If the mean of 9, 10, 15, `x`, 6, 8 and 12 is 11 and is 11. The median of the observations is

Options

A. 4
B. 10
C. 13
D. 5

Answer

To find the median of the given data set, we first need to determine the value of \( x \) using the information provided about the mean. Let's break down the solution step by step:

### Step 1: Understand the Mean Formula
The mean is calculated using the formula:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]
Given that the mean is 11 and there are 7 observations (9, 10, 15, \( x \), 6, 8, and 12), we can set up the equation.

### Step 2: Set Up the Equation
We can express the sum of the observations as:
\[ 9 + 10 + 15 + x + 6 + 8 + 12 \]
The number of observations is 7. Therefore, we can write:
\[ \frac{9 + 10 + 15 + x + 6 + 8 + 12}{7} = 11 \]

### Step 3: Calculate the Total Sum
First, calculate the sum of the known numbers:
\[ 9 + 10 = 19 \]
\[ 19 + 15 = 34 \]
\[ 34 + 6 = 40 \]
\[ 40 + 8 = 48 \]
\[ 48 + 12 = 60 \]
So, the sum of the known observations is 60. Thus, we can rewrite the equation as:
\[ \frac{60 + x}{7} = 11 \]

### Step 4: Solve for \( x \)
To eliminate the fraction, multiply both sides by 7:
\[ 60 + x = 77 \]
Now, isolate \( x \) by subtracting 60 from both sides:
\[ x = 77 - 60 \]
\[ x = 17 \]

### Step 5: List All Observations
Now that we have \( x \), we can list all the observations:
\[ 9, 10, 15, 17, 6, 8, 12 \]

### Step 6: Arrange the Observations in Ascending Order
Next, we need to arrange the observations in ascending order:
\[ 6, 8, 9, 10, 12, 15, 17 \]

### Step 7: Find the Median
Since there are 7 observations (an odd number), the median is the middle value. The median can be found using the formula:
\[ \text{Median} = \text{(n + 1) / 2} \text{th observation} \]
Where \( n \) is the number of observations. Here, \( n = 7 \):
\[ \text{Median} = \frac{7 + 1}{2} = 4 \text{th observation} \]
Counting the observations:
1. 6
2. 8
3. 9
4. 10 (this is the 4th observation)

Thus, the median is:
\[ \text{Median} = 10 \]

### Final Answer
The median of the observations is **10**.

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