Before we can apply the concept of a standard normal distribution to a real data set we need to discuss the concept of standardization of a normal distribution. We know that a normal distribution is parametrized by two parameters, its mean \(\mu \in \mathbb R\) and its standard deviation \(\sigma \in \mathbb R>0\), \(X \sim N(\mu,\sigma)\). The actual value of these parameters depends on the population and the metrics used to describe its features. In order to transform any particular \(\mu\) and \(\sigma\), related to a particular random variable \(X\), to \(\mu=0\) and \(\sigma=1\), we have to convert the \(x\)-value to a \(z\)-value, by applying the equation below.

\[z = \frac{x-\mu}{\sigma}\]

As a result we get a standard normal distribution for any particular normal distribution. This procedure is essential if you need to determine the \(z\)-scores or any particular probability related to a \(z\)-score \((P(z))\) by looking them up in a table. We will see later on, that R is such a powerful tool, which makes the step of standardization dispensable.