Answer :
To address the question, we need to create the 'less than' and 'more than' ogives and determine the median of the given distribution. Here's a detailed step-by-step solution for both parts:
### Step-by-Step Solution:
#### 1. Calculating Cumulative Frequencies for 'Less Than' Ogive:
The 'less than' ogive displays the cumulative frequency distribution where each class is less than the upper limit of the class interval.
From the data:
- Number of workers for Rs 50-55: 4
- Number of workers for Rs 55-60: 6
- Number of workers for Rs 60-65: 10
- Number of workers for Rs 65-70: 12
- Number of workers for Rs 70-75: 5
- Number of workers for Rs 75-80: 3
The cumulative frequencies are:
- Less than Rs 55: 4
- Less than Rs 60: 4 + 6 = 10
- Less than Rs 65: 10 + 10 = 20
- Less than Rs 70: 20 + 12 = 32
- Less than Rs 75: 32 + 5 = 37
- Less than Rs 80: 37 + 3 = 40
So, the cumulative frequencies for 'less than' ogive are:
[tex]\[ \text{[4, 10, 20, 32, 37, 40]} \][/tex]
#### 2. Calculating Cumulative Frequencies for 'More Than' Ogive:
The 'more than' ogive shows the cumulative frequency of workers earning more than the lower limit of each class interval.
First, we calculate the total number of workers, which is 40.
Now, the cumulative frequencies for 'more than' are calculated as follows:
- More than Rs 50: 40 (all workers)
- More than Rs 55: 40 - 4 = 36
- More than Rs 60: 40 - (4 + 6) = 30
- More than Rs 65: 40 - (4 + 6 + 10) = 20
- More than Rs 70: 40 - (4 + 6 + 10 + 12) = 8
- More than Rs 75: 40 - (4 + 6 + 10 + 12 + 5) = 3
- More than Rs 80: 40 - (4 + 6 + 10 + 12 + 5 + 3) = 0
So, the cumulative frequencies for 'more than' ogive are:
[tex]\[ \text{[36, 30, 20, 8, 3, 0]} \][/tex]
#### 3. Plotting the Ogives:
To plot the ogives, we'll use the respective cumulative frequencies calculated above.
'Less than' Ogive Points:
- (55, 4), (60, 10), (65, 20), (70, 32), (75, 37), (80, 40)
'More than' Ogive Points:
- (50, 40), (55, 36), (60, 30), (65, 20), (70, 8), (75, 3), (80, 0)
Draw both sets of points on the same set of axes and connect them to form two cumulative frequency curves. The point where the two curves intersect will help verify the median found algebraically.
#### 4. Finding the Median:
To find the median of the distribution, we will use the cumulative frequency and frequency of the corresponding classes.
Total number of workers, [tex]\( N = 40 \)[/tex].
Median is the value which separates the higher half from the lower half, which in this case is the [tex]\(\frac{N}{2} = \frac{40}{2} = 20\)[/tex]th value.
From the 'less than' cumulative frequencies:
- The 20th value lies in the 60-65 class interval (since 20 is included in the interval with cumulative frequency = 20).
We apply the formula for the median:
[tex]\[ \text{Median} = L + \left( \frac{\frac{N}{2} - CF}{f} \right) \times h \][/tex]
Where:
- [tex]\( L\)[/tex] = Lower boundary of the median class = 60 (lower boundary of the 60-65 class)
- [tex]\( CF\)[/tex] = Cumulative frequency of the class preceding the median class = 10
- [tex]\( f\)[/tex] = Frequency of the median class = 10
- [tex]\( h\)[/tex] = Class interval size = 65 - 60 = 5
Substituting the values we get:
[tex]\[ \text{Median} = 60 + \left( \frac{20 - 10}{10} \right) \times 5 \][/tex]
[tex]\[ \text{Median} = 60 + \left( \frac{10}{10} \right) \times 5 \][/tex]
[tex]\[ \text{Median} = 60 + 1 \times 5 \][/tex]
[tex]\[ \text{Median} = 60 + 5 \][/tex]
[tex]\[ \text{Median} = 65 \][/tex]
Therefore, the median wage of the distribution is Rs 65.
### Step-by-Step Solution:
#### 1. Calculating Cumulative Frequencies for 'Less Than' Ogive:
The 'less than' ogive displays the cumulative frequency distribution where each class is less than the upper limit of the class interval.
From the data:
- Number of workers for Rs 50-55: 4
- Number of workers for Rs 55-60: 6
- Number of workers for Rs 60-65: 10
- Number of workers for Rs 65-70: 12
- Number of workers for Rs 70-75: 5
- Number of workers for Rs 75-80: 3
The cumulative frequencies are:
- Less than Rs 55: 4
- Less than Rs 60: 4 + 6 = 10
- Less than Rs 65: 10 + 10 = 20
- Less than Rs 70: 20 + 12 = 32
- Less than Rs 75: 32 + 5 = 37
- Less than Rs 80: 37 + 3 = 40
So, the cumulative frequencies for 'less than' ogive are:
[tex]\[ \text{[4, 10, 20, 32, 37, 40]} \][/tex]
#### 2. Calculating Cumulative Frequencies for 'More Than' Ogive:
The 'more than' ogive shows the cumulative frequency of workers earning more than the lower limit of each class interval.
First, we calculate the total number of workers, which is 40.
Now, the cumulative frequencies for 'more than' are calculated as follows:
- More than Rs 50: 40 (all workers)
- More than Rs 55: 40 - 4 = 36
- More than Rs 60: 40 - (4 + 6) = 30
- More than Rs 65: 40 - (4 + 6 + 10) = 20
- More than Rs 70: 40 - (4 + 6 + 10 + 12) = 8
- More than Rs 75: 40 - (4 + 6 + 10 + 12 + 5) = 3
- More than Rs 80: 40 - (4 + 6 + 10 + 12 + 5 + 3) = 0
So, the cumulative frequencies for 'more than' ogive are:
[tex]\[ \text{[36, 30, 20, 8, 3, 0]} \][/tex]
#### 3. Plotting the Ogives:
To plot the ogives, we'll use the respective cumulative frequencies calculated above.
'Less than' Ogive Points:
- (55, 4), (60, 10), (65, 20), (70, 32), (75, 37), (80, 40)
'More than' Ogive Points:
- (50, 40), (55, 36), (60, 30), (65, 20), (70, 8), (75, 3), (80, 0)
Draw both sets of points on the same set of axes and connect them to form two cumulative frequency curves. The point where the two curves intersect will help verify the median found algebraically.
#### 4. Finding the Median:
To find the median of the distribution, we will use the cumulative frequency and frequency of the corresponding classes.
Total number of workers, [tex]\( N = 40 \)[/tex].
Median is the value which separates the higher half from the lower half, which in this case is the [tex]\(\frac{N}{2} = \frac{40}{2} = 20\)[/tex]th value.
From the 'less than' cumulative frequencies:
- The 20th value lies in the 60-65 class interval (since 20 is included in the interval with cumulative frequency = 20).
We apply the formula for the median:
[tex]\[ \text{Median} = L + \left( \frac{\frac{N}{2} - CF}{f} \right) \times h \][/tex]
Where:
- [tex]\( L\)[/tex] = Lower boundary of the median class = 60 (lower boundary of the 60-65 class)
- [tex]\( CF\)[/tex] = Cumulative frequency of the class preceding the median class = 10
- [tex]\( f\)[/tex] = Frequency of the median class = 10
- [tex]\( h\)[/tex] = Class interval size = 65 - 60 = 5
Substituting the values we get:
[tex]\[ \text{Median} = 60 + \left( \frac{20 - 10}{10} \right) \times 5 \][/tex]
[tex]\[ \text{Median} = 60 + \left( \frac{10}{10} \right) \times 5 \][/tex]
[tex]\[ \text{Median} = 60 + 1 \times 5 \][/tex]
[tex]\[ \text{Median} = 60 + 5 \][/tex]
[tex]\[ \text{Median} = 65 \][/tex]
Therefore, the median wage of the distribution is Rs 65.
