To solve the problem of finding the mean mass of the group of students from the given data, we can follow these steps:

Step 1: Identify the midpoints of each class

The midpoint ($x_i$) of each mass interval can be calculated as: $x_i = \frac{\text{Lower bound} + \text{Upper bound}}{2}$

The midpoints for the intervals are:

• $40-44$: $\frac{40+44}{2} = 42$
• $45-49$: $\frac{45+49}{2} = 47$
• $50-54$: $\frac{50+54}{2} = 52$
• $55-59$: $\frac{55+59}{2} = 57$
• $60-64$: $\frac{60+64}{2} = 62$

Step 2: Multiply midpoints by their respective frequencies

Next, we multiply each midpoint by its corresponding frequency ($f_i$) to find $f_i x_i$:

• $42 \times 5 = 210$
• $47 \times 8 = 376$
• $52 \times 12 = 624$
• $57 \times 12 = 684$
• $62 \times 3 = 186$

Step 3: Sum up $f_i x_i$ and $f_i$

Now, we calculate the totals: $\text{Sum of } f_i x_i = 210 + 376 + 624 + 684 + 186 = 2080$ $\text{Sum of } f_i = 5 + 8 + 12 + 12 + 3 = 40$

Step 4: Calculate the mean

The formula for the mean is: $\text{Mean} = \frac{\text{Sum of } f_i x_i}{\text{Sum of } f_i}$ $\text{Mean} = \frac{2080}{40} = 52$

Final Answer:

The mean mass for this group of students is: $\mathbf{c. \, 52 \, \text{kg}}$

Do you want further details or have any questions?
Here are 5 related questions to deepen understanding:

1. How can you find the mode of the data from the given table?
2. What is the median class of this grouped data?
3. Can you explain why the midpoints are used in calculating the mean for grouped data?
4. How would the mean change if one more student with a mass of 65 kg was added?
5. What is the total weight of all the students combined?

Tip: In grouped data, using midpoints allows us to approximate the values in a continuous interval when the actual data points are unknown!