If `sum_(i=1)^(5) (x_(i) - 6) = 5` and `sum_(i=1)^(5)(x_(i)-6)^(2) = 25`, then the standard deviation of observations

Answer

To find the standard deviation of the observations given the conditions, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Information**:
- We have two equations:
\[ \sum_{i=1}^{5} (x_i - 6) = 5 \]
\[ \sum_{i=1}^{5} (x_i - 6)^2 = 25 \]

2. **Determine the Number of Observations (n)**:
- The number of observations \( n \) is given as 5 (since \( i \) ranges from 1 to 5).

3. **Calculate the Mean (a)**:
- The mean \( a \) is given as 6 (since we are considering \( x_i - 6 \)).

4. **Calculate the Variance**:
- The formula for variance \( \sigma^2 \) is:
\[ \sigma^2 = \frac{\sum_{i=1}^{n} (x_i - a)^2}{n} - \left(\frac{\sum_{i=1}^{n} (x_i - a)}{n}\right)^2 \]
- Substituting the values:
- \( \sum_{i=1}^{5} (x_i - 6)^2 = 25 \)
- \( n = 5 \)
- \( \sum_{i=1}^{5} (x_i - 6) = 5 \)

5. **Substituting into Variance Formula**:
- First, calculate the first term:
\[ \frac{\sum_{i=1}^{5} (x_i - 6)^2}{n} = \frac{25}{5} = 5 \]
- Now calculate the second term:
\[ \frac{\sum_{i=1}^{5} (x_i - 6)}{n} = \frac{5}{5} = 1 \]
- Now square the second term:
\[ \left(\frac{\sum_{i=1}^{5} (x_i - 6)}{n}\right)^2 = 1^2 = 1 \]

6. **Final Calculation of Variance**:
- Now substitute back into the variance formula:
\[ \sigma^2 = 5 - 1 = 4 \]

7. **Calculate the Standard Deviation**:
- The standard deviation \( \sigma \) is the square root of the variance:
\[ \sigma = \sqrt{\sigma^2} = \sqrt{4} = 2 \]

### Final Answer:
The standard deviation of the observations is \( 2 \).