We are given the following frequency table:
- Class intervals: \( 1 - 3, 3 - 5, 5 - 7, 7 - 9, 9 - 11 \)
- Frequencies: \( 7, 8, 2, 2, 1 \)
Step 1: Identify the modal class.
The modal class is the class interval with the highest frequency. From the table, the highest frequency is 8, which corresponds to the class interval \( 3 - 5 \). Therefore, the modal class is \( 3 - 5 \).
Step 2: Apply the mode formula.
The formula for the mode is:
\[
\text{Mode} = L + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h
\]
where:
- \( L \) = lower limit of the modal class = 3
- \( f_1 \) = frequency of the modal class = 8
- \( f_0 \) = frequency of the class preceding the modal class = 7
- \( f_2 \) = frequency of the class succeeding the modal class = 2
- \( h \) = class width = 2 (since the class interval width is 2 for all intervals)
Substituting the values into the formula:
\[
\text{Mode} = 3 + \frac{8 - 7}{2 \times 8 - 7 - 2} \times 2
\]
Step 3: Simplify the expression.
First, simplify the numerator and denominator:
\[
\text{Mode} = 3 + \frac{1}{16 - 7} \times 2 = 3 + \frac{1}{9} \times 2
\]
Next, simplify further:
\[
\text{Mode} = 3 + \frac{2}{9}
\]
Step 4: Final Calculation.
\[
\text{Mode} = 3 + 0.22 = 3.22
\]
Conclusion:
The mode of the given data is \( \boxed{3.22} \).
