To determine the mean for the given frequency distribution, follow these steps:
1. List the values and their corresponding frequencies:
[tex]\[
\begin{array}{c|c}
\text{Value} & \text{Frequency} \\
\hline
12 & 2 \\
17 & 12 \\
22 & 20 \\
31 & 13 \\
33 & 10 \\
\end{array}
\][/tex]
2. Calculate the total number of data points by summing up all the frequencies:
[tex]\[
2 + 12 + 20 + 13 + 10 = 57
\][/tex]
3. Calculate the weighted sum of values by multiplying each value by its corresponding frequency and then summing these products:
[tex]\[
12 \times 2 + 17 \times 12 + 22 \times 20 + 31 \times 13 + 33 \times 10
\][/tex]
Performing the multiplications first:
[tex]\[
12 \times 2 = 24 \\
17 \times 12 = 204 \\
22 \times 20 = 440 \\
31 \times 13 = 403 \\
33 \times 10 = 330
\][/tex]
Now sum these results:
[tex]\[
24 + 204 + 440 + 403 + 330 = 1401
\][/tex]
4. Calculate the mean by dividing the weighted sum by the total number of data points:
[tex]\[
\text{Mean} = \frac{\text{Weighted Sum}}{\text{Total Number of Data Points}} = \frac{1401}{57} = 24.57894736842105
\][/tex]
5. Round the mean to one decimal place if required:
[tex]\[
24.6
\][/tex]
Thus, the mean of the given frequency distribution is [tex]\(24.6\)[/tex].
