The AM of n observations is M. If the sum of `n-4` observations is a, then the mean of remaining 4 observations is

Options

A. `(nM-a)/(4)`
B. `(nM+a)/(2)`
C. `(nM-a)/(2)`
D. `nM+a`

Answer

To solve the problem, we need to find the mean of the remaining 4 observations given the arithmetic mean of n observations and the sum of n-4 observations.

### Step-by-Step Solution:

1. **Understanding the Arithmetic Mean (AM)**:
The arithmetic mean (AM) of n observations is given by the formula:
\[ M = \frac{\text{Sum of all observations}}{n} \]
Therefore, if we denote the sum of all observations as \( S \), we can write:
\[ S = nM \]

2. **Sum of n - 4 Observations**:
We are given that the sum of \( n - 4 \) observations is \( a \). This means:
\[ \text{Sum of } (n - 4) \text{ observations} = a \]

3. **Finding the Sum of the Remaining 4 Observations**:
The sum of the remaining 4 observations can be found by subtracting the sum of the \( n - 4 \) observations from the total sum \( S \):
\[ \text{Sum of remaining 4 observations} = S - a \]
Substituting \( S = nM \):
\[ \text{Sum of remaining 4 observations} = nM - a \]

4. **Calculating the Mean of the Remaining 4 Observations**:
The mean of the remaining 4 observations is given by:
\[ \text{Mean of remaining 4 observations} = \frac{\text{Sum of remaining 4 observations}}{4} \]
Substituting the expression we found for the sum:
\[ \text{Mean of remaining 4 observations} = \frac{nM - a}{4} \]

### Final Answer:
Thus, the mean of the remaining 4 observations is:
\[ \frac{nM - a}{4} \]