Scaling a distribution is as important as centering or demeaning considered here. The question we want to find an answer for is this: What can you do to a random variable to obtain another random variable, say, , whose variance is one? Like in case of centering, geometric considerations can be used but I want to follow the algebraic approach, which is more powerful.
Hint: in case of centering, we subtract the mean, . For the problem at hand the suggestion is to use scaling: , where is a number to be determined.
Using the fact that variance is homogeneous of degree 2, we have
We want to be 1, so solving for gives . Thus, division by the standard deviation answers our question: the variable has variance and standard deviation equal to 1.
Note. Always use the notation for standard deviation with its argument .