To approximate the mean of the given grouped frequency distribution table (GFDT), we need to follow these steps:

Step 1: Find the Midpoint (Class Mark) for Each Class Interval

The midpoint for each class interval is calculated as: $\text{Midpoint} = \frac{\text{Lower Bound} + \text{Upper Bound}}{2}$

Step 2: Multiply Each Midpoint by its Corresponding Frequency

Next, for each class interval, we multiply the midpoint by its frequency.

Step 3: Sum the Products and Frequencies

We will sum the products and also sum the frequencies.

Step 4: Calculate the Mean

The formula for the mean of a grouped frequency distribution is: $\text{Mean} = \frac{\sum (f \times x)}{\sum f}$ Where:

• $f$ is the frequency of each class,
• $x$ is the midpoint of the class,
• $\sum f$ is the total frequency.

Now, let’s go ahead and calculate it.

Step 1: Midpoints

Data Range	Frequency (f)	Midpoint (x)	f × x
60 - 64	1	62	62
65 - 69	1	67	67
70 - 74	2	72	144
75 - 79	7	77	539
80 - 84	6	82	492
85 - 89	12	87	1044
90 - 94	10	92	920
95 - 99	21	97	2037
100 - 104	13	102	1326

Step 2: Sum the Products and Frequencies

$\sum (f \times x) = 62 + 67 + 144 + 539 + 492 + 1044 + 920 + 2037 + 1326 = 5531$ $\sum f = 1 + 1 + 2 + 7 + 6 + 12 + 10 + 21 + 13 = 73$

Step 3: Calculate the Mean

$\text{Mean} = \frac{\sum (f \times x)}{\sum f} = \frac{5531}{73} \approx 75.7$

Thus, the approximate mean for the given data is 75.7.

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Would you like further details or clarification? Here are some related questions for you to explore:

1. How do you calculate the variance for a grouped frequency distribution?
2. What is the difference between a grouped and ungrouped frequency distribution?
3. How does the median of a grouped frequency distribution differ from the mean?
4. How would you approximate the standard deviation from this grouped data?
5. What are some limitations of using midpoints in grouped frequency distributions?

Tip: When dealing with grouped data, always use the midpoint for each class interval, as it serves as a representative value for that interval.