Question

The standard deviation of the first \(10\) multiples of \(4\) is

• A. \(7\)
• B. \(8\)
• C. \(11.5\)
• D. \(14\)

Hint:

If every observation of a data set is multiplied by a constant \(k\), then the standard deviation is also multiplied by \(|k|\).

Answer 1

The Correct Option is C

Step 1: Write the first \(10\) multiples of \(4\).
The observations are \[ 4,8,12,16,20,24,28,32,36,40. \] This is an arithmetic progression with \[ a=4,\qquad d=4,\qquad n=10. \]

Step 2: Find the mean.
The mean of an arithmetic progression is \[ \bar{x}=\frac{\text{first term}+\text{last term}}{2}. \] Hence, \[ \bar{x}=\frac{4+40}{2} \] \[ \bar{x}=22. \]

Step 3: Compute deviations from the mean.
\[ -18,-14,-10,-6,-2,2,6,10,14,18. \] Their squares are \[ 324,196,100,36,4,4,36,100,196,324. \] Therefore, \[ \sum (x-\bar{x})^2 = 324+196+100+36+4+4+36+100+196+324 \] \[ =1320. \]

Step 4: Calculate the variance.
\[ \sigma^2 = \frac{\sum (x-\bar{x})^2}{n} = \frac{1320}{10} = 132. \] Thus, \[ \sigma=\sqrt{132} = 2\sqrt{33}. \] \[ \sigma \approx 11.49. \]

Step 5: Final conclusion.
Therefore, \[ \boxed{\sigma \approx 11.5} \] Hence, the correct option is \[ \boxed{(3)}. \]