To calculate the mode for the given grouped data, we use the mode formula for grouped data:
[tex]\[
\text{Mode} = L + \left( \frac{f_1 - f_0}{(2f_1 - f_0 - f_2)} \right) \times h
\][/tex]
where:
- [tex]\( L \)[/tex] is the lower boundary of the modal class
- [tex]\( f_1 \)[/tex] is the frequency of the modal class
- [tex]\( f_0 \)[/tex] is the frequency of the class before the modal class
- [tex]\( f_2 \)[/tex] is the frequency of the class after the modal class
- [tex]\( h \)[/tex] is the class width
Given the class intervals and their frequencies:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Class Interval} & \text{Frequency} \\
\hline
10-20 & 4 \\
20-30 & 6 \\
30-40 & 5 \\
40-50 & 10 \\
50-60 & 20 \\
60-70 & 22 \\
70-80 & 24 \\
80-90 & 6 \\
90-100 & 2 \\
100-110 & 1 \\
\hline
\end{array}
\][/tex]
We identify the modal class as the class interval with the highest frequency. Here, the highest frequency is [tex]\( 24 \)[/tex], which corresponds to the class interval [tex]\( 70-80 \)[/tex].
Thus:
- [tex]\( L \)[/tex] (the lower boundary of the modal class) = 70
- [tex]\( f_1 \)[/tex] (the frequency of the modal class) = 24
- [tex]\( f_0 \)[/tex] (the frequency of the class before the modal class) = 22 (for the class [tex]\( 60-70 \)[/tex])
- [tex]\( f_2 \)[/tex] (the frequency of the class after the modal class) = 6 (for the class [tex]\( 80-90 \)[/tex])
- [tex]\( h \)[/tex] (the class width) = 10 (since the width of each class interval is uniform)
Using the mode formula:
[tex]\[
\text{Mode} = 70 + \left( \frac{24 - 22}{(2 \times 24 - 22 - 6)} \right) \times 10
\][/tex]
First, calculate the numerator:
[tex]\[
24 - 22 = 2
\][/tex]
Next, calculate the denominator:
[tex]\[
(2 \times 24 - 22 - 6) = 48 - 22 - 6 = 20
\][/tex]
Now, compute the value inside the parentheses:
[tex]\[
\frac{2}{20} = 0.1
\][/tex]
Finally, we calculate the mode:
[tex]\[
\text{Mode} = 70 + (0.1 \times 10) = 70 + 1 = 71
\][/tex]
Therefore, the mode of the given frequency distribution is [tex]\( 71 \)[/tex].
