MHB Stat.02 Find the value of the new variance

[karush]

There are 8 items in a data set. The sum of the items is 48.
a. Find the mean.
$\qquad\textit{mean}=\dfrac{\textit{sum}}{\textit{data set}}=\dfrac{48}{8}=\textbf{6}$
The variance of this data set is 2. Each value in the set is multiplied by 3.
b. Write down the value of the new mean.
$\qquad \textit{new mean} =\textit{current mean}\cdot \textit{scalar}=6\cdot 3=\textbf{18}$
c. Find the value of the new variance

OK I didn't know how to get the new variance

 

[HOI]

The variance is, by definition, $\sum_{i= 1}^8 (x_i- 6)^2= 2$.
If each member of the data is multiplied by 3 we replace $x_i$ by $3x_i$ and, yes, the mean is 3(6)= 18 so the variance is $\sum_{i=1}^8 (3x- 18)^2= 9\sum_{i= 1}^8 (x_i- 6)^2$.

 

[HOI]

Notice that when every datum is multiplied by "a" the mean is multiplied by a but the variance, because it involves a square, is multiplied by $a^2$. Of course, the standard deviation, the square root of the variance, is also multiplied by a.