If `sum_(i=1)^(9) (x_(i)-5) " and" sum__(i=1)^(9) (x_(i)-5)^(2)=45`, then the standard deviation of the 9 items `x_(1),x_(2),..,x_(9)` is

Options

A. 3
B. 9
C. 4
D. 2

Explanation

To find the standard deviation of the nine items \( x_1, x_2, \ldots, x_9 \), we can follow these steps:

### Step 1: Define the variables
Let \( y_i = x_i - 5 \). This transformation centers the data around zero.

### Step 2: Write down the given information
We are given:
\[ \sum_{i=1}^{9} y_i = 9 \]
\[ \sum_{i=1}^{9} y_i^2 = 45 \]

### Step 3: Calculate the variance
The formula for variance \( \sigma^2 \) is given by:
\[ \sigma^2 = \frac{\sum_{i=1}^{n} y_i^2}{n} - \left(\frac{\sum_{i=1}^{n} y_i}{n}\right)^2 \]
where \( n \) is the number of items, which in this case is 9.

### Step 4: Substitute the values into the variance formula
Substituting the known values:
\[ \sigma^2 = \frac{45}{9} - \left(\frac{9}{9}\right)^2 \]

### Step 5: Simplify the expression
Calculating the first term:
\[ \frac{45}{9} = 5 \]
Calculating the second term:
\[ \left(\frac{9}{9}\right)^2 = 1 \]
Now substituting these values back into the variance formula:
\[ \sigma^2 = 5 - 1 = 4 \]

### Step 6: Calculate the standard deviation
The standard deviation \( \sigma \) is the square root of the variance:
\[ \sigma = \sqrt{4} = 2 \]

### Final Answer
The standard deviation of the nine items \( x_1, x_2, \ldots, x_9 \) is \( 2 \).

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