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To calculate the arithmetic mean from the given frequency distribution, we use the following formula for the mean of a frequency distribution:

\[
\text{Mean} = \frac{\sum (x \cdot f)}{\sum f}
\]

Where:
- \(x\) is the value of the variable,
- \(f\) is the frequency associated with each \(x\).

### Step 1: Calculate \(x \cdot f\) for each value of \(x\)
We will calculate the product of \(x\) and \(f\) for each pair in the table:

\[
\begin{aligned}
5 \cdot 25 &= 125, \\
6 \cdot 45 &= 270, \\
7 \cdot 90 &= 630, \\
8 \cdot 165 &= 1320, \\
9 \cdot 112 &= 1008, \\
10 \cdot 96 &= 960, \\
11 \cdot 81 &= 891, \\
12 \cdot 26 &= 312, \\
13 \cdot 18 &= 234, \\
14 \cdot 12 &= 168.
\end{aligned}
\]

### Step 2: Sum \(x \cdot f\)
Now, sum all the \(x \cdot f\) values:

\[
125 + 270 + 630 + 1320 + 1008 + 960 + 891 + 312 + 234 + 168 = 5918.
\]

### Step 3: Sum \(f\)
Next, sum all the frequencies \(f\):

\[
25 + 45 + 90 + 165 + 112 + 96 + 81 + 26 + 18 + 12 = 670.
\]

### Step 4: Calculate the arithmetic mean
Finally, apply the formula for the mean:

\[
\text{Mean} = \frac{5918}{670} \approx 8.83.
\]

Thus, the arithmetic mean is approximately **8.83**.

Would you like any further explanation or details?

Here are some related questions to expand on this concept:
1. How would the arithmetic mean change if the highest frequency was removed?
2. What is the median of this frequency distribution?
3. How can you calculate the mode for this data set?
4. How does the arithmetic mean relate to other measures of central tendency?
5. Can we apply weighted averages in this context, and how would that differ?

**Tip:** The arithmetic mean is sensitive to extreme values (outliers), so it might not always represent the data's central tendency well in skewed distributions.

Find the arithmetic mean from the following frequency distribution: x = [5, 6, 7, 8, 9, 10, 11, 12, 13, 14], f = [25, 45, 90, 165, 112, 96, 81, 26, 18, 12]

Learn how to calculate the arithmetic mean from a frequency distribution table where x ranges from 5 to 14, and corresponding frequencies range from 25 to 12. This problem covers key statistical concepts such as the arithmetic mean and frequency distribution. Suitable for students in Grades 9-12.

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