To find the mode of a frequency distribution, we use the following formula:
\[
\text{Mode} = L + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h
\]
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 preceding the modal class.
- \( f_2 \) is the frequency of the class succeeding the modal class.
- \( h \) is the class width.
Step 1: Identify the modal class.
The highest frequency is 23, which corresponds to the class interval \( 30-40 \). Hence, the modal class is \( 30-40 \).
Step 2: Apply the formula.
- \( L = 30 \) (the lower boundary of the modal class)
- \( f_1 = 23 \) (the frequency of the modal class)
- \( f_0 = 15 \) (the frequency of the class preceding the modal class)
- \( f_2 = 7 \) (the frequency of the class succeeding the modal class)
- \( h = 10 \) (the class width)
Substituting these values into the formula:
\[
\text{Mode} = 30 + \left( \frac{23 - 15}{2(23) - 15 - 7} \right) \times 10
\text{Mode} = 30 + \left( \frac{8}{46 - 22} \right) \times 10
\text{Mode} = 30 + \left( \frac{8}{24} \right) \times 10
\text{Mode} = 30 + \left( \frac{1}{3} \right) \times 10 = 30 + \frac{10}{3} = 30 + 3.33 = 33.33.
Conclusion:
The mode of the given data is \( 33.33 \).
