Question

Find the mean of the following data:

Hint:

To calculate the mean from a frequency distribution, multiply each midpoint by its frequency, sum the products, and divide by the total frequency.

Answer 1

Step 1: Find the midpoints of each class interval.
The midpoints for each class interval are calculated as the average of the lower and upper limits:
- Midpoint for \( 10 - 25 \): \( \frac{10 + 25}{2} = 17.5 \)
- Midpoint for \( 25 - 40 \): \( \frac{25 + 40}{2} = 32.5 \)
- Midpoint for \( 40 - 55 \): \( \frac{40 + 55}{2} = 47.5 \)
- Midpoint for \( 55 - 70 \): \( \frac{55 + 70}{2} = 62.5 \)
- Midpoint for \( 70 - 85 \): \( \frac{70 + 85}{2} = 77.5 \)
- Midpoint for \( 85 - 100 \): \( \frac{85 + 100}{2} = 92.5 \)

Step 2: Multiply the midpoints by their respective frequencies.
- \( 17.5 \times 2 = 35 \)
- \( 32.5 \times 3 = 97.5 \)
- \( 47.5 \times 7 = 332.5 \)
- \( 62.5 \times 6 = 375 \)
- \( 77.5 \times 6 = 465 \)
- \( 92.5 \times 6 = 555 \)

Step 3: Find the sum of the products.
Now, we sum the products: \[ 35 + 97.5 + 332.5 + 375 + 465 + 555 = 1860 \]
Step 4: Find the sum of the frequencies.
The sum of the frequencies is: \[ 2 + 3 + 7 + 6 + 6 + 6 = 30 \]
Step 5: Calculate the mean.
The formula for the arithmetic mean is: \[ \text{Mean} = \frac{\sum f \cdot x}{\sum f} \] Substituting the values: \[ \text{Mean} = \frac{1860}{30} = 62 \]
Conclusion:
The mean of the given frequency distribution is \( \boxed{62} \).