The median of grouped data is a measure of central tendency that represents the middle value of a dataset when it is organized into classes or groups. It is particularly useful in understanding the distribution of data, especially when the dataset is large.

To calculate the median of grouped data, follow these steps:

1. Construct a cumulative frequency table: Create a frequency distribution table and calculate the cumulative frequency for each class interval. The cumulative frequency is the running total of frequencies.

In this example, the cumulative frequency for each class is calculated. The total number of observations $N$ is the cumulative frequency of the last class.

Total $N = 41$

2. Determine the median class: The median class is the class interval where the median lies. It can be found using the formula:

Median position $= \dfrac{N}{2} = \dfrac{41}{2} = 20.5$

Since the cumulative frequency just exceeds $20.5$ is $23$, the median class is $30 - 40$.

3. Identify the necessary values: For the median class, identify the following:

• Lower limit of the median class $L$
• Cumulative frequency of the class preceding the median class $C_f$
• Frequency of the median class $f$
• Class width $h$

From the median class $30 - 40$:

• Lower limit $L = 30$
• Cumulative frequency of the previous class $C_f = 15$
• Frequency of the median class $f = 8$
• Width of the class $h = 10$

4. Apply the median formula: The median $M_d$ can be calculated using the following formula:

$M_d = L + \left( \frac{\frac{N}{2} - C_f}{f} \right) \cdot h

Substituting the values into the formula:

$M_d = 30 + \left( \frac{20.5 - 15}{8} \right) \cdot 10

Calculating step-by-step:

$M_d = 30 + \left( \frac{5.5}{8} \right) \cdot 10

$M_d = 30 + 6.875 = 36.875

The median of the grouped data is approximately $36.88$. This value provides insight into the central tendency of the dataset, indicating that half of the observations lie below this value and half lie above it.