The mean and median of 100 items are 50 and 52, respectively. The value of largest item is 100. It was later found that, it is 110 not 100. The true mean and median are

Options

A. `50.10,51.5`
B. `50.10,52`
C. `50,52`
D. None of these

Answer

To solve the problem, we need to find the new mean and median after correcting the largest item from 100 to 110.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Mean of 100 items = 50
- Median of 100 items = 52
- Largest item initially reported = 100
- Corrected largest item = 110

2. **Calculate the Total Sum of the Items:**
- Since the mean is given by the formula:
\[ \text{Mean} = \frac{\text{Total Sum}}{\text{Number of Items}} \]
- We can rearrange this to find the total sum:
\[ \text{Total Sum} = \text{Mean} \times \text{Number of Items} = 50 \times 100 = 5000 \]

3. **Adjust the Total Sum for the Corrected Largest Item:**
- The initially reported largest item was 100, but it is actually 110. Therefore, we need to adjust the total sum:
\[ \text{New Total Sum} = \text{Old Total Sum} - \text{Old Largest Item} + \text{New Largest Item} \]
- Substituting the values:
\[ \text{New Total Sum} = 5000 - 100 + 110 = 5000 + 10 = 5010 \]

4. **Calculate the New Mean:**
- Now, we can find the new mean using the new total sum:
\[ \text{New Mean} = \frac{\text{New Total Sum}}{\text{Number of Items}} = \frac{5010}{100} = 50.1 \]

5. **Determine the New Median:**
- The median is the middle value when the data is arranged in order. Since the largest value has changed from 100 to 110, and it is still the largest value in the dataset, it does not affect the position of the median.
- The median remains the same at 52, as it is the average of the 50th and 51st items in the ordered list.

### Final Results:
- **New Mean:** 50.1
- **New Median:** 52