What are the steps to verify a frequency distribution table and construct a histogram, and what is the correct total frequency?

The table correctly lists the class intervals, their corresponding frequencies, class boundaries, and class marks (midpoints).

• Class Boundaries: Calculated by subtracting 0.5 from the lower limit and adding 0.5 to the upper limit of each class interval. For example, for the class $40-49$, the boundaries are $40 - 0.5 = 39.5$ and $49 + 0.5 = 49.5$. This is correctly applied for all classes.
• Class Marks (Midpoints): Calculated as the average of the lower and upper limits of each class interval. For example, for the class $40-49$, the midpoint is $\frac{40+49}{2} = \frac{89}{2} = 44.5$. This is correctly applied for all classes.

The sum of the frequencies listed in the table is:

$$13 + 9 + 6 + 8 + 8 + 1 = 45$$

However, the total written at the bottom of the frequency column is 50. There is a discrepancy between the sum of the individual frequencies (45) and the written total (50). Based on the listed frequencies, the correct total frequency is 45.

A histogram visually represents a frequency distribution.

• The x-axis represents the class boundaries. The bars are drawn adjacent to each other, with their edges at the class boundaries.
• The y-axis represents the frequency. The height of each bar corresponds to the frequency of its respective class interval.

Based on the table:

• For the class $40-49$ (boundaries $39.5-49.5$), the frequency is 13. The histogram shows a bar of height 13.
• For the class $50-59$ (boundaries $49.5-59.5$), the frequency is 9. The histogram shows a bar of height 9.
• For the class $60-69$ (boundaries $59.5-69.5$), the frequency is 6. The histogram shows a bar of height 6.
• For the class $70-79$ (boundaries $69.5-79.5$), the frequency is 8. The histogram shows a bar of height 8.
• For the class $80-89$ (boundaries $79.5-89.5$), the frequency is 8. The histogram shows a bar of height 8.
• For the class $90-99$ (boundaries $89.5-99.5$), the frequency is 1. The histogram shows a bar of height 1.

The partially drawn histogram accurately reflects the frequencies and class boundaries from the table. The only correction needed is for the total frequency, which should be 45 if the individual frequencies are correct.

The frequency distribution table and the histogram are correctly constructed based on the given data, with the exception of the sum of frequencies.