To find the mode of grouped data, identify the modal class and use the formula for the mode, which involves the frequencies of the modal class and its adjacent classes.
The given data is:
Step 1:
Identify the modal class, which is the class interval with the highest frequency. From the given data, the highest frequency is 20, which corresponds to the class interval \( 40-50 \).
Step 2:
Let the modal class be \( 40-50 \), where:
- \( l = 40 \) (lower boundary of the modal class),
- \( f_1 = 20 \) (frequency of the modal class),
- \( f_0 = 13 \) (frequency of the class preceding the modal class, i.e., \( 30-40 \)),
- \( f_2 = 11 \) (frequency of the class succeeding the modal class, i.e., \( 50-60 \)),
- \( h = 10 \) (class width).
Step 3:
Now, apply the formula for the mode:
\[
\text{Mode} = l + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h
\]
Substitute the values into the formula:
\[
\text{Mode} = 40 + \frac{20 - 13}{2 \times 20 - 13 - 11} \times 10
\]
Simplify:
\[
\text{Mode} = 40 + \frac{7}{40 - 24} \times 10 = 40 + \frac{7}{16} \times 10 = 40 + 4.375 = 44.375.
\]
Conclusion:
The mode of the given data is \( 44.375 \).
