Normla distribution help

I am really confused on how and when to use the formula z=x-the mean/standard deviation ? Can someone please help explain this. What does z actually stand for? Also, I am unclear as to how this is coded. Thanks

$Z$ is the standardised normal distribution with parameters $N(0,1)$.

What's the point? Well one of the points is that in different contexts you're going to get different parameters on your normal distribution; i.e. $N(12,4)$ or $N(98,44)$, etc... and without a calculator with normal dist. functions you would need to result to a normal dist. table, i.e. the one at the back of your formula booklet.

But if you didn't standardise your result, that table would need to be different for every different context you encounter. Obviously you don't feel like printing off infinitely many tables, so just print one off corresponding to $N(0,1)$ and then every context can be reduced to this form exactly by the transformation

$Z = \dfrac{X-\mu}{\sigma}$



So, $Z$ is exactly like your $X$, but it's standardised. So if $X \sim N(\mu, \sigma^2)$ then $Z \sim N(0,1)$.


To see how the transformation changes the parameters, I suggest you go over that chapter in your book and see what shifting data by a constant does to the parameters, and what dividing by a constant does to them as well. The transformation above is simple;

First you shift all your data by $\mu$.

Then you divide all the data by $\sigma$.

@Bertybassett Just adding to this a bit for new spec A Level (assuming that's what you're doing):

You have a calculator that can evaluate any P(X<x) without using tables if you have the mean/sd. The only time you ever need to standardise in new spec is if you have an unknown mu/sigma.

Thanks for the reply. So you only ever standardise and use that formula for z if you have unknown values of the SD or mean? And in the new spec do we never have to use the tables for the normal distribution (I know that my classwiz can calculate the normal distribution as well as inverses, but would the tables be required in hypothesis testing)? Thanks