The following diagram describes the probability density function of any normal distribution.

It can be used to add a scale appropriate to any values of \(\mu\) and \(\sigma\). For example, the pdf of a \(\NormalDistn(\mu=180, \sigma=10)\) distribution is

Z-scores

The number of standard deviations from the mean is called a z-score.

\[ Z = \frac {X-\mu} {\sigma} \]

Z-scores have a standard normal distribution,

\[ Z \;\; \sim \; \; \NormalDistn(0,\; 1) \]