The mode of grouped data is the value that appears most frequently within a dataset organized into classes. It is a measure of central tendency that helps identify the most common value or interval in a given dataset.

To calculate the mode of grouped data, we follow these steps:

1. Identify the modal class: The modal class is the class interval with the highest frequency. It can be found by analyzing the frequency distribution table.
2. Determine the necessary values: For the modal class, identify the following values:

• Frequency of the modal class $f_1$
• Frequency of the class preceding the modal class $f_0$
• Frequency of the class succeeding the modal class $f_2$
• Lower limit of the modal class $L$
• Width of the class interval $h$

3. Construct the frequency distribution table: Create a table that lists the class intervals, frequencies, and identify the modal class.

In this example, the modal class is the interval $40 - 50$ since it has the highest frequency of $15$.

4. Apply the mode formula: The mode $M$ of grouped data can be calculated using the following formula:

$M = L + \left( \frac{{f_1 - f_0}}{{(2f_1 - f_0 - f_2)}} \right) \cdot h$

Where:

• $L$ = Lower limit of the modal class
• $f_1$ = Frequency of the modal class
• $f_0$ = Frequency of the class preceding the modal class
• $f_2$ = Frequency of the class succeeding the modal class
• $h$ = Width of the class interval

In this case:

• Lower limit $L = 40$
• $f_1 = 15$
• $f_0 = 8$
• $f_2 = 10$
• Width $h = 10$

Substituting the values into the formula:

$M = 40 + \left( \frac{{15 - 8}}{{(2 \cdot 15 - 8 - 10)}} \right) \cdot 10$

Now, calculating step-by-step:

$M = 40 + \left( \frac{7}{12} \right) \cdot 10$

$M = 40 + \frac{70}{12} \approx 40 + 5.83 \approx 45.83$

The mode of the grouped data is approximately $45.83$. This value indicates the most frequently occurring range of values within the dataset, providing insights into the data's distribution.