The median of the following observation 11, 12, 14, (x - 2), (x + 4), (x + 9), 32, 38, 47 arranged in ascending order is 24. Find the value of x and hence find the mean.

Answer

To solve the problem step by step, we will first find the value of \( x \) using the information about the median, and then we will calculate the mean of the observations.

### Step 1: Identify the Observations
The given observations are:
\[ 11, 12, 14, (x - 2), (x + 4), (x + 9), 32, 38, 47 \]

### Step 2: Count the Number of Observations
We count the total number of observations:
- There are 9 observations in total.

### Step 3: Determine the Median Position
Since the number of observations \( n = 9 \) (which is odd), the median is the value at the position:
\[ \text{Median Position} = \frac{n + 1}{2} = \frac{9 + 1}{2} = 5 \]
Thus, the median is the 5th observation when arranged in ascending order.

### Step 4: Identify the 5th Observation
Now, we need to find the 5th observation in terms of \( x \):
- The 5th observation is \( (x + 4) \).

### Step 5: Set Up the Median Equation
We know from the problem that the median is given as 24:
\[ x + 4 = 24 \]

### Step 6: Solve for \( x \)
To find \( x \), we rearrange the equation:
\[ x = 24 - 4 = 20 \]

### Step 7: Substitute \( x \) Back into the Observations
Now we substitute \( x = 20 \) back into the observations:
- \( x - 2 = 20 - 2 = 18 \)
- \( x + 4 = 20 + 4 = 24 \)
- \( x + 9 = 20 + 9 = 29 \)

So the updated observations are:
\[ 11, 12, 14, 18, 24, 29, 32, 38, 47 \]

### Step 8: Calculate the Mean
To find the mean, we use the formula:
\[ \text{Mean} = \frac{\text{Sum of Observations}}{\text{Total Number of Observations}} \]
Calculating the sum:
\[ \text{Sum} = 11 + 12 + 14 + 18 + 24 + 29 + 32 + 38 + 47 \]
Calculating this step by step:
- \( 11 + 12 = 23 \)
- \( 23 + 14 = 37 \)
- \( 37 + 18 = 55 \)
- \( 55 + 24 = 79 \)
- \( 79 + 29 = 108 \)
- \( 108 + 32 = 140 \)
- \( 140 + 38 = 178 \)
- \( 178 + 47 = 225 \)

Thus, the sum of the observations is \( 225 \).

Now, we calculate the mean:
\[ \text{Mean} = \frac{225}{9} = 25 \]

### Final Answers
- The value of \( x \) is \( 20 \).
- The mean of the observations is \( 25 \).