If `barx` is the mean of `x_1, x_2, x_3, ... .. , x_n`, then `sum_(i=1)^(n)(x_i-barx)=`

Options

A. `23 /25`
B. 0
C. `28/25`
D. `4/5`

Answer

To solve the problem, we need to evaluate the expression:

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

where \(\bar{x}\) is the mean of the observations \(x_1, x_2, x_3, \ldots, x_n\).

### Step-by-Step Solution:

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

2. **Substitute the Mean into the Summation**:
We can rewrite the summation:
\[ \sum_{i=1}^{n} (x_i - \bar{x}) = \sum_{i=1}^{n} x_i - \sum_{i=1}^{n} \bar{x} \]
Since \(\bar{x}\) is a constant (the mean), we can simplify the second term:
\[ \sum_{i=1}^{n} \bar{x} = n \cdot \bar{x} \]

3. **Combine the Terms**:
Now, substituting back into the equation gives:
\[ \sum_{i=1}^{n} (x_i - \bar{x}) = \sum_{i=1}^{n} x_i - n \cdot \bar{x} \]

4. **Substitute the Mean**:
We know that \(\sum_{i=1}^{n} x_i = n \cdot \bar{x}\) (by the definition of the mean), so we can substitute this into our equation:
\[ \sum_{i=1}^{n} (x_i - \bar{x}) = n \cdot \bar{x} - n \cdot \bar{x} \]

5. **Final Result**:
Therefore, we find that:
\[ \sum_{i=1}^{n} (x_i - \bar{x}) = 0 \]

### Conclusion:
The final answer is:
\[ \sum_{i=1}^{n} (x_i - \bar{x}) = 0 \]