To calculate the mode from the given data, we follow the steps below:
1. Identify the modal class: The modal class is the class interval with the highest frequency.
- Given frequencies are: [tex]\(3, 9, 10, 14, 15, 21, 25, 13, 15, 10, 8, 6\)[/tex].
- The highest frequency is 25, which corresponds to the class interval [tex]\(18 - 20\)[/tex].
2. Extract the necessary values:
- [tex]\(l\)[/tex]: Lower limit of the modal class, which is 18.
- [tex]\(h\)[/tex]: Class width, which is the difference between the upper and lower boundaries of the modal class. For [tex]\(18-20\)[/tex], [tex]\(h = 20 - 18 = 2\)[/tex].
- [tex]\(f_1\)[/tex]: Frequency of the modal class, which is 25.
- [tex]\(f_0\)[/tex]: Frequency of the class interval before the modal class, which is [tex]\(21\)[/tex] (For the [tex]\(15 - 18\)[/tex] class interval).
- [tex]\(f_2\)[/tex]: Frequency of the class interval after the modal class, which is [tex]\(13\)[/tex] (For the [tex]\(20 - 24\)[/tex] class interval).
3. Apply the mode formula for continuous data:
[tex]\[
\text{Mode} = l + \left( \frac{f_1 - f_0}{(f_1 - f_0) + (f_1 - f_2)} \right) \times h
\][/tex]
4. Plug in the values into the formula:
[tex]\[
\text{Mode} = 18 + \left( \frac{25 - 21}{(25 - 21) + (25 - 13)} \right) \times 2
\][/tex]
5. Simplify the expression:
[tex]\[
\text{Mode} = 18 + \left( \frac{4}{4 + 12} \right) \times 2
\][/tex]
[tex]\[
\text{Mode} = 18 + \left( \frac{4}{16} \right) \times 2
\][/tex]
[tex]\[
\text{Mode} = 18 + 0.25 \times 2
\][/tex]
[tex]\[
\text{Mode} = 18 + 0.5
\][/tex]
[tex]\[
\text{Mode} = 18.5
\][/tex]
Therefore, the mode of the given data is [tex]\( \mathbf{18.5} \)[/tex].
