In a data 10 numbers are arranged in ascending order. If the 8th entry is increased by 6 the median increases by

Options

A. 6
B. 3
C. 2
D. 0

Answer

To solve the problem, we need to understand how the median is affected when the 8th entry in a sorted list of 10 numbers is increased by 6.

### Step-by-Step Solution:

1. **Understanding the Median**:
- For a dataset with an even number of observations (in this case, 10), the median is calculated as the average of the two middle numbers.
- Specifically, for 10 numbers arranged in ascending order, the median will be the average of the 5th and 6th entries.

2. **Identifying the Entries**:
- Let’s denote the 10 numbers as \( a_1, a_2, a_3, a_4, a_5, a_6, a_7, a_8, a_9, a_{10} \) where \( a_1 \leq a_2 \leq ... \leq a_{10} \).
- The 5th and 6th entries are \( a_5 \) and \( a_6 \).

3. **Calculating the Initial Median**:
- The initial median is given by:
\[ \text{Median} = \frac{a_5 + a_6}{2} \]

4. **Increasing the 8th Entry**:
- Now, we increase the 8th entry \( a_8 \) by 6. This means we now have \( a_8' = a_8 + 6 \).
- Since the numbers are arranged in ascending order, we need to check if this change affects the positions of \( a_5 \) and \( a_6 \).

5. **Effect on the Median**:
- The 8th entry \( a_8 \) is greater than both \( a_5 \) and \( a_6 \) because the numbers are in ascending order.
- Increasing \( a_8 \) does not affect the values of \( a_5 \) and \( a_6 \) since they are still the same.
- Therefore, the median remains:
\[ \text{New Median} = \frac{a_5 + a_6}{2} \]

6. **Conclusion**:
- Since the median does not change after increasing the 8th entry, the increase in the median is:
\[ \text{Increase in Median} = 0 \]

### Final Answer:
The median increases by **0**.