If `barx` is the mean of n observations `x_(1),x_(2),x_(3)……x_(n)`, then the value of `sum_(i=1)^(n)(x_(i)-barx)` is (i) `-1` (ii) 0 (iii) 1 (iv) n-1

Options

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

Answer

To solve the problem, we need to find the value of the expression:

\[ \sum_{i=1}^{n} (x_i - \bar{x}) \]

where \(\bar{x}\) is the mean of the observations \(x_1, x_2, \ldots, x_n\).

### Step-by-Step Solution:

1. **Understand the Mean**:
The mean \(\bar{x}\) of \(n\) observations \(x_1, x_2, \ldots, x_n\) is defined as:
\[ \bar{x} = \frac{x_1 + x_2 + x_3 + \ldots + x_n}{n} \]

2. **Express the Summation**:
We need to evaluate the summation:
\[ \sum_{i=1}^{n} (x_i - \bar{x}) = (x_1 - \bar{x}) + (x_2 - \bar{x}) + (x_3 - \bar{x}) + \ldots + (x_n - \bar{x}) \]

3. **Rewrite the Summation**:
This can be rewritten as:
\[ \sum_{i=1}^{n} x_i - \sum_{i=1}^{n} \bar{x} \]
Since \(\bar{x}\) is a constant (the mean), we can express the second summation as:
\[ \sum_{i=1}^{n} \bar{x} = n \cdot \bar{x} \]

4. **Substitute the Mean**:
Now, substituting the expression for \(\bar{x}\):
\[ \sum_{i=1}^{n} (x_i - \bar{x}) = \sum_{i=1}^{n} x_i - n \cdot \bar{x} \]

5. **Use the Definition of Mean**:
From the definition of the mean, we know:
\[ \sum_{i=1}^{n} x_i = n \cdot \bar{x} \]
Therefore, substituting this into our equation gives:
\[ \sum_{i=1}^{n} (x_i - \bar{x}) = n \cdot \bar{x} - n \cdot \bar{x} = 0 \]

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

### Final Answer:
The correct option is (ii) 0.