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Sagot :
To compute the median of the given frequency distribution, we will follow the steps of finding the median for a grouped frequency distribution.
### Step 1: Calculate the cumulative frequency
We need to calculate the cumulative frequency for each class interval. Cumulative frequency is the sum of the frequencies of all classes up to and including the current class.
Let's tabulate the data first:
| Marks | Students (Frequency) | Cumulative Frequency (C.F.) |
|--------|----------------------|-----------------------------|
| 0-10 | 15 | 15 |
| 10-20 | 20 | 15 + 20 = 35 |
| 20-30 | 25 | 35 + 25 = 60 |
| 30-40 | 24 | 60 + 24 = 84 |
| 40-50 | 12 | 84 + 12 = 96 |
| 50-60 | 31 | 96 + 31 = 127 |
| 60-70 | 71 | 127 + 71 = 198 |
| 70-80 | 52 | 198 + 52 = 250 |
### Step 2: Determine the median class
The total number of students (n) is 250. The median is located in the class where the cumulative frequency just exceeds [tex]\( \frac{n}{2} \)[/tex] (i.e., 125).
From the cumulative frequency table, we can see that 127 is the first cumulative frequency to exceed 125. Hence, the median class is [tex]\( 50-60 \)[/tex].
### Step 3: Apply the median formula
The formula for finding the median in a grouped frequency distribution is:
[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 number of students.
- [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.
For the median class [tex]\( 50-60 \)[/tex]:
- [tex]\( L = 50 \)[/tex]
- [tex]\( n = 250 \)[/tex]
- [tex]\( CF = 96 \)[/tex] (cumulative frequency of the class preceding 50-60)
- [tex]\( f = 31 \)[/tex]
- [tex]\( h = 10 \)[/tex] (difference between the upper and lower boundaries of any class)
### Step 4: Calculate the median
Plugging the values into the formula:
[tex]\[ \text{Median} = 50 + \left( \frac{125 - 96}{31} \right) \times 10 \][/tex]
[tex]\[ \text{Median} = 50 + \left( \frac{29}{31} \right) \times 10 \][/tex]
[tex]\[ \text{Median} = 50 + \left( 0.935 \right) \times 10 \][/tex]
[tex]\[ \text{Median} = 50 + 9.35 \][/tex]
[tex]\[ \text{Median} = 59.35 \][/tex]
### Conclusion
The median mark obtained by the students is 59.35.
### Step 1: Calculate the cumulative frequency
We need to calculate the cumulative frequency for each class interval. Cumulative frequency is the sum of the frequencies of all classes up to and including the current class.
Let's tabulate the data first:
| Marks | Students (Frequency) | Cumulative Frequency (C.F.) |
|--------|----------------------|-----------------------------|
| 0-10 | 15 | 15 |
| 10-20 | 20 | 15 + 20 = 35 |
| 20-30 | 25 | 35 + 25 = 60 |
| 30-40 | 24 | 60 + 24 = 84 |
| 40-50 | 12 | 84 + 12 = 96 |
| 50-60 | 31 | 96 + 31 = 127 |
| 60-70 | 71 | 127 + 71 = 198 |
| 70-80 | 52 | 198 + 52 = 250 |
### Step 2: Determine the median class
The total number of students (n) is 250. The median is located in the class where the cumulative frequency just exceeds [tex]\( \frac{n}{2} \)[/tex] (i.e., 125).
From the cumulative frequency table, we can see that 127 is the first cumulative frequency to exceed 125. Hence, the median class is [tex]\( 50-60 \)[/tex].
### Step 3: Apply the median formula
The formula for finding the median in a grouped frequency distribution is:
[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 number of students.
- [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.
For the median class [tex]\( 50-60 \)[/tex]:
- [tex]\( L = 50 \)[/tex]
- [tex]\( n = 250 \)[/tex]
- [tex]\( CF = 96 \)[/tex] (cumulative frequency of the class preceding 50-60)
- [tex]\( f = 31 \)[/tex]
- [tex]\( h = 10 \)[/tex] (difference between the upper and lower boundaries of any class)
### Step 4: Calculate the median
Plugging the values into the formula:
[tex]\[ \text{Median} = 50 + \left( \frac{125 - 96}{31} \right) \times 10 \][/tex]
[tex]\[ \text{Median} = 50 + \left( \frac{29}{31} \right) \times 10 \][/tex]
[tex]\[ \text{Median} = 50 + \left( 0.935 \right) \times 10 \][/tex]
[tex]\[ \text{Median} = 50 + 9.35 \][/tex]
[tex]\[ \text{Median} = 59.35 \][/tex]
### Conclusion
The median mark obtained by the students is 59.35.
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