To solve these questions, let's break them down one by one.

1. Calculating the Mean Age

The table provided shows age ranges and their corresponding frequencies:

• $10 - 20: 4$
• $20 - 30: 6$
• $30 - 40: 8$
• $40 - 50: 2$
• $50 - 60: 4$

To find the mean age, we use the formula:

$\text{Mean} = \frac{\sum f X_m}{n}$

where:

• $f$ is the frequency,
• $X_m$ is the midpoint (class mark) of each age range,
• $n$ is the total number of people (24 in this case).

Steps:

1. Calculate the midpoints ($X_m$) for each age range:

   • $10 - 20$: $X_m = \frac{10 + 20}{2} = 15$
   • $20 - 30$: $X_m = \frac{20 + 30}{2} = 25$
   • $30 - 40$: $X_m = \frac{30 + 40}{2} = 35$
   • $40 - 50$: $X_m = \frac{40 + 50}{2} = 45$
   • $50 - 60$: $X_m = \frac{50 + 60}{2} = 55$

2. Multiply each midpoint by its frequency and then sum the results: $\sum f X_m = (4 \times 15) + (6 \times 25) + (8 \times 35) + (2 \times 45) + (4 \times 55)$ $= 60 + 150 + 280 + 90 + 220 = 800$

3. Divide by the total frequency (24): $\text{Mean} = \frac{800}{24} \approx 33.33$

Thus, the mean age of this group is approximately 33.3.

Answer for the Mean Age Question:

The correct answer is:

• 33.3

2. Interpretation of $X_m$ in the Formula

In the formula for the mean, $X_m$ represents the class mark, which is the midpoint of each age range.

Answer for the Meaning of $X_m$ Question:

The correct answer is:

• Class mark

Would you like further details or have any questions on this solution?

Related Questions

1. How do you calculate the midpoint (class mark) for a given range?
2. What is the purpose of using midpoints in a grouped data mean calculation?
3. How would the mean change if the age ranges were different?
4. How does frequency affect the calculation of the mean?
5. What is the difference between class mark and class boundaries?

Tip

When calculating means for grouped data, always verify that you correctly calculate each midpoint and check your frequency totals.