Sure, let's find the mean, median, and mode for the given data step-by-step.
### Given Data:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f \\
\hline
12 & 4 \\
\hline
18 & 5 \\
\hline
21 & 5 \\
\hline
24 & 6 \\
\hline
25 & 8 \\
\hline
29 & 2 \\
\hline
30 & 1 \\
\hline
\end{array}
\][/tex]
### 1. Finding the Mean:
The mean is calculated using the formula:
[tex]\[
\text{Mean} = \frac{\sum (x \cdot f)}{\sum f}
\][/tex]
First, we calculate [tex]\(\sum (x \cdot f)\)[/tex]:
[tex]\[
\begin{align*}
12 \cdot 4 & = 48 \\
18 \cdot 5 & = 90 \\
21 \cdot 5 & = 105 \\
24 \cdot 6 & = 144 \\
25 \cdot 8 & = 200 \\
29 \cdot 2 & = 58 \\
30 \cdot 1 & = 30 \\
\end{align*}
\][/tex]
Summing these products:
[tex]\[
48 + 90 + 105 + 144 + 200 + 58 + 30 = 675
\][/tex]
Next, we calculate [tex]\(\sum f\)[/tex]:
[tex]\[
4 + 5 + 5 + 6 + 8 + 2 + 1 = 31
\][/tex]
Finally, we find the mean:
[tex]\[
\text{Mean} = \frac{675}{31} \approx 21.774
\][/tex]
### 2. Finding the Median:
To find the median, we need to determine the cumulative frequency and find where the middle value lies. The cumulative frequency ([tex]\(cf\)[/tex]) is calculated by summing the frequencies up to each point.
[tex]\[
\begin{array}{|c|c|c|}
\hline
x & f & cf \\
\hline
12 & 4 & 4 \\
\hline
18 & 5 & 9 \\
\hline
21 & 5 & 14 \\
\hline
24 & 6 & 20 \\
\hline
25 & 8 & 28 \\
\hline
29 & 2 & 30 \\
\hline
30 & 1 & 31 \\
\hline
\end{array}
\][/tex]
Since the total frequency [tex]\(\sum f = 31\)[/tex] is an odd number, the median will be the value corresponding to the middle cumulative frequency. The middle position is given by:
[tex]\[
\frac{31 + 1}{2} = 16
\][/tex]
We locate the 16th term in the cumulative frequency column, which falls in the class where [tex]\(x = 24\)[/tex].
So, the median is [tex]\(24\)[/tex].
### 3. Finding the Mode:
The mode is the value of [tex]\(x\)[/tex] that has the highest frequency [tex]\(f\)[/tex].
From the data:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f \\
\hline
12 & 4 \\
\hline
18 & 5 \\
\hline
21 & 5 \\
\hline
24 & 6 \\
\hline
25 & 8 \\
\hline
29 & 2 \\
\hline
30 & 1 \\
\hline
\end{array}
\][/tex]
The highest frequency is [tex]\(8\)[/tex], which corresponds to [tex]\(x = 25\)[/tex].
So, the mode is [tex]\(25\)[/tex].
### Summary of Results:
[tex]\[
\begin{align*}
\text{Mean} &\approx 21.774 \\
\text{Median} &= 24 \\
\text{Mode} &= 25 \\
\end{align*}
\][/tex]
