To find the mean of the given data set, we first need to calculate the midpoint of each class, then multiply each midpoint by its corresponding frequency, sum these products, and finally divide by the total frequency. Here’s the step-by-step process:

1. Calculate the midpoints of each class interval:

   • For 40-44, midpoint = $(40 + 44) / 2 = 42$
   • For 45-49, midpoint = $(45 + 49) / 2 = 47$
   • For 50-54, midpoint = $(50 + 54) / 2 = 52$
   • For 55-59, midpoint = $(55 + 59) / 2 = 57$
   • For 60-64, midpoint = $(60 + 64) / 2 = 62$
   • For 65-69, midpoint = $(65 + 69) / 2 = 67$
   • For 70-74, midpoint = $(70 + 74) / 2 = 72$
   • For 75-79, midpoint = $(75 + 79) / 2 = 77$
   • For 80-84, midpoint = $(80 + 84) / 2 = 82$

2. Multiply each midpoint by its frequency:

   • $42 \times 13 = 546$
   • $47 \times 18 = 846$
   • $52 \times 15 = 780$
   • $57 \times 8 = 456$
   • $62 \times 7 = 434$
   • $67 \times 6 = 402$
   • $72 \times 2 = 144$
   • $77 \times 1 = 77$
   • $82 \times 1 = 82$

3. Sum the products:

   $546 + 846 + 780 + 456 + 434 + 402 + 144 + 77 + 82$

4. Find the total frequency:

   $13 + 18 + 15 + 8 + 7 + 6 + 2 + 1 + 1$

5. Divide the sum of the products by the total frequency to find the mean.

Let's calculate these values:The mean of the given data set is approximately 53.06.

Do you want details or have any questions?

Here are some related questions for further exploration:

1. How do you calculate the median for this grouped data?
2. What is the mode of this data set, and how is it determined?
3. How would the mean change if the frequencies of the higher intervals increased?
4. How can we calculate the standard deviation for this data set?
5. What are the implications of a higher mean in this context?

Tip: When calculating the mean of grouped data, always ensure you use the midpoint of each class to approximate the distribution correctly.