Here is an analysis of the four frequency distributions and the reasons why each is incorrectly constructed:

a. Class:

Issues:

• The class widths are inconsistent. For example:
  • $27-32$ has a width of $32 - 27 = 6$,
  • $33-38$ has a width of $38 - 33 = 6$,
  • $39-44$ also has a width of $6$,
    but $45-49$ has a width of $5$, and $50-55$ has a width of $6$.
• The class intervals are overlapping. For example:
  • $33-38$ overlaps with $39-44$, which makes the boundaries unclear.

b. Class:

Issues:

• The class widths are inconsistent:
  • $5-9$ has a width of $9 - 5 = 5$,
  • $9-13$ has a width of $13 - 9 = 5$,
  • $13-17$ also has a width of $5$,
    but $17-20$ has a width of $4$, and $20-24$ has a width of $5$.
• The classes do not have consistent boundaries. For instance, there is a gap between $17-20$ and $20-24$, making it unclear where $20$ should belong.

c. Class:

Issues:

• The class widths are inconsistent:
  • $123-127$ has a width of $127 - 123 = 5$,
  • $128-132$ has a width of $5$,
    but $133-142$ has a width of $10$, and $143-147$ has a width of $5$.
• The class intervals do not follow a sequence:
  • There is a gap between $127-128$, and $132-133$.

d. Class:

Issues:

• The class boundaries are inconsistent. For example:
  • $9-13$ has a width of $5$,
  • $14-19$ has a width of $6$,
    and other intervals like $20-25$ also have different widths.
• The classes are not continuous:
  • There is a gap between $13-14$ and $19-20$, causing ambiguity for data points that fall within those gaps.

General Explanation:

For a frequency distribution to be correctly constructed:

1. The class widths must be consistent.
2. The class intervals should not overlap.
3. The class intervals should be continuous (no gaps between intervals).

Would you like detailed steps for correcting these distributions or further clarification?

5 Follow-up Questions:

1. What is the significance of consistent class widths in frequency distributions?
2. How do overlapping intervals affect data interpretation?
3. Why is it important for class intervals to be continuous?
4. How can we calculate the appropriate class width for a given dataset?
5. What are the consequences of inconsistent frequency distributions in real-world data analysis?

Tip:

When designing frequency distributions, ensure you calculate class width using the formula:
$\text{Class Width} = \frac{\text{Range}}{\text{Number of Classes}}$
and round up to maintain consistency!