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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.