION: The ages of visitors to a cultural festival are given below. Determine the modal class using the grouping method.

Age Group | Number of Visitors
----------|--------------------
5-15 | 12
15-25 | 18
25-35 | 25
35-45 | 20
45-55 | 15
55-65 | 10

| ANSWER: The modal class is 25-35.

MCQ

Quick Quiz

Why is the Grouping Method used for finding the mode in grouped data?

To calculate the mean of the data

To find the highest frequency directly without any extra steps

To accurately identify the modal class when the highest frequency isn't clearly dominant

To determine the range of the data

The Correct Answer Is:

The Grouping Method helps when simply looking for the highest frequency isn't enough to find the mode. It considers neighboring frequencies to give a more accurate modal class. Options A, B, and D are incorrect as they describe other statistical measures or a simplified view of mode.

Real World Connection

In the Real World

Imagine a company like Zomato or Swiggy analyzing delivery times. They might group delivery times (e.g., 0-15 mins, 15-30 mins, etc.). If the highest number of deliveries fall into 15-30 mins, but many also fall into 0-15 mins and 30-45 mins, they might use the Grouping Method to confirm the 'most common' delivery time range to optimize their logistics or offer better service guarantees.

Key Vocabulary

Key Terms

Mode: The value that appears most frequently in a data set. | Grouped Data: Data organized into classes or intervals. | Class Interval: A range of values in which data is grouped. | Frequency: The number of times a particular value or class interval occurs. | Modal Class: The class interval that contains the mode.

What's Next

What to Learn Next

Great job learning about the Grouping Method for Mode! Next, you should explore how to calculate the actual Mode within the identified modal class using the 'Modal Formula'. This will help you find a single, precise value for the mode, building on what you've learned here.