To solve the questions based on the table provided:

Part (a): Modal Class

The modal class is the class with the highest frequency.

From the table:

• The frequencies are: 18, 14, 3, 16, 21, 7.
• The highest frequency is 21, corresponding to the class $40 < x \leq 50$.

Thus, the modal class is: $40 < x \leq 50$.

Part (b): Median Class

To find the median class, we need to determine where the middle value of the data lies.

1. Add up the frequencies to find the total number of data points: $\text{Total frequency} = 18 + 14 + 3 + 16 + 21 + 7 = 79$
2. The median is the middle value. For 79 data points, the median is the $\frac{79+1}{2} = 40$-th value.
3. Determine the class containing the 40th value by calculating the cumulative frequencies:
   • Cumulative frequency up to $0 < x \leq 10$: $18$
   • Cumulative frequency up to $10 < x \leq 20$: $18 + 14 = 32$
   • Cumulative frequency up to $20 < x \leq 30$: $32 + 3 = 35$
   • Cumulative frequency up to $30 < x \leq 40$: $35 + 16 = 51$

The 40th value lies within the cumulative frequency range $35 < 40 \leq 51$, corresponding to the class $30 < x \leq 40$.

Thus, the median class is: $30 < x \leq 40$.

Final Answers:

• (a) Modal class: $40 < x \leq 50$
• (b) Median class: $30 < x \leq 40$

Would you like further clarification on how to calculate these values?

Related Questions:

1. How is the cumulative frequency table constructed?
2. What is the mean class interval of the data provided?
3. How can we calculate the exact median using interpolation?
4. How is the range of this data determined?
5. Why is the modal class useful in analyzing data distributions?

Tip:

For grouped data, always determine whether a cumulative frequency column or a frequency polygon