To calculate the mode of the given frequency distribution using the grouping method, I'll guide you through the process step by step. Here's the frequency distribution provided:

Class Interval	Frequency (f)
0 - 5	1
5 - 10	2
10 - 15	10
15 - 20	6
20 - 25	9
25 - 30	2

Steps to Calculate the Mode Using the Grouping Method:

1. Identify the Modal Class: The modal class is the class interval with the highest frequency. Here, the class interval with the highest frequency is 10-15, with a frequency of 10.

2. Create a Grouping Table:

   • We create a table to analyze the frequency distribution further, breaking it into different columns to help with mode calculation.

Column 1	Column 2 (1st Differences)	Column 3 (2nd Differences)
1
2	1
10	8	7
6	-4	-12
9	3	7
2	-7

• Column 1 contains the frequencies.
• Column 2 represents the first differences of the frequencies.
• Column 3 represents the second differences.

3. Determine the Mode:

   • The mode is located within the modal class. To calculate it precisely, we use the formula:

   $\text{Mode} = L + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h$

   Where:

   • $L$ = lower boundary of the modal class = 10 (since the modal class is 10-15)
   • $f_1$ = frequency of the modal class = 10
   • $f_0$ = frequency of the class preceding the modal class = 2 (class 5-10)
   • $f_2$ = frequency of the class succeeding the modal class = 6 (class 15-20)
   • $h$ = class width = 5

   Plugging in the values:

   $\text{Mode} = 10 + \left(\frac{10 - 2}{2(10) - 2 - 6}\right) \times 5$ $\text{Mode} = 10 + \left(\frac{8}{20 - 8}\right) \times 5$ $\text{Mode} = 10 + \left(\frac{8}{12}\right) \times 5$ $\text{Mode} = 10 + \left(\frac{2}{3}\right) \times 5$ $\text{Mode} = 10 + 3.33 \approx 13.33$

Thus, the mode of the distribution is approximately 13.33.

Would you like further details on any of the steps, or do you have any questions?

Here are some related questions to explore further:

1. How is the class interval determined in a frequency distribution?
2. What are the different methods to calculate the mode?
3. Why is the mode important in statistics, and in what scenarios is it most useful?
4. Can a dataset have more than one mode, and how is it calculated in such cases?
5. How does the grouping method differ from other methods for finding the mode?

Tip: When calculating the mode, it's crucial to accurately identify the modal class, as it directly influences the result.