Seven of the eight numbers in a distribution are 11, 16, 6, 10, 13, 11, 13
Given that the median of the distribution is 12, find the mean of the distribution.

Options

A. 12
B. 11
C. `11.6`
D. `12.2`

Answer

To solve the problem step-by-step, we will follow the reasoning laid out in the video transcript.

### Step 1: Identify the given numbers and the unknown
We are given seven numbers: 11, 16, 6, 10, 13, 11, and 13. We need to find the eighth number, which we will denote as \( x \).

### Step 2: Arrange the numbers in ascending order
To find the median, we need to arrange the numbers in ascending order. Including \( x \), the numbers will be:
- 6, 10, 11, 11, \( x \), 13, 13, 16

### Step 3: Understand the median condition
We know that the median of the distribution is 12. Since there are 8 numbers, the median will be the average of the 4th and 5th numbers in the ordered list.

### Step 4: Set up the equation for the median
The 4th and 5th numbers in the ordered list are:
- 4th number: 11
- 5th number: \( x \)

According to the median formula:
\[ \text{Median} = \frac{\text{4th number} + \text{5th number}}{2} = 12 \]
Substituting the known values:
\[ \frac{11 + x}{2} = 12 \]

### Step 5: Solve for \( x \)
Multiply both sides by 2 to eliminate the fraction:
\[ 11 + x = 24 \]
Now, subtract 11 from both sides:
\[ x = 24 - 11 = 13 \]

### Step 6: List all the numbers including \( x \)
Now that we have found \( x = 13 \), the complete list of numbers is:
- 6, 10, 11, 11, 13, 13, 13, 16

### Step 7: Calculate the mean
To find the mean, we sum all the numbers and divide by the total number of observations (which is 8):
\[ \text{Mean} = \frac{6 + 10 + 11 + 11 + 13 + 13 + 13 + 16}{8} \]

### Step 8: Perform the addition
Calculating the sum:
\[ 6 + 10 + 11 + 11 + 13 + 13 + 13 + 16 = 93 \]

### Step 9: Divide by the number of observations
Now divide the sum by 8:
\[ \text{Mean} = \frac{93}{8} = 11.625 \]

### Final Answer
The mean of the distribution is \( 11.625 \).

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