The mean of 16 observations is 16 . If one observation 16 is deleted and three observations 5 , 5 and 6 are included , then find the mean of the final observations .

Options

A. 16
B. 15.5
C. 13.5
D. none of these

Explanation

To find the mean of the final observations after deleting one observation and adding three new ones, we can follow these steps:

### Step 1: Calculate the sum of the original observations
Given that the mean of 16 observations is 16, we can calculate the sum of these observations using the formula for the mean:

\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]

Rearranging this gives us:

\[ \text{Sum of observations} = \text{Mean} \times \text{Number of observations} \]

Substituting the values:

\[ \text{Sum of observations} = 16 \times 16 = 256 \]

### Step 2: Adjust the sum after deleting one observation
We need to delete one observation, which is 16. Therefore, we subtract 16 from the total sum:

\[ \text{New sum} = 256 - 16 = 240 \]

### Step 3: Add the new observations
Now, we add the three new observations: 5, 5, and 6. The sum of these new observations is:

\[ \text{Sum of new observations} = 5 + 5 + 6 = 16 \]

Now, we add this to the new sum:

\[ \text{Final sum} = 240 + 16 = 256 \]

### Step 4: Calculate the new number of observations
After deleting one observation and adding three new ones, the total number of observations becomes:

\[ \text{New number of observations} = 16 - 1 + 3 = 18 \]

### Step 5: Calculate the mean of the final observations
Finally, we calculate the mean of the final observations using the new sum and the new number of observations:

\[ \text{Mean of final observations} = \frac{\text{Final sum}}{\text{New number of observations}} = \frac{256}{18} \]

Calculating this gives:

\[ \text{Mean of final observations} \approx 14.22 \]

### Final Answer
The mean of the final observations is approximately **14.22**.

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