TITLE: Standardizing A Random Variable That is Normally Distributed
QUESTION [14 upvotes]: To standardize a random variable that is normally distributed, it makes absolute sense to subtract the expected value $\mu$ , from each value that the random variable can assume--it shifts all of the values such that the expected value is centered at the origin. But how does dividing by the standard deviation play a role in the standardization of a random variable? That part is not as intuitive to me as is subtracting $\mu$.

REPLY [5 votes]: Recall that $\operatorname{Var}{X}=E(X-\mu)^2$, where $\mu$ is the mean of $X$. So $\operatorname{Var}(X)$ is an average of squares. Thus if we scale $X$ by a factor $\rho$, then the variance gets multiplied by $\rho^2$. 
To get a feeling for this, recall that if we scale a geometric figure, such as a square or a triangle, by the factor $\rho$, then area gets scaled by a factor of $\rho^2$.
Now suppose that the variance of $X$ is $\sigma^2$. Then we must scale $X$ by the factor $\dfrac{1}{\sigma}$ to bring the variance to $1$.