To find the mean of the given frequency distribution, we can follow these steps:
1. List the Scores and Their Corresponding Frequencies:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|c|}
\hline
\text{Score}, x & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline
\text{Frequency}, f & 1 & 2 & 3 & 5 & 6 & 4 & 6 & 3 \\
\hline
\end{array}
\][/tex]
2. Find the Total Number of Data Items:
The total number of data items is calculated by summing the frequencies.
[tex]\[
\text{Total data items} = 1 + 2 + 3 + 5 + 6 + 4 + 6 + 3 = 30
\][/tex]
3. Calculate the Sum of (Score * Frequency):
We will multiply each score by its corresponding frequency and then sum these products.
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Score}, x & \text{Frequency}, f & x \times f \\
\hline
1 & 1 & 1 \times 1 = 1 \\
\hline
2 & 2 & 2 \times 2 = 4 \\
\hline
3 & 3 & 3 \times 3 = 9 \\
\hline
4 & 5 & 4 \times 5 = 20 \\
\hline
5 & 6 & 5 \times 6 = 30 \\
\hline
6 & 4 & 6 \times 4 = 24 \\
\hline
7 & 6 & 7 \times 6 = 42 \\
\hline
8 & 3 & 8 \times 3 = 24 \\
\hline
\end{array}
\][/tex]
[tex]\[
\text{Sum of } (x \times f) = 1 + 4 + 9 + 20 + 30 + 24 + 42 + 24 = 154
\][/tex]
4. Calculate the Mean:
The mean is given by the formula:
[tex]\[
\text{Mean} = \frac{\sum (x \times f)}{\sum f}
\][/tex]
Plugging in the values we have:
[tex]\[
\text{Mean} = \frac{154}{30} \approx 5.133
\][/tex]
Thus, the mean, rounded to three decimal places, is:
[tex]\[
\boxed{5.133}
\][/tex]
