The following observations have been arranged in acending order. If the median of the data is 78, find the value of x.
44, 47, 63, 65, `x+13`, 87, 93, 99, 110.

Answer

To find the value of \( x \) in the given observations, we will follow these steps:

1. **Identify the number of observations**: The given observations are:
\[ 44, 47, 63, 65, x + 13, 87, 93, 99, 110 \]
There are 9 observations in total.

2. **Determine the position of the median**: Since there are 9 observations (an odd number), the median is the middle value. The position of the median can be calculated using the formula:
\[ \text{Median position} = \frac{n + 1}{2} \]
where \( n \) is the number of observations. Thus,
\[ \text{Median position} = \frac{9 + 1}{2} = \frac{10}{2} = 5 \]
This means the median is the 5th term in the ordered list.

3. **Set up the equation for the median**: The 5th term in the list is \( x + 13 \). According to the problem, the median is given as 78. Therefore, we can set up the equation:
\[ x + 13 = 78 \]

4. **Solve for \( x \)**: To find \( x \), we will isolate it in the equation:
\[ x = 78 - 13 \]
\[ x = 65 \]

5. **Conclusion**: The value of \( x \) is 65.