All normal distributions have the same basic shape, but with their centre and spread determined by the parameters \(\mu\) and \(\sigma\). The following diagram describes the probability density function of any normal distribution.
Examples
In the following examples, the values of the parameters \(\mu\) and \(\sigma\) are used to add a numerical scale to the diagram.
Z-scores
Since the number of standard deviations from the mean is so important for normal distributions, we call it a z-score. Z-scores can be defined mathematically as
\[ Z = \frac {X-\mu} {\sigma} \]Z-scores have the same distribution, irrespective of the distribution of \(X\). We will prove later that z-scores have a standard normal distribution,
\[ Z \;\; \sim \; \; \NormalDistn(0,\; 1) \]