Mean of n observations `x_1,x_2`,….,`x_n` is `barx`. If an observation `x_q` is replaced by `x'_q`, then the new mean is :

Options

A. `barx-x_q+x'_q`
B. `((n-1)barx+x'_q)/n`
C. `((n-1)barx-x'_q)/n`
D. `(nbarx-x_q+x'_q)/n`

Answer

To find the new mean after replacing an observation in a set of observations, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Mean**: The mean of n observations \( x_1, x_2, \ldots, x_n \) is given by:
\[ \bar{x} = \frac{x_1 + x_2 + \ldots + x_n}{n} \]

2. **Express the Sum of Observations**: The sum of all observations can be expressed as:
\[ \sum_{i=1}^{n} x_i = n \cdot \bar{x} \]

3. **Replace an Observation**: Suppose we replace the observation \( x_q \) with a new observation \( x'_q \). The new sum of observations becomes:
\[ \text{New Sum} = \sum_{i=1}^{n} x_i - x_q + x'_q \]

4. **Substitute the Old Sum**: We can substitute the old sum into the equation:
\[ \text{New Sum} = n \cdot \bar{x} - x_q + x'_q \]

5. **Calculate the New Mean**: The new mean \( \bar{x}' \) can be calculated as follows:
\[ \bar{x}' = \frac{\text{New Sum}}{n} = \frac{n \cdot \bar{x} - x_q + x'_q}{n} \]

6. **Simplify the Expression**: This simplifies to:
\[ \bar{x}' = \bar{x} + \frac{x'_q - x_q}{n} \]

### Final Result:
Thus, the new mean after replacing \( x_q \) with \( x'_q \) is:
\[ \bar{x}' = \bar{x} + \frac{x'_q - x_q}{n} \]