The distribution below gives the weights of 30 students of a class. Find the median weight of the students. Weight (in kg) 40 - 45 45 - 50 50 - 55 55 - 60 60 - 65 65 - 70 70 - 75 Number of students 2 3 8 6 6 3 2
Step-by-Step Solution
Key Concept: For grouped data, the median is found using the formula \(\displaystyle \text{Median}=L+\left(\frac{\frac{N}{2}-c_f}{f}\right)h\), where \(L\) is the lower class boundary of the median class, \(N\) is the total frequency, \(c_f\) is the cumulative frequency of the class preceding the median class, \(f\) is the frequency of the median class, and \(h\) is the class width.
1. Total number of students: \(N = 30\).
2. Find \(\frac{N}{2}\): \(\frac{N}{2}=\frac{30}{2}=15\).
3. Prepare cumulative frequency table:
| Weight (kg) | Frequency (f) | Cumulative Frequency (c_f) |
|-------------|---------------|----------------------------|
| 40 – 45 | 2 | 2 |
| 45 – 50 | 3 | 2 + 3 = 5 |
| 50 – 55 | 8 | 5 + 8 = 13 |
| 55 – 60 | 6 | 13 + 6 = 19 |
| 60 – 65 | 6 | 19 + 6 = 25 |
| 65 – 70 | 3 | 25 + 3 = 28 |
| 70 – 75 | 2 | 28 + 2 = 30 |
4. Locate the median class: The median class is the first class whose cumulative frequency \(\ge 15\). Here, \(c_f = 19\) for the class 55 – 60 kg, so this is the median class.
5. Identify required values:
- Lower class boundary \(L = 55\) (since the class is 55–60 kg, the lower boundary is 55).
- Class width \(h = 60 - 55 = 5\) kg.
- Frequency of median class \(f = 6\).
- Cumulative frequency of the preceding class \(c_f = 13\).
6. Apply the median formula:
\[\text{Median}= L + \left(\frac{\frac{N}{2} - c_f}{f}\right) h\]
Substituting the values:
\[\text{Median}= 55 + \left(\frac{15 - 13}{6}\right) \times 5\]
\[\text{Median}= 55 + \left(\frac{2}{6}\right) \times 5\]
\[\text{Median}= 55 + \frac{1}{3} \times 5\]
\[\text{Median}= 55 + \frac{5}{3}\]
\[\text{Median}= 55 + 1.666\ldots \approx 56.7\text{ kg}\]
7. Result: The median weight of the 30 students is approximately 56.7 kg.
Correct Answer: ≈ 56.7 kg