To find the mean for the data items in the given frequency distribution, follow these steps:
1. Identify the Scores and Frequencies:
- Scores ([tex]$x$[/tex]): [tex]\(1, 2, 3, 4, 5, 6, 7, 8, 9, 10\)[/tex]
- Frequencies ([tex]$f$[/tex]): [tex]\(2, 2, 1, 5, 4, 8, 6, 5, 4, 3\)[/tex]
2. Calculate the Total Number of Data Items:
- The total number of data items is the sum of the frequencies.
[tex]\[
\text{Total number of data items} = 2 + 2 + 1 + 5 + 4 + 8 + 6 + 5 + 4 + 3 = 40
\][/tex]
3. Calculate the Weighted Sum of Scores:
- Multiply each score by its corresponding frequency and sum all the products.
[tex]\[
\begin{align*}
\text{Weighted Sum} & = (1 \cdot 2) + (2 \cdot 2) + (3 \cdot 1) + (4 \cdot 5) + (5 \cdot 4) \\
&\quad + (6 \cdot 8) + (7 \cdot 6) + (8 \cdot 5) + (9 \cdot 4) + (10 \cdot 3) \\
& = 2 + 4 + 3 + 20 + 20 + 48 + 42 + 40 + 36 + 30 \\
& = 245
\end{align*}
\][/tex]
4. Calculate the Mean:
- The mean is the weighted sum of scores divided by the total number of data items.
[tex]\[
\text{Mean} = \frac{\text{Weighted Sum}}{\text{Total number of data items}} = \frac{245}{40} = 6.125
\][/tex]
5. Round the Mean to 3 Decimal Places:
- In this case, the mean is already accurate to three decimal places, so it remains 6.125.
Thus, the mean for the data items in the given frequency distribution is [tex]\( \boxed{6.125} \)[/tex].
