Answer :
Let's address each part of the question step-by-step:
### (क) Modal Class
The modal class is the class interval that contains the highest frequency of individuals. From the provided table:
| उमेर (Age in years) | Number of people |
|----------------------|------------------|
| 10-20 | 2 |
| 20-30 | 5 |
| 30-40 | 6 |
| 40-50 | 3 |
| 50-60 | 4 |
| 60-70 | 4 |
From the table, we observe that the highest frequency is 6, which corresponds to the age group 30-40. Thus, the modal class is:
30-40
### (ख) Median Class
To find the median class, we first need to determine the cumulative frequency distribution and then identify the median position.
The frequencies and cumulative frequencies can be computed as follows:
| उमेर (Age in years) | Number of People | Cumulative Frequency |
|----------------------|------------------|-----------------------|
| 10-20 | 2 | 2 |
| 20-30 | 5 | 2 + 5 = 7 |
| 30-40 | 6 | 7 + 6 = 13 |
| 40-50 | 3 | 13 + 3 = 16 |
| 50-60 | 4 | 16 + 4 = 20 |
| 60-70 | 4 | 20 + 4 = 24 |
The total number of people is 24. The median position is given by:
[tex]\[ \text{Median position} = \frac{N}{2} = \frac{24}{2} = 12 \][/tex]
The interval where the cumulative frequency just exceeds the median position (12) is 30-40. Hence, the median class is:
30-40
### (ग) Calculating the Median
To calculate the actual median value, we use the formula for the median in a frequency distribution:
[tex]\[ \text{Median} = L + \left(\frac{\frac{N}{2} - CF}{f}\right) \times h \][/tex]
where:
- [tex]\( L \)[/tex] is the lower boundary of the median class,
- [tex]\( N \)[/tex] is the total frequency,
- [tex]\( CF \)[/tex] is the cumulative frequency of the class preceding the median class,
- [tex]\( f \)[/tex] is the frequency of the median class,
- [tex]\( h \)[/tex] is the class interval.
From the cumulative frequency table:
- [tex]\( L = 30 \)[/tex] (lower boundary for the 30-40 class)
- [tex]\( N = 24 \)[/tex]
- [tex]\( CF = 7 \)[/tex] (cumulative frequency before 30-40 class)
- [tex]\( f = 6 \)[/tex] (frequency of 30-40 class)
- [tex]\( h = 10 \)[/tex] (assuming uniform class interval)
Plugging in these values:
[tex]\[ \text{Median} = 30 + \left(\frac{12 - 7}{6}\right) \times 10 \][/tex]
[tex]\[ \text{Median} = 30 + \left(\frac{5}{6}\right) \times 10 \][/tex]
[tex]\[ \text{Median} = 30 + 8.\overline{3} \][/tex]
[tex]\[ \text{Median} = 38.\overline{3} \][/tex]
So, the median is approximately:
38.33 years
### (घ) Comparing the Ratio
To compare the number of people above and below the median class, we sum the frequencies:
- Below median class:
- Frequencies: 2 (10-20), 5 (20-30)
- Total below = 2 + 5 = 7
- Above median class:
- Frequencies: 3 (40-50), 4 (50-60), 4 (60-70)
- Total above = 3 + 4 + 4 = 11
Now, we find the ratio of the number of people above to the number of people below the median class:
[tex]\[ \text{Ratio} = \frac{11}{7} \approx 1.57 \][/tex]
Thus, the ratio of the number of people whose ages are above and below the median class is approximately:
1.57
In summary:
- (क) Modal Class: 30-40
- (ख) Median Class: 30-40
- (ग) Median: 38.33 years
- (घ) Ratio of people above to below the median class: 1.57
### (क) Modal Class
The modal class is the class interval that contains the highest frequency of individuals. From the provided table:
| उमेर (Age in years) | Number of people |
|----------------------|------------------|
| 10-20 | 2 |
| 20-30 | 5 |
| 30-40 | 6 |
| 40-50 | 3 |
| 50-60 | 4 |
| 60-70 | 4 |
From the table, we observe that the highest frequency is 6, which corresponds to the age group 30-40. Thus, the modal class is:
30-40
### (ख) Median Class
To find the median class, we first need to determine the cumulative frequency distribution and then identify the median position.
The frequencies and cumulative frequencies can be computed as follows:
| उमेर (Age in years) | Number of People | Cumulative Frequency |
|----------------------|------------------|-----------------------|
| 10-20 | 2 | 2 |
| 20-30 | 5 | 2 + 5 = 7 |
| 30-40 | 6 | 7 + 6 = 13 |
| 40-50 | 3 | 13 + 3 = 16 |
| 50-60 | 4 | 16 + 4 = 20 |
| 60-70 | 4 | 20 + 4 = 24 |
The total number of people is 24. The median position is given by:
[tex]\[ \text{Median position} = \frac{N}{2} = \frac{24}{2} = 12 \][/tex]
The interval where the cumulative frequency just exceeds the median position (12) is 30-40. Hence, the median class is:
30-40
### (ग) Calculating the Median
To calculate the actual median value, we use the formula for the median in a frequency distribution:
[tex]\[ \text{Median} = L + \left(\frac{\frac{N}{2} - CF}{f}\right) \times h \][/tex]
where:
- [tex]\( L \)[/tex] is the lower boundary of the median class,
- [tex]\( N \)[/tex] is the total frequency,
- [tex]\( CF \)[/tex] is the cumulative frequency of the class preceding the median class,
- [tex]\( f \)[/tex] is the frequency of the median class,
- [tex]\( h \)[/tex] is the class interval.
From the cumulative frequency table:
- [tex]\( L = 30 \)[/tex] (lower boundary for the 30-40 class)
- [tex]\( N = 24 \)[/tex]
- [tex]\( CF = 7 \)[/tex] (cumulative frequency before 30-40 class)
- [tex]\( f = 6 \)[/tex] (frequency of 30-40 class)
- [tex]\( h = 10 \)[/tex] (assuming uniform class interval)
Plugging in these values:
[tex]\[ \text{Median} = 30 + \left(\frac{12 - 7}{6}\right) \times 10 \][/tex]
[tex]\[ \text{Median} = 30 + \left(\frac{5}{6}\right) \times 10 \][/tex]
[tex]\[ \text{Median} = 30 + 8.\overline{3} \][/tex]
[tex]\[ \text{Median} = 38.\overline{3} \][/tex]
So, the median is approximately:
38.33 years
### (घ) Comparing the Ratio
To compare the number of people above and below the median class, we sum the frequencies:
- Below median class:
- Frequencies: 2 (10-20), 5 (20-30)
- Total below = 2 + 5 = 7
- Above median class:
- Frequencies: 3 (40-50), 4 (50-60), 4 (60-70)
- Total above = 3 + 4 + 4 = 11
Now, we find the ratio of the number of people above to the number of people below the median class:
[tex]\[ \text{Ratio} = \frac{11}{7} \approx 1.57 \][/tex]
Thus, the ratio of the number of people whose ages are above and below the median class is approximately:
1.57
In summary:
- (क) Modal Class: 30-40
- (ख) Median Class: 30-40
- (ग) Median: 38.33 years
- (घ) Ratio of people above to below the median class: 1.57
