Question

Each observation in a raw data is first multiplied by 3, then 6 is added. It is then divided by 3 and subsequently reduced by 6. Which of the following statements is true?

• A. The new mean is equal to the original mean
• B. The new mean is 4 more than the original mean
• C. The new mean is 4 less than the original mean
• D. The new mean is 2 more than the original mean

Answer 1

The correct option is C The new mean is 4 less than the original mean

Let the observations be $x_1,x_2\dots x_n$
and mean be $\overline{x}$
$\Rightarrow \frac{x_1+x_2+\dots +x_n}{n}=\overline{x}$
Each observation is first multiplied by 3
then 6 is added. It is then divided by 3
and reduced by 6 which gives the data:

$\frac{3x_1+6}{3}-6,\frac{3x_2+6}{3}-6,\dots ,\frac{3x_n+6}{3}-6,$
Here,
$\frac{3x_1+6}{3}-6=\frac{3x_1+6-18}{3}=\frac{3x_1-12}{3}=x_1-4$
Similarly, $\frac{3x_n+6}{3}-6=x_n-4$
$\Rightarrow$ the observations can be written
as: $x_1-4,x_2-4,\dots x_n-4$
$\therefore$ New mean
$=\frac{x_1-4+x_2-4+\dots +x_n-4}{n}=\overline{x}-4$.