Let's analyze and solve the two problems presented in the image:

1. Mode of the Age Data:

We are given the age distribution of patients admitted to a hospital.

The mode is the value or interval with the highest frequency. Here, the interval with the most patients is 35-45 years, with 23 patients. Therefore, the mode is: $\boxed{35-45 \text{ years}}$

2. Median and Upper Quartile of Heights:

We are given the heights of 60 students in a frequency distribution table.

(i) Median: To find the median, we need to find the middle value. Since there are 60 students, the median will be the 30.5th student.

First, we calculate the cumulative frequency:

The 30.5th student lies in the 155 cm group (since the cumulative frequency at 155 cm is 46, and the median falls between the 30th and 46th students). Thus, the median height is 155 cm.

(ii) Upper Quartile: The upper quartile (Q3) is the value at the 75th percentile, which corresponds to the 45th student (75% of 60 = 45).

From the cumulative frequency table, the 45th student also lies in the 155 cm group. Therefore, the upper quartile is also 155 cm.

Summary of Solutions:

1. Mode of age data: 35-45 years.
2. Median height: 155 cm.
3. Upper quartile height: 155 cm.

Would you like more details on any part of the calculations or further explanations?

5 Related Questions:

1. How is the mode calculated for continuous data sets like age distributions?
2. What is the significance of quartiles in a data set?
3. How do we calculate the mean from a frequency distribution?
4. What are the key differences between median and mean in a skewed distribution?
5. How can we estimate percentiles from cumulative frequency tables?

Tip:

When working with cumulative frequency tables, ensure you correctly identify where the required data point lies by checking the cumulative total just before and after the target value.