Mean of n observations `x_(1), x_(2), ....,x_(n)` is `bar(x)`. If an observation `x_(q)` is replaced by `x'_(q)` then the new mean is

Options

A. `bar(x)=x_(q)+x'_(q)`
B. `((n-1)bar(x)+x'_(q))/(n)`
C. `((n-1)bar(x)-x'_(q))/(n)`
D. `(n bar(x)-x_(q)+x'_(q))/(n)`

Answer

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

### Step-by-Step Solution:

1. **Understand the Mean Formula**:
The mean (average) of n observations \( x_1, x_2, \ldots, x_n \) is given by:
\[ \bar{x} = \frac{x_1 + x_2 + \ldots + x_n}{n} \]
This can also be expressed in summation notation as:
\[ \bar{x} = \frac{\sum_{i=1}^{n} x_i}{n} \]

2. **Calculate the Sum of Observations**:
From the mean formula, we can express the sum of the observations as:
\[ \sum_{i=1}^{n} x_i = n \cdot \bar{x} \]

3. **Replace the Observation**:
Suppose we replace the observation \( x_q \) with a new observation \( x'_q \). The new sum of observations will be:
\[ \sum_{i=1}^{n} x_i' = \sum_{i=1}^{n} x_i - x_q + x'_q \]
Substituting the expression for the sum:
\[ \sum_{i=1}^{n} x_i' = n \cdot \bar{x} - x_q + x'_q \]

4. **Calculate the New Mean**:
The new mean \( \bar{x}' \) after the replacement can be calculated as:
\[ \bar{x}' = \frac{\sum_{i=1}^{n} x_i'}{n} \]
Substituting the new sum into this formula gives:
\[ \bar{x}' = \frac{n \cdot \bar{x} - x_q + x'_q}{n} \]

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

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