interpretation of a theorem of normal distribution

Question

Theorem: If X is $N(\mu, \sigma^2)$, then $Z=\frac{(X-\mu)}{\sigma}$ is $N(0,1)$

Question: What is Z and the significance of it?

Answer 1

In many cases you cannot evaluate a probability on a random variable distributed with a normal distribution with mean $\mu$ and variance $\sigma^2$. So what we do is use a new variable, call it $Z$, which is distributed according to a standard normal distribution for which cumulative probabilities are tabulated. In a time in which computers where not so accessible, tabulated values of the cumulative of a standard normal distribution where used a lot.

Clearly a standard normal distribution is just a gaussian distribution with mean $\mu=0$ and variance $\sigma^2 = 1$. It's just an all around easier distribution to work with. Whenever you've found what you want using the variable $Z$ you can go back to your initial random variable $X$ using the same formula $$X = \sigma Z + \mu$$