To calculate the Mode of the given grouped frequency distribution, we use the mode formula:

$\text{Mode} = L + \left( \frac{f_m - f_{1}}{2f_m - f_{1} - f_{2}} \right) \times h$

Where:

• $L$: Lower boundary of the modal class
• $f_m$: Frequency of the modal class
• $f_1$: Frequency of the class before the modal class
• $f_2$: Frequency of the class after the modal class
• $h$: Width of the class intervals

Step-by-step Solution:

1. Identify the modal class: The modal class is the class with the highest frequency. From the table:

   • Frequencies: $4, 10, 16, 8, 2$
   • The highest frequency is $16$, so the modal class is $30-40$.

2. Extract values:

   • $L = 30$ (lower boundary of the modal class)
   • $f_m = 16$ (frequency of the modal class)
   • $f_1 = 10$ (frequency of the class before the modal class: $20-30$)
   • $f_2 = 8$ (frequency of the class after the modal class: $40-50$)
   • $h = 10$ (width of the class intervals)

3. Substitute into the formula: $\text{Mode} = 30 + \left( \frac{16 - 10}{2(16) - 10 - 8} \right) \times 10$

   Simplify step-by-step:

   • $f_m - f_1 = 16 - 10 = 6$
   • $2f_m - f_1 - f_2 = 2(16) - 10 - 8 = 32 - 18 = 14$
   • Fraction: $\frac{6}{14} = 0.4286$ (approx)

   Substituting back: $\text{Mode} = 30 + (0.4286 \times 10)$ $\text{Mode} = 30 + 4.286 = 34.29$

Final Answer:

The mode is approximately 34.29.

Do you want a detailed explanation of any step? Let me know!

Here are 5 related questions you might consider:

1. How is the mode different from the mean and median in grouped data?
2. Can a dataset have more than one modal class? If yes, how is it handled?
3. What is the significance of class width ($h$) in finding the mode?
4. How would the result change if the highest frequency occurred in a different class?
5. How can we calculate the mode if the data is ungrouped?

Tip: Always check the class boundaries and widths carefully when solving grouped data problems to avoid small errors in substitution!