To find the mean of the given frequency distribution, we follow these steps:
1. List the Scores and Frequencies:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|c|}
\hline
\text{Score, } x & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline
\text{Frequency, } f & 3 & 3 & 6 & 6 & 6 & 2 & 6 & 3 \\
\hline
\end{array}
\][/tex]
2. Calculate the Total Frequency:
To determine the total frequency, sum up all the frequencies:
[tex]\[
\text{Total Frequency} = 3 + 3 + 6 + 6 + 6 + 2 + 6 + 3 = 35
\][/tex]
3. Calculate the Weighted Sum of Scores:
Each score needs to be multiplied by its corresponding frequency, and then all these products should be summed:
[tex]\[
\text{Weighted Sum} = (1 \times 3) + (2 \times 3) + (3 \times 6) + (4 \times 6) + (5 \times 6) + (6 \times 2) + (7 \times 6) + (8 \times 3)
\][/tex]
Calculating each individually:
[tex]\[
\begin{align*}
1 \times 3 & = 3 \\
2 \times 3 & = 6 \\
3 \times 6 & = 18 \\
4 \times 6 & = 24 \\
5 \times 6 & = 30 \\
6 \times 2 & = 12 \\
7 \times 6 & = 42 \\
8 \times 3 & = 24 \\
\end{align*}
\][/tex]
Summing these products:
[tex]\[
\text{Weighted Sum} = 3 + 6 + 18 + 24 + 30 + 12 + 42 + 24 = 159
\][/tex]
4. Calculate the Mean:
The mean ([tex]\(\bar{x}\)[/tex]) can be calculated by dividing the weighted sum by the total frequency:
[tex]\[
\bar{x} = \frac{\text{Weighted Sum}}{\text{Total Frequency}} = \frac{159}{35} \approx 4.543
\][/tex]
Therefore, the mean of the given frequency distribution is:
[tex]\[
\boxed{4.543}
\][/tex]
