Sure, let's analyze the given data step by step to find the answers to the questions.
Given Data:
| उमेर वर्षमा (Age in years) | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 |
|----------------------------|------|-------|-------|-------|-------|
| खेलाडी सङ्या (Number of players) | 6 | 4 | 5 | 4 | 11 |
a) Modal Class of the Given Data:
The modal class is the class interval with the highest frequency. In our data:
- Number of players in 0-10 years: 6
- Number of players in 10-20 years: 4
- Number of players in 20-30 years: 5
- Number of players in 30-40 years: 4
- Number of players in 40-50 years: 11
The highest frequency is 11, which corresponds to the class interval 40-50 years. Therefore, the modal class is:
(40, 50)
b) Median Class of the Given Data:
To find the median class, we need to determine the class interval that contains the median.
- The total number of players: 6 + 4 + 5 + 4 + 11 = 30
- The median position (n/2) = 30/2 = 15
We need to find the class interval where the cumulative frequency reaches or exceeds 15:
1. Cumulative frequency for 0-10 years: 6
2. Cumulative frequency for 10-20 years: 6 + 4 = 10
3. Cumulative frequency for 20-30 years: 10 + 5 = 15
4. Cumulative frequency for 30-40 years: 15 + 4 = 19
5. Cumulative frequency for 40-50 years: 19 + 11 = 30
The cumulative frequency first reaches 15 in the class interval 20-30 years. Therefore, the median class is:
(20, 30)
c) First Quartile (Q1) of the Given Data:
The first quartile (Q1) position is at (n/4) = 30/4 = 7.5
We need to find the class interval where the cumulative frequency reaches or exceeds 7.5:
1. Cumulative frequency for 0-10 years: 6
2. Cumulative frequency for 10-20 years: 6 + 4 = 10
3. Cumulative frequency for 20-30 years: 10 + 5 = 15
4. Cumulative frequency for 30-40 years: 15 + 4 = 19
5. Cumulative frequency for 40-50 years: 19 + 11 = 30
The cumulative frequency first reaches or exceeds 7.5 in the class interval 10-20 years. Therefore, the first quartile (Q1) lies in:
(10, 20)
d) Average Age of Players Under 20 Years:
To find the average age of players under 20 years, we need to consider the players in the class intervals 0-10 and 10-20 years:
- Number of players in 0-10 years: 6
- Number of players in 10-20 years: 4
Total number of players under 20 years = 6 + 4 = 10
We calculate the midpoint (average age within each class interval) and then find the weighted average:
- Midpoint of 0-10 years = (0 + 10) / 2 = 5 years
- Midpoint of 10-20 years = (10 + 20) / 2 = 15 years
Weighted average age:
[tex]\( \text{Average age} = \frac{(6 \times 5) + (4 \times 15)}{6 + 4} \)[/tex]
= [tex]\( \frac{30 + 60}{10} \)[/tex]
= [tex]\( \frac{90}{10} \)[/tex]
= 9 years
Therefore, the average age of players under 20 years is:
9 years
