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, when 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 in the following section, that, being such a powerful tool, R makes the step of standardization dispensable.

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Citation

The E-Learning project SOGA-R was developed at the Department of Earth Sciences by Kai Hartmann, Joachim Krois and Annette Rudolph. You can reach us via mail by soga[at]zedat.fu-berlin.de.

You may use this project freely under the Creative Commons Attribution-ShareAlike 4.0 International License.

Please cite as follow: Hartmann, K., Krois, J., Rudolph, A. (2023): Statistics and Geodata Analysis using R (SOGA-R). Department of Earth Sciences, Freie Universitaet Berlin.