The sum of 10 items is 12 and sum of their squares is 18, then standard deviaiton is

Options

A. `1/5`
B. `2/5`
C. `3/5`
D. `4/5`

Answer

To find the standard deviation given the sum of 10 items and the sum of their squares, we can follow these steps:

### Step 1: Identify the given values
We know:
- The sum of the items (ΣX) = 12
- The sum of the squares of the items (ΣX²) = 18
- The number of items (n) = 10

### Step 2: Use the formula for standard deviation
The formula for standard deviation (σ) is given by:

\[ \sigma = \sqrt{\frac{\Sigma X^2}{n} - \left(\frac{\Sigma X}{n}\right)^2} \]

### Step 3: Substitute the values into the formula
Now, we substitute the known values into the formula:

\[ \sigma = \sqrt{\frac{18}{10} - \left(\frac{12}{10}\right)^2} \]

### Step 4: Calculate each part
First, calculate \(\frac{18}{10}\):

\[ \frac{18}{10} = 1.8 \]

Next, calculate \(\frac{12}{10}\):

\[ \frac{12}{10} = 1.2 \]

Now, square this value:

\[ \left(\frac{12}{10}\right)^2 = (1.2)^2 = 1.44 \]

### Step 5: Substitute back into the equation
Now substitute these results back into the standard deviation formula:

\[ \sigma = \sqrt{1.8 - 1.44} \]

### Step 6: Simplify the expression
Calculate \(1.8 - 1.44\):

\[ 1.8 - 1.44 = 0.36 \]

### Step 7: Take the square root
Now take the square root of \(0.36\):

\[ \sigma = \sqrt{0.36} = 0.6 \]

### Step 8: Express the result in fractional form
Since \(0.6\) can be expressed as a fraction:

\[ 0.6 = \frac{3}{5} \]

### Conclusion
Thus, the standard deviation is:

\[ \sigma = \frac{3}{5} \]

### Final Answer
The correct option is the third one: \(\frac{3}{5}\).

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