Question

Find the arithmetic mean from the following frequency table:

Hint:

To find the arithmetic mean for a frequency distribution, calculate the sum of the products of midpoints and frequencies, then divide by the total frequency.

Answer 1

We are given the following frequency table:
- Class intervals: \( 0 - 10, 10 - 20, 20 - 30, 30 - 40, 40 - 50 \)
- Frequencies: \( 5, 12, 25, 10, 8 \)

Step 1: Find the midpoints.
The midpoints for each class interval are calculated as the average of the lower and upper limits of the interval.
For each class interval, we calculate:
- Midpoint for \( 0 - 10 \): \( \frac{0 + 10}{2} = 5 \)
- Midpoint for \( 10 - 20 \): \( \frac{10 + 20}{2} = 15 \)
- Midpoint for \( 20 - 30 \): \( \frac{20 + 30}{2} = 25 \)
- Midpoint for \( 30 - 40 \): \( \frac{30 + 40}{2} = 35 \)
- Midpoint for \( 40 - 50 \): \( \frac{40 + 50}{2} = 45 \)

Step 2: Multiply the midpoints by their respective frequencies.
Next, we multiply each midpoint by the corresponding frequency:
- \( 5 \times 5 = 25 \)
- \( 15 \times 12 = 180 \)
- \( 25 \times 25 = 625 \)
- \( 35 \times 10 = 350 \)
- \( 45 \times 8 = 360 \)

Step 3: Find the sum of the products.
Now, we sum the products:
\[ 25 + 180 + 625 + 350 + 360 = 1540 \]
Step 4: Find the sum of the frequencies.
The sum of the frequencies is:
\[ 5 + 12 + 25 + 10 + 8 = 60 \]
Step 5: Calculate the arithmetic mean.
The formula for the arithmetic mean is:
\[ \text{Mean} = \frac{\sum (f \cdot x)}{\sum f} \] Where \( \sum (f \cdot x) \) is the sum of the products of frequencies and midpoints, and \( \sum f \) is the sum of the frequencies.
Substituting the values, we get:
\[ \text{Mean} = \frac{1540}{60} = 25.67 \]
Conclusion:
The arithmetic mean of the given frequency distribution is \( \boxed{25.67} \).