If `barx` represents the mean of n observations `x_(1), x_(2),………., x_(n)`, then values of `Sigma_(i=1)^(n) (x_(i)-barx)`

Options

A. `-1`
B. 0
C. 1
D. `n-1`

Explanation

To solve the question, we need to evaluate the expression \( \Sigma_{i=1}^{n} (x_{i} - \bar{x}) \), where \( \bar{x} \) is the mean of the observations \( x_{1}, x_{2}, \ldots, x_{n} \).

### Step-by-Step Solution:

1. **Understanding the Mean**:
The mean \( \bar{x} \) of the observations \( x_{1}, x_{2}, \ldots, x_{n} \) is defined as:
\[ \bar{x} = \frac{\Sigma_{i=1}^{n} x_{i}}{n} \]

2. **Substituting the Mean into the Summation**:
We want to evaluate:
\[ \Sigma_{i=1}^{n} (x_{i} - \bar{x}) \]
We can rewrite this as:
\[ \Sigma_{i=1}^{n} x_{i} - \Sigma_{i=1}^{n} \bar{x} \]

3. **Calculating Each Part**:
- The first part \( \Sigma_{i=1}^{n} x_{i} \) is simply the sum of all observations.
- The second part \( \Sigma_{i=1}^{n} \bar{x} \) can be simplified. Since \( \bar{x} \) is a constant (the mean), we have:
\[ \Sigma_{i=1}^{n} \bar{x} = n \cdot \bar{x} \]

4. **Substituting Back**:
Now substituting these back into our expression, we get:
\[ \Sigma_{i=1}^{n} (x_{i} - \bar{x}) = \Sigma_{i=1}^{n} x_{i} - n \cdot \bar{x} \]

5. **Using the Definition of Mean**:
From the definition of the mean, we know that:
\[ n \cdot \bar{x} = \Sigma_{i=1}^{n} x_{i} \]
Therefore, we can substitute this into our expression:
\[ \Sigma_{i=1}^{n} (x_{i} - \bar{x}) = \Sigma_{i=1}^{n} x_{i} - \Sigma_{i=1}^{n} x_{i} = 0 \]

6. **Conclusion**:
Thus, the value of \( \Sigma_{i=1}^{n} (x_{i} - \bar{x}) \) is:
\[ 0 \]

### Final Answer:
The value of \( \Sigma_{i=1}^{n} (x_{i} - \bar{x}) \) is **0**.