Question

If the mean of the set of values \( (3,6,9,10,n) \) is \( 9 \), then the variance of the given set of values is:

• A. \(19\)
• B. \(22\)
• C. \(24\)
• D. \(15\)

Hint:

For variance problems: \[ \sigma^2 = \frac{\sum x_i^2}{n}-\mu^2 \] Always:

• Find missing values first using mean.

• Compute squares carefully.

• Avoid arithmetic mistakes in summation.

Answer 1

The Correct Option is B

Step 1: Use the mean to find the missing value.
The five values are $3,6,9,10,n$ with mean $9$. So $\dfrac{3+6+9+10+n}{5}=9$, giving $28+n=45$, so $n=17$.
Step 2: List the full data.
The set is $3,6,9,10,17$ with mean $9$.
Step 3: Square each value.
$9,36,81,100,289$.
Step 4: Add the squares.
$9+36+81+100+289=515$.
Step 5: Apply the variance formula.
Variance $=\dfrac{1}{n}\sum x_i^2-\bar x^2=\dfrac{515}{5}-9^2=103-81$.
Step 6: Get the answer.
So the variance is $22$. \[ \boxed{22} \]