To construct a grouped frequency distribution table (GFDT) for this data set, we need to follow these steps:

Steps:

1. Determine the range of the dataTo construct a grouped frequency distribution table (GFDT) for this data set, we need to follow these steps:

Steps:

1. Determine the range of the

   • The smallest value in the data set: 267
   • The largest value in the data set: 729
   • Range = Largest value - Smallest value = $729 - 267 = 462$

2. Select the number of classes: Given that we want 10 classes.

3. Determine the class width: We can calculate the class width using the formula:

   $\text{Class width} = \frac{\text{Range}}{\text{Number of classes}} = \frac{462}{10} = 46.2$

   To make the class width a "nice" number, we round up to the nearest whole number, which is 47.

4. Determine class limits: Start from the smallest value (267) and add the class width (47) to create each subsequent class. Since the data is discrete, we will use closed intervals.

5. Tally the frequencies: Count how many data points fall within each class.

Grouped Frequency Distribution Table (GFDT):

Class Interval	Frequency
$[267, 313]$	4
$[314, 360]$	3
$[361, 407]$	5
$[408, 454]$	7
$[455, 501]$	6
$[502, 548]$	4
$[549, 595]$	2
$[596, 642]$	1
$[643, 689]$	0
$[690, 736]$	1

Explanation:

• Class width: We used 47 to ensure that the intervals are wide enough to cover the entire range of the data set.
• Closed intervals: Since the data is discrete, each class is written as a closed interval $[a, b]$, meaning that both the lower and upper class limits are included in the interval.
• Frequencies: The frequency column represents how many data points fall within each class.

Let me know if you'd like further details on any of the steps!

Follow-up Questions:

1. How do you determine the midpoint of each class for a more detailed frequency analysis?
2. How can we represent this data using a histogram?
3. What other class widths could we choose for different grouped distributions?
4. What is the cumulative frequency for each class?
5. How would the distribution change if we grouped into 5 classes instead of 10?

Tip:

To ensure intervals are meaningful, always round the class width up to avoid missing any data values.