How to scale a signal to get desired variance
standardizing a Normal random variable $x$
$$ 
\frac{x -m}{\sigma} \sim \mathcal{N}(0,1)
$$
so if $y$ is a random variable that is Normal with mean zero, with standard deviation $1$, 
$$
\sqrt{\sigma^2}y \sim \mathcal{N}(0, \sigma^2)
$$
This standardization actually works for more than just Normal Distributions.