To calculate the class width and create a frequency distribution for the provided data set, let's go through the steps systematically.

Step 1: Identify the class width

1. Find the range of the data:
   The range is the difference between the maximum and minimum values in the data set.

   • Maximum value = 75
   • Minimum value = 21

   So, the range = $75 - 21 = 54$.

2. Determine the class width:
   You are asked to use six classes. To find the class width, divide the range by the number of classes.

   $\text{Class width} = \frac{\text{Range}}{\text{Number of classes}} = \frac{54}{6} = 9$

Step 2: Construct the frequency distribution

• First Class: Start with the lowest value in the data set, which is 21. The first class will range from 21 to 29 (21 + 9 = 30, but 29 is the last value for this class).
• Subsequent classes: Add the class width (9) to the upper limit of the previous class to determine the next class range.

Thus, the class ranges will be:

• Class 1: 21-29
• Class 2: 30-38
• Class 3: 39-47
• Class 4: 48-56
• Class 5: 57-65
• Class 6: 66-74

Step 3: Count the frequencies for each class

Now, let's count how many data points fall within each class:

• Class 1 (21-29): 21, 24, 25, 27 → 4 values
• Class 2 (30-38): 32, 33, 34, 37 → 4 values
• Class 3 (39-47): 41, 47 → 2 values
• Class 4 (48-56): 52, 53 → 2 values
• Class 5 (57-65): 61, 61, 62 → 3 values
• Class 6 (66-74): 66, 65, 72, 75 → 4 values

Final Frequency Distribution

Class Range	Frequency
21 - 29	4
30 - 38	4
39 - 47	2
48 - 56	2
57 - 65	3
66 - 74	4

Summary:

• Class width: 9
• Class limits and frequencies as shown in the table.

Would you like to explore more details or have further questions about this process?

Additional Questions to Explore:

1. How do you interpret the frequency distribution?
2. How can you calculate the cumulative frequency for this data set?
3. How would you calculate the mean and median for this data set?
4. What are the advantages of using a frequency distribution?
5. How does the choice of class width affect the shape of the frequency distribution?

Tip:

When creating a frequency distribution, it's important to ensure that the class limits are consistent and that each data point falls within exactly one class.