The weights (in kg) of 15 students are : 31, 35, 27, 29, 32, 43, 37, 41, 34, 28, 36, 44, 45, 42, 30. Find the median. If the weight 44 kg is replaced by 46 kg and 27 kg by 25 kg, find the new median.

Answer

To find the median of the given weights of 15 students, follow these steps:

### Step 1: List the Weights
The weights of the 15 students are:
31, 35, 27, 29, 32, 43, 37, 41, 34, 28, 36, 44, 45, 42, 30

### Step 2: Arrange the Weights in Ascending Order
Arrange the weights from the smallest to the largest:
27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 41, 42, 43, 44, 45

### Step 3: Determine the Number of Observations
Count the number of weights:
- There are 15 weights (n = 15).

### Step 4: Find the Median
Since the number of observations (n) is odd, the median is the middle value. The median can be found using the formula:
\[ \text{Median} = \frac{n + 1}{2} \text{th term} \]
Substituting n = 15:
\[ \text{Median} = \frac{15 + 1}{2} = \frac{16}{2} = 8 \]
The 8th term in the ordered list is 35.

### Conclusion for the First Part
Thus, the median of the weights is **35 kg**.

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### Step 5: Update the Weights
Now, we replace 44 kg with 46 kg and 27 kg with 25 kg:
- New weights: 25, 28, 29, 30, 31, 32, 34, 35, 36, 37, 41, 42, 43, 45, 46

### Step 6: Arrange the New Weights in Ascending Order
The updated weights in ascending order are:
25, 28, 29, 30, 31, 32, 34, 35, 36, 37, 41, 42, 43, 45, 46

### Step 7: Find the New Median
Again, the number of observations is still 15 (n = 15). Using the same median formula:
\[ \text{Median} = \frac{n + 1}{2} \text{th term} \]
Substituting n = 15:
\[ \text{Median} = \frac{15 + 1}{2} = \frac{16}{2} = 8 \]
The 8th term in the new ordered list is still 35.

### Conclusion for the Second Part
Thus, the new median after the replacements is also **35 kg**.

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