Answer :
Let's address each part of the question step by step.
### (a) What is the modal class of the given data?
The modal class is the class interval with the highest frequency (i.e., the one with the most workers).
- Wages (in Rs.): [tex]\(500-600\)[/tex], [tex]\(600-700\)[/tex], [tex]\(700-800\)[/tex], [tex]\(800-900\)[/tex], [tex]\(900-1000\)[/tex], [tex]\(1000-1100\)[/tex]
- Number of workers: 3, 5, 6, 2, 3, 1
From the given data, the class interval [tex]\(700-800\)[/tex] has the highest number of workers (6 workers).
Modal Class: [tex]\(700-800\)[/tex]
### (b) Find the median class of the given data.
To find the median class, we first need to determine the median position in the data set:
1. Calculate the total number of workers.
[tex]\[ 3 + 5 + 6 + 2 + 3 + 1 = 20 \][/tex]
2. The median position is given by:
[tex]\[ \frac{N}{2} = \frac{20}{2} = 10 \][/tex]
3. Use cumulative frequencies to find where the median position lies:
- Cumulative frequency for [tex]\((500-600)\)[/tex] is [tex]\(3\)[/tex]
- Cumulative frequency for [tex]\((600-700)\)[/tex] is [tex]\(3 + 5 = 8\)[/tex]
- Cumulative frequency for [tex]\(700-800\)[/tex] is [tex]\(8 + 6 = 14\)[/tex]
The 10th worker lies within the cumulative frequency of [tex]\(14\)[/tex], which corresponds to the [tex]\(700-800\)[/tex] wage class.
Median Class: [tex]\(700-800\)[/tex]
### (c) What is the average daily income of a worker? Calculate it.
To calculate the average daily income, we will use the midpoint of each wage range. The formula to calculate the average daily income is:
[tex]\[ \text{Average daily income} = \frac{\sum (\text{Midpoint of interval} \times \text{Number of workers})}{\text{Total number of workers}} \][/tex]
The midpoints of the wage intervals are:
- [tex]\( \frac{500 + 600}{2} = 550\)[/tex]
- [tex]\( \frac{600 + 700}{2} = 650\)[/tex]
- [tex]\( \frac{700 + 800}{2} = 750\)[/tex]
- [tex]\( \frac{800 + 900}{2} = 850\)[/tex]
- [tex]\( \frac{900 + 1000}{2} = 950\)[/tex]
- [tex]\( \frac{1000 + 1100}{2} = 1050\)[/tex]
Now we calculate the total income:
[tex]\[ (550 \times 3) + (650 \times 5) + (750 \times 6) + (850 \times 2) + (950 \times 3) + (1050 \times 1) \][/tex]
[tex]\[ = 1650 + 3250 + 4500 + 1700 + 2850 + 1050 = 15000 \][/tex]
Thus, the average daily income:
[tex]\[ = \frac{15000}{20} = 750 \][/tex]
Average daily income: Rs. 750
### (d) Do the median and modal class always lie in the same class? Write with reason.
From our calculations in parts (a) and (b), we see that both the modal class and median class are the same for this particular data set, i.e., [tex]\(700-800\)[/tex].
However, the median and modal class do not always lie in the same class. This is because:
- The modal class is determined by the highest frequency, which is purely about the count of occurrences in each class.
- The median class is determined by the position of the middle value, which is dependent on the cumulative distribution of the data.
Therefore, there are cases where the modal class and median class can be different.
Reason: While in this specific dataset the median and modal class are the same, they do not always align because they measure different aspects of the data: frequency vs. position.
### (a) What is the modal class of the given data?
The modal class is the class interval with the highest frequency (i.e., the one with the most workers).
- Wages (in Rs.): [tex]\(500-600\)[/tex], [tex]\(600-700\)[/tex], [tex]\(700-800\)[/tex], [tex]\(800-900\)[/tex], [tex]\(900-1000\)[/tex], [tex]\(1000-1100\)[/tex]
- Number of workers: 3, 5, 6, 2, 3, 1
From the given data, the class interval [tex]\(700-800\)[/tex] has the highest number of workers (6 workers).
Modal Class: [tex]\(700-800\)[/tex]
### (b) Find the median class of the given data.
To find the median class, we first need to determine the median position in the data set:
1. Calculate the total number of workers.
[tex]\[ 3 + 5 + 6 + 2 + 3 + 1 = 20 \][/tex]
2. The median position is given by:
[tex]\[ \frac{N}{2} = \frac{20}{2} = 10 \][/tex]
3. Use cumulative frequencies to find where the median position lies:
- Cumulative frequency for [tex]\((500-600)\)[/tex] is [tex]\(3\)[/tex]
- Cumulative frequency for [tex]\((600-700)\)[/tex] is [tex]\(3 + 5 = 8\)[/tex]
- Cumulative frequency for [tex]\(700-800\)[/tex] is [tex]\(8 + 6 = 14\)[/tex]
The 10th worker lies within the cumulative frequency of [tex]\(14\)[/tex], which corresponds to the [tex]\(700-800\)[/tex] wage class.
Median Class: [tex]\(700-800\)[/tex]
### (c) What is the average daily income of a worker? Calculate it.
To calculate the average daily income, we will use the midpoint of each wage range. The formula to calculate the average daily income is:
[tex]\[ \text{Average daily income} = \frac{\sum (\text{Midpoint of interval} \times \text{Number of workers})}{\text{Total number of workers}} \][/tex]
The midpoints of the wage intervals are:
- [tex]\( \frac{500 + 600}{2} = 550\)[/tex]
- [tex]\( \frac{600 + 700}{2} = 650\)[/tex]
- [tex]\( \frac{700 + 800}{2} = 750\)[/tex]
- [tex]\( \frac{800 + 900}{2} = 850\)[/tex]
- [tex]\( \frac{900 + 1000}{2} = 950\)[/tex]
- [tex]\( \frac{1000 + 1100}{2} = 1050\)[/tex]
Now we calculate the total income:
[tex]\[ (550 \times 3) + (650 \times 5) + (750 \times 6) + (850 \times 2) + (950 \times 3) + (1050 \times 1) \][/tex]
[tex]\[ = 1650 + 3250 + 4500 + 1700 + 2850 + 1050 = 15000 \][/tex]
Thus, the average daily income:
[tex]\[ = \frac{15000}{20} = 750 \][/tex]
Average daily income: Rs. 750
### (d) Do the median and modal class always lie in the same class? Write with reason.
From our calculations in parts (a) and (b), we see that both the modal class and median class are the same for this particular data set, i.e., [tex]\(700-800\)[/tex].
However, the median and modal class do not always lie in the same class. This is because:
- The modal class is determined by the highest frequency, which is purely about the count of occurrences in each class.
- The median class is determined by the position of the middle value, which is dependent on the cumulative distribution of the data.
Therefore, there are cases where the modal class and median class can be different.
Reason: While in this specific dataset the median and modal class are the same, they do not always align because they measure different aspects of the data: frequency vs. position.
