Q:

Statistics - scaling sets of data - mean & S.d

A set of numbers has a mean of 22 and a standard deviation of 6. If 3 is added to each number of the set, and each resulting number is then doubled, find the mean(50) and standard deviation(12) of the new set.
Ans in brackets.
For the mean; 
Ex=22n+3n 
Ex=25n*2 
Ex=50n 
X(mean)=50 
But i can't figured out for the S.d, do i need to use y=ax+b or other method ??

A:

You are considering a set of numbers $x_1,\ldots,x_n$ with the property that
$$
\bar{x}=\frac1n\sum_{i=1}^n x_i=22,\quad \sigma_x=\sqrt{\frac1n\sum_{i=1}^n(x_i-\bar{x})^2}=6.
$$
Now we construct a new set of numbers $y_1,\ldots,y_n$ by letting $y_i=2(x_i+3)=2x_i+6$ for all $i$. Then $\bar{y}=2\bar{x}+6$ and hence
$$
y_i-\bar{y}=2x_i+6-(2\bar{x}+6)=2(x_i-\bar{x}).
$$
Thus the standard deviation of $y_1,\ldots,y_n$ is
$$
\sigma_y=\sqrt{\frac1n\sum_{i=1}^n(y_i-\bar{y})^2}=\sqrt{4\frac1n\sum_{i=1}^n(x_i-\bar{x})^2}=2\sigma_x.
$$

