# Using the Z-Table in Reverse: from the regions to the Z-values

In a "forward direction" use of the Z-Table, you'll often

1. Start with a normal distribution with mean ${\displaystyle \left.\mu \right.}$ and standard deviation ${\displaystyle \left.\sigma \right.}$
2. Transform a value ${\displaystyle \left.X\right.}$ to a standard normal, to a standard ${\displaystyle \left.Z\right.}$, via
${\displaystyle Z={\frac {X-\mu }{\sigma }}}$
3. Find an area value from the Z-Table, corresponding to ${\displaystyle P(0\leq z\leq Z)}$.

In the "backward direction", you do these three steps in reverse:

1. Starting with an area, you
2. Find the corresponding ${\displaystyle \left.Z\right.}$ of the area, considered as ${\displaystyle P(0\leq z\leq Z)}$
3. Transform the Z-values from the standard normal to the original normal values, via the transformation

${\displaystyle \left.X=\mu +Z\sigma \right.}$