To approximate the mean for a grouped frequency distribution table (GFDT), we follow these steps:

1. Identify the midpoints of each class: The midpoint for a class is calculated as $\text{Midpoint} = \frac{\text{Lower limit} + \text{Upper limit}}{2}$.
2. Multiply each midpoint by the frequency: This gives the weighted value for each class.
3. Calculate the total frequency ($N$) and the sum of all weighted midpoints ($\Sigma fx$).
4. Divide the sum of weighted midpoints by the total frequency:
   $\text{Mean} = \frac{\Sigma fx}{N}$

Given Data:

Calculations:

1. Sum of frequencies ($N$):
   $N = 1 + 4 + 4 + 14 + 21 + 13 + 7 + 3 + 1 = 68$

2. Sum of weighted midpoints ($\Sigma fx$):
   $\Sigma fx = 52 + 228 + 248 + 938 + 1512 + 1001 + 574 + 261 + 92 = 4906$

3. Mean:
   $\text{Mean} = \frac{\Sigma fx}{N} = \frac{4906}{68} \approx 72.2$

Final Answer:

The approximate mean is 72.2.

Would you like me to clarify further or explain any of the steps?

Related Questions:

1. What is the significance of the mean in a dataset?
2. How would the calculation change if one of the frequencies doubled?
3. Why do we use midpoints in grouped data?
4. Can you explain how the mean compares to the median in this distribution?
5. How could we approximate the variance for this data?

Tip: Always ensure the class intervals are continuous before performing grouped data calculations.