To find the mean of the data items given in the frequency distribution, we will follow these steps:
1. List the scores ([tex]\(x\)[/tex]) and their corresponding frequencies ([tex]\(f\)[/tex]):
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|c|}
\hline
\text{Score } x & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline
\text{Frequency } f & 2 & 2 & 4 & 4 & 6 & 3 & 4 & 2 \\
\hline
\end{array}
\][/tex]
2. Calculate the product of each score and its corresponding frequency ([tex]\(x \times f\)[/tex]):
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|c|}
\hline
\text{Score } x & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline
\text{Frequency } f & 2 & 2 & 4 & 4 & 6 & 3 & 4 & 2 \\
\hline
\text{Product } (x \times f) & 2 & 4 & 12 & 16 & 30 & 18 & 28 & 16 \\
\hline
\end{array}
\][/tex]
3. Sum up all the products ([tex]\(\sum x \times f\)[/tex]):
[tex]\[
2 + 4 + 12 + 16 + 30 + 18 + 28 + 16 = 126
\][/tex]
4. Sum up all the frequencies ([tex]\(\sum f\)[/tex]):
[tex]\[
2 + 2 + 4 + 4 + 6 + 3 + 4 + 2 = 27
\][/tex]
5. Calculate the mean ([tex]\(\bar{x}\)[/tex]), which is the total sum of the products divided by the total sum of the frequencies:
[tex]\[
\bar{x} = \frac{\sum (x \times f)}{\sum f} = \frac{126}{27} \approx 4.667
\][/tex]
So, the mean of the data items in the given frequency distribution, rounded to three decimal places, is:
[tex]\[
\boxed{4.667}
\][/tex]
