Question

Find the modal class and mode of the following data: 

Hint:

To find the mode for grouped data, use the modal class, and apply the mode formula to account for the frequencies of adjacent classes.

Answer 1

Step 1: Identify the modal class.
The modal class is the class with the highest frequency. From the table, we observe that the highest frequency is 18, which corresponds to the class \( 40 - 60 \). Therefore, the modal class is \( 40 - 60 \).
Step 2: Apply the formula for mode.
The formula for calculating the mode in a grouped frequency distribution is: \[ \text{Mode} = L + \frac{(f_1 - f_0)}{(2f_1 - f_0 - f_2)} \times h \] where:
- \( L \) is the lower boundary of the modal class,
- \( f_1 \) is the frequency of the modal class,
- \( f_0 \) is the frequency of the class before the modal class,
- \( f_2 \) is the frequency of the class after the modal class,
- \( h \) is the class width.
For the modal class \( 40 - 60 \), we have:
- \( L = 40 \),
- \( f_1 = 18 \),
- \( f_0 = 10 \),
- \( f_2 = 14 \),
- \( h = 20 \).
Substitute the values into the formula: \[ \text{Mode} = 40 + \frac{(18 - 10)}{(2 \times 18 - 10 - 14)} \times 20 \] \[ \text{Mode} = 40 + \frac{8}{(36 - 10 - 14)} \times 20 \] \[ \text{Mode} = 40 + \frac{8}{12} \times 20 \] \[ \text{Mode} = 40 + \frac{160}{12} = 40 + 13.33 = 53.33. \]
Step 3: Conclusion.
The modal class is \( 40 - 60 \), and the mode is approximately 53.33.