Answer :
To calculate the median of a grouped frequency distribution, we follow several steps. Let's go through the process using the given data:
### Step 1: Identify the given data
The data consists of the "Marks (Above)" and corresponding "Frequency (Freq)":
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline \text{Marks (Above)} & 0 & 15 & 30 & 45 & 60 & 75 & 90 & 105 \\ \hline \text{Frequency} & 150 & 140 & 100 & 80 & 70 & 30 & 14 & 0 \\ \hline \end{array} \][/tex]
### Step 2: Convert Frequencies to Less Than Cumulative Frequencies
To find the median, we need the cumulative frequencies running in reverse order, transforming "frequency over" into "frequency less than."
Calculating cumulative frequencies from the last class to the first:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline \text{Marks (Above)} & 0 & 15 & 30 & 45 & 60 & 75 & 90 & 105 \\ \hline \text{Frequency} & 150 & 140 & 100 & 80 & 70 & 30 & 14 & 0 \\ \hline \text{Cumulative Frequency (Less Than)} & 584 & 434 & 294 & 194 & 114 & 44 & 14 & 0 \\ \hline \end{array} \][/tex]
### Step 3: Find the Total Number of Observations (N)
The total number of observations, [tex]\(N\)[/tex], is the sum of all frequencies:
[tex]\[ N = 150 + 140 + 100 + 80 + 70 + 30 + 14 + 0 = 584 \][/tex]
### Step 4: Determine the Median Class
The median is located in the class whose cumulative frequency just exceeds [tex]\( \frac{N}{2} \)[/tex].
Calculate [tex]\( \frac{N}{2} \)[/tex]:
[tex]\[ \frac{N}{2} = \frac{584}{2} = 292 \][/tex]
Locate the median class by finding the first cumulative frequency greater than 292. From our cumulative frequencies:
[tex]\[ 584, 434, 294, 194, 114, 44, 14, 0 \][/tex]
The first cumulative frequency exceeding 292 is 434, corresponding to the class "0 to 15". Hence, the median class is [tex]\( 0-15 \)[/tex].
### Step 5: Apply the Median Formula for Grouped Data
The median formula for grouped data is:
[tex]\[ \text{Median} = L + \left(\frac{\frac{N}{2} - F}{f}\right) \cdot h \][/tex]
Where:
- [tex]\( L \)[/tex] is the lower boundary of the median class
- [tex]\( F \)[/tex] is the cumulative frequency of the class before the median class
- [tex]\( f \)[/tex] is the frequency of the median class
- [tex]\( h \)[/tex] is the class width
For our median class [tex]\( 0-15 \)[/tex]:
- The class boundaries are [tex]\(0\)[/tex] (lower boundary) and [tex]\(15\)[/tex] (upper boundary), so [tex]\( L = 0 \)[/tex]
- The cumulative frequency of the class before the median class is 434 (the class itself), [tex]\( F = 434 - 150 = 434 \)[/tex]
- Frequency of the median class ([tex]\( f \)[/tex]) is 150
- Class width ([tex]\( h \)[/tex]) is [tex]\( 15 - 0 = 15 \)[/tex]
Substitute these values into the formula:
[tex]\[ \text{Median} = 0 + \left(\frac{292 - 434}{150}\right) \cdot 15 \][/tex]
### Step 6: Calculate the Median
Compute the value step-by-step:
[tex]\[ \frac{292 - 434}{150} = \frac{-142}{150} = -0.9467 \][/tex]
Multiplying by the class width:
[tex]\[ -0.9467 \times 15 = -14.2 \][/tex]
Add to the lower boundary of the median class:
[tex]\[ \text{Median} = 0 - 14.2 = -14.2 \][/tex]
Thus, the median for the given data is:
[tex]\[ \boxed{-14.2} \][/tex]
### Step 1: Identify the given data
The data consists of the "Marks (Above)" and corresponding "Frequency (Freq)":
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline \text{Marks (Above)} & 0 & 15 & 30 & 45 & 60 & 75 & 90 & 105 \\ \hline \text{Frequency} & 150 & 140 & 100 & 80 & 70 & 30 & 14 & 0 \\ \hline \end{array} \][/tex]
### Step 2: Convert Frequencies to Less Than Cumulative Frequencies
To find the median, we need the cumulative frequencies running in reverse order, transforming "frequency over" into "frequency less than."
Calculating cumulative frequencies from the last class to the first:
[tex]\[ \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline \text{Marks (Above)} & 0 & 15 & 30 & 45 & 60 & 75 & 90 & 105 \\ \hline \text{Frequency} & 150 & 140 & 100 & 80 & 70 & 30 & 14 & 0 \\ \hline \text{Cumulative Frequency (Less Than)} & 584 & 434 & 294 & 194 & 114 & 44 & 14 & 0 \\ \hline \end{array} \][/tex]
### Step 3: Find the Total Number of Observations (N)
The total number of observations, [tex]\(N\)[/tex], is the sum of all frequencies:
[tex]\[ N = 150 + 140 + 100 + 80 + 70 + 30 + 14 + 0 = 584 \][/tex]
### Step 4: Determine the Median Class
The median is located in the class whose cumulative frequency just exceeds [tex]\( \frac{N}{2} \)[/tex].
Calculate [tex]\( \frac{N}{2} \)[/tex]:
[tex]\[ \frac{N}{2} = \frac{584}{2} = 292 \][/tex]
Locate the median class by finding the first cumulative frequency greater than 292. From our cumulative frequencies:
[tex]\[ 584, 434, 294, 194, 114, 44, 14, 0 \][/tex]
The first cumulative frequency exceeding 292 is 434, corresponding to the class "0 to 15". Hence, the median class is [tex]\( 0-15 \)[/tex].
### Step 5: Apply the Median Formula for Grouped Data
The median formula for grouped data is:
[tex]\[ \text{Median} = L + \left(\frac{\frac{N}{2} - F}{f}\right) \cdot h \][/tex]
Where:
- [tex]\( L \)[/tex] is the lower boundary of the median class
- [tex]\( F \)[/tex] is the cumulative frequency of the class before the median class
- [tex]\( f \)[/tex] is the frequency of the median class
- [tex]\( h \)[/tex] is the class width
For our median class [tex]\( 0-15 \)[/tex]:
- The class boundaries are [tex]\(0\)[/tex] (lower boundary) and [tex]\(15\)[/tex] (upper boundary), so [tex]\( L = 0 \)[/tex]
- The cumulative frequency of the class before the median class is 434 (the class itself), [tex]\( F = 434 - 150 = 434 \)[/tex]
- Frequency of the median class ([tex]\( f \)[/tex]) is 150
- Class width ([tex]\( h \)[/tex]) is [tex]\( 15 - 0 = 15 \)[/tex]
Substitute these values into the formula:
[tex]\[ \text{Median} = 0 + \left(\frac{292 - 434}{150}\right) \cdot 15 \][/tex]
### Step 6: Calculate the Median
Compute the value step-by-step:
[tex]\[ \frac{292 - 434}{150} = \frac{-142}{150} = -0.9467 \][/tex]
Multiplying by the class width:
[tex]\[ -0.9467 \times 15 = -14.2 \][/tex]
Add to the lower boundary of the median class:
[tex]\[ \text{Median} = 0 - 14.2 = -14.2 \][/tex]
Thus, the median for the given data is:
[tex]\[ \boxed{-14.2} \][/tex]
