Section 5.6 Standard Units
Definition 5.6.1. Conversion to Standard Units.
Any distribution variable can be converted to “standard units” using the linear translation
\begin{equation*}
\displaystyle z = \frac{x-\mu}{\sigma}.
\end{equation*}
In doing so, values of z will always represent the number of standard deviations x is from the mean and will provide “dimensionless” comparisons.
Example 5.6.2.
Consider our earlier continuous example 5.4.5 in which we found \mu = \frac{5}{4} and \sigma = \sqrt{\frac{51}{80}}\text{.} Then,
\begin{equation*}
P(0 < X < 1) = P \left ( \frac{0-\frac{5}{4}}{\sqrt{\frac{51}{80}}} < \frac{X - \frac{5}{4}}{\sqrt{\frac{51}{80}}} < \frac{1-\frac{5}{4}}{\sqrt{\frac{51}{80}}} \right )
\end{equation*}
gives the middle term is Z and the other endpoints are now in standard units that indicate the number of standard deviations from the mean rather than actual problem units.