If the mean of the observation a , a + 6 , a + 2 , a + 8 and a + 4 is 11 . Find .
the median.

Answer

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

### Step 1: Set up the equation for the mean.
The observations are:
- \( a \)
- \( a + 6 \)
- \( a + 2 \)
- \( a + 8 \)
- \( a + 4 \)

The mean of these observations is given by the formula:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]
Here, the number of observations is 5.

### Step 2: Calculate the sum of the observations.
The sum of the observations can be calculated as follows:
\[ \text{Sum} = a + (a + 6) + (a + 2) + (a + 8) + (a + 4) \]
Combining like terms:
\[ \text{Sum} = a + a + 6 + a + 2 + a + 8 + a + 4 = 5a + 20 \]

### Step 3: Set the mean equal to 11.
According to the problem, the mean is equal to 11:
\[ \frac{5a + 20}{5} = 11 \]

### Step 4: Solve for \( a \).
Multiply both sides by 5 to eliminate the denominator:
\[ 5a + 20 = 55 \]
Now, subtract 20 from both sides:
\[ 5a = 35 \]
Finally, divide by 5:
\[ a = 7 \]

### Step 5: Substitute \( a \) back into the observations.
Now we can find the actual observations by substituting \( a = 7 \):
- \( a = 7 \)
- \( a + 6 = 7 + 6 = 13 \)
- \( a + 2 = 7 + 2 = 9 \)
- \( a + 8 = 7 + 8 = 15 \)
- \( a + 4 = 7 + 4 = 11 \)

So the observations are:
\[ 7, 13, 9, 15, 11 \]

### Step 6: Arrange the observations in ascending order.
The ordered observations are:
\[ 7, 9, 11, 13, 15 \]

### Step 7: Find the median.
Since there are 5 observations (an odd number), the median is the middle value. The median can be found using the formula:
\[ \text{Median} = \text{the } \left(\frac{n + 1}{2}\right)^{th} \text{ term} \]
where \( n \) is the number of observations. Here, \( n = 5 \):
\[ \text{Median} = \left(\frac{5 + 1}{2}\right)^{th} \text{ term} = \text{3rd term} \]
The 3rd term in the ordered list \( 7, 9, 11, 13, 15 \) is:
\[ 11 \]

### Conclusion
Thus, the median of the observations is:
\[ \text{Median} = 11 \]