To find the median for the grouped frequency distribution table provided, let's go through the steps.

Steps to Approximate the Median in a Grouped Frequency Distribution

1. Identify the total frequency: Sum all the frequencies to find the total number of data points.

   Given: $\text{Total Frequency (N)} = 1 + 2 + 4 + 6 + 15 + 25 + 11 + 6 + 6 + 5 = 81$

2. Find $\frac{N}{2}$: Divide the total frequency by 2 to determine the median position.

   $\frac{N}{2} = \frac{81}{2} = 40.5$

3. Locate the median class: The median class is the class interval where the cumulative frequency reaches or exceeds 40.5.

   Calculating cumulative frequencies:

   • 30-34: $1$
   • 35-39: $1 + 2 = 3$
   • 40-44: $3 + 4 = 7$
   • 45-49: $7 + 6 = 13$
   • 50-54: $13 + 15 = 28$
   • 55-59: $28 + 25 = 53$
   • 60-64: $53 + 11 = 64$

   The cumulative frequency exceeds 40.5 for the first time in the class interval 55-59, making it the median class.

4. Find the midpoint of the median class: The median class is 55-59, so its midpoint is:

   $\text{Midpoint} = \frac{55 + 59}{2} = 57$

Answer

The approximate median, reported as the midpoint of the class containing the median, is 57.

Would you like a deeper explanation of any part of this process?

Here are five related questions to deepen your understanding:

1. How is the median class determined in a grouped frequency distribution?
2. Why is the midpoint of the median class used as an estimate of the median?
3. How would this process change if there were unequal class intervals?
4. Can this method of median calculation be used for non-numeric grouped data?
5. How does the median differ from the mean in terms of sensitivity to outliers?

Tip: In grouped data, always check the cumulative frequency carefully to accurately locate the median class.