To calculate the mode for grouped data, we follow 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
• $f_1$ = frequency of the modal class
• $f_0$ = frequency of the class preceding the modal class
• $f_2$ = frequency of the class following the modal class
• $h$ = class width (assuming uniform class intervals)

Step 1: Identify the Modal Class

The modal class is the class with the highest frequency.

From the table:

• Class: $5 - 10, 10 - 15, 15 - 20, 20 - 25, 25 - 30, 30 - 35, 35 - 40$
• Frequencies: $4, 5, 3, 2, 6, 7, 3$

The highest frequency is 7, so the modal class is $30 - 35$.

Step 2: Extract Values for the Formula

• $L$ = 30 (lower boundary of the modal class)
• $f_1$ = 7 (frequency of the modal class)
• $f_0$ = 6 (frequency of the previous class $25 - 30$)
• $f_2$ = 3 (frequency of the next class $35 - 40$)
• $h$ = 5 (class width)

Step 3: Apply the Formula

$\text{Mode} = 30 + \left( \frac{7 - 6}{2(7) - 6 - 3} \right) \times 5$

Simplify the terms:

$\text{Mode} = 30 + \left( \frac{1}{14 - 9} \right) \times 5 = 30 + \left( \frac{1}{5} \right) \times 5$

$\text{Mode} = 30 + 1 = 31$

Final Answer

The mode of the given data is 31.

Would you like more details or have further questions?

Related Questions:

1. What is the difference between mode, mean, and median in grouped data?
2. How do you find the class width if intervals are not uniform?
3. Can a dataset have more than one modal class?
4. How do you calculate mode if the frequency distribution is ungrouped?
5. What if two or more classes have the same highest frequency? How is mode determined?

Tip: Mode is most useful for categorical or discrete datasets, providing the most frequent occurrence.