The standard normal distribution is a special case of the normal distribution. For the standard normal distribution the value of the mean is equal to zero (\(\mu = 0\)) and the value of the standard deviation is equal to 1 (\(\sigma = 1\)).
Thus, by plugging in \(\mu = 0\) and \(\sigma = 1\) in the PDF of the normal distribution, the equation simplifies to:
\[\begin{align} f(x)& = \frac{1}{\sigma \sqrt{2 \pi}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2} \\ & =\frac{1}{1 \times \sqrt{2 \pi}}e^{-\frac{1}{2}\left(\frac{x-0}{1}\right)^2} \\ & = \frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2} \end{align}\]
A random variable that follows the standard normal distribution is denoted by \(z\). Consequently, the units for the standard normal distribution curve are denoted by \(z\) and are called the \(z\)-values or \(z\)-scores. They are also called standard units or standard scores.
The cumulative distribution function (CDF) of the standard normal distribution, corresponding to the area under the curve for the interval \((-\infty, z]\), is usually denoted with the capital Greek letter \(\phi\) and given by:
\[F(x<z) = \phi (z) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{z}e^{-\frac{1}{2}x^2}dx\]
where \(e \approx 2.71828\) and \(\pi \approx 3.14159\).
Basic Properties of the Standard Normal Curve
The standard normal curve is a special case of the normal distribution and thus also a probability distribution curve. Therefore, basic properties of the normal distribution hold true for the standard normal curve as well (Weiss 2010).
- The total area under the standard normal curve is 1 (this property is shared by all density curves).
- The standard normal curve extends indefinitely in both directions, approaching, but never touching, the horizontal axis as it does so.
- The standard normal curve is bell shaped and centered at \(z=0\). Almost the whole area under the standard normal curve lies between \(z=-3\) and \(z=3\).
The \(z\)-values on the right side of the mean are positive and those on the left side are negative. The \(z\)-value for a point on the horizontal axis gives the distance between the mean (\(z=0\)) and that point in terms of the standard deviation. For example, a point with a value of \(z=2\) is two standard deviations to the right of the mean. Similarly, a point with a value of \(z=-2\) is two standard deviations to the left of the mean.
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.