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 will follow these steps:

### Step 1: Understand the Given Information
We have two equations:
1. \(\sum_{i=1}^{5} (x_i - 6) = 5\)
2. \(\sum_{i=1}^{5} (x_i - 6)^2 = 25\)

### Step 2: Calculate the Mean of the Adjusted Values
From the first equation, we can find the mean of the adjusted values \( (x_i - 6) \):
\[ \text{Mean} = \frac{\sum_{i=1}^{5} (x_i - 6)}{5} = \frac{5}{5} = 1 \]

### Step 3: Calculate the Variance
Variance is calculated using the formula:
\[ \text{Variance} = \frac{\sum_{i=1}^{n} (x_i - a)^2}{n} - \left(\frac{\sum_{i=1}^{n} (x_i - a)}{n}\right)^2 \]
Here, \( n = 5 \), \( a = 6 \).

Using the second equation, we substitute:
\[ \text{Variance} = \frac{25}{5} - \left(\frac{5}{5}\right)^2 \]
Calculating this gives:
\[ \text{Variance} = 5 - 1 = 4 \]

### Step 4: Calculate the Standard Deviation
The standard deviation is the square root of the variance:
\[ \text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{4} = 2 \]

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

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