To approximate the mean for a grouped frequency distribution table (GFDT), we use the following formula:

$\text{Mean} = \frac{\sum f_i \cdot x_i}{\sum f_i}$

Where:

• $f_i$ is the frequency of the $i$-th group.
• $x_i$ is the midpoint of the $i$-th class interval.

Step 1: Calculate the midpoints ($x_i$) for each class interval.

The midpoint is calculated as:

$\text{Midpoint} = \frac{\text{Lower Bound} + \text{Upper Bound}}{2}$

Step 2: Multiply each midpoint by its corresponding frequency.

Step 3: Calculate the sum of the frequency ($\sum f_i$) and the sum of the product of midpoint and frequency ($\sum f_i \cdot x_i$).

Now, let's compute this.

Data:

Class Interval	Frequency $f_i$	Midpoint $x_i$	$f_i \cdot x_i$
40 - 44	1	42	42
45 - 49	1	47	47
50 - 54	4	52	208
55 - 59	5	57	285
60 - 64	5	62	310
65 - 69	10	67	670
70 - 74	10	72	720
75 - 79	24	77	1848
80 - 84	15	82	1230

Step 4: Calculate the totals

$\sum f_i = 1 + 1 + 4 + 5 + 5 + 10 + 10 + 24 + 15 = 75$

$\sum f_i \cdot x_i = 42 + 47 + 208 + 285 + 310 + 670 + 720 + 1848 + 1230 = 5360$

Step 5: Apply the formula for the mean

$\text{Mean} = \frac{5360}{75} = 71.47$

Thus, the approximate mean is:

$\boxed{71.47}$

Would you like further details or have any questions?

5 Related Questions:

1. How is the midpoint of a class interval calculated?
2. What is the difference between a grouped and ungrouped frequency distribution?
3. How can we use the mean to describe data sets?
4. How would you calculate the median for a grouped frequency distribution?
5. How does the number of class intervals affect the mean?

Tip: Always ensure the class intervals are consistent when calculating a mean for grouped data to avoid calculation errors.