The normal distribution is used extensively in probability theory, statistics as well as the natural and social sciences. It is also called the Gaussian distribution, because Carl Friedrich Gauss (1777-1855) was one of the first to apply it for the analysis of astronomical data (Lovric 2011).
The normal probability distribution or the normal curve is a bell-shaped (symmetric) curve. Its mean is denoted by $\mu$ and its standard deviation by $\sigma$. A continuous random variable $x$ that has a normal distribution is called a normal random variable.
The notation for a normal distribution is $X \sim N(\mu,\sigma)$. The probability density function (PDF) is written as
$$f(x) = \frac{1}{\sigma \sqrt{2 \pi}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}\, ,$$
where $e \approx 2.7182818$ and $\pi \approx 3.14159$. The probability density function $f(x)$ gives the vertical distance between the horizontal axis and the normal curve at point $x$.
The normal distribution is described by two parameters, the mean, $\mu$, and the standard deviation, $\sigma$. Each different set of values of $\mu$ and $\sigma$ gives a different normal distribution. The value of $\mu$ determines the center of a normal distribution curve on the horizontal axis, and the value of $\sigma$ gives the spread of the normal distribution curve.
A normal distribution is characterized, among others, by the following characteristics (Lovric 2011, Mann 2012):
- The total area under a normal distribution curve is 1.0 or 100 %.
- A normal distribution curve is symmetric about the mean. Consequently, 50% of the total area under a normal distribution curve lies on the left side of the mean, and 50% lies on the right side of the mean.
- The tails of a normal distribution curve extend indefinitely in both directions without touching or crossing the horizontal axis. Although a normal distribution curve never meets the horizontal axis, beyond the points represented by $\mu - 3\sigma$ and $\mu + 3\sigma$ it gets so close to this axis that the area under the curve beyond these points in both directions can be considered zero.
NOTE:
The normal distribution is a theoretical but fundamental statistical model that underlies most data reporting standards. Many statistical methods — such as correlation analysis, linear regression, and common hypothesis tests — assume resp. expect normally distributed input data, uncertainties, and residuals.
The assumption of normality is obligatory for a wide range of methods to be applicable. Commonly, this can be legitimately achieved by transformations.
Two important characteristics of continuous frequency distributions, related to normality are skewness and excess kurtosis.
Skewness ($3^{rd}$ moment $g$) measures the magnitude and direction of asymmetry relative to the symmetric normal distribution ($g=0$).
Excess kurtosis ($4^{th}$ moment $\kappa$) describes the "tailedness" or peakedness—how much probability density is concentrated in the tails (leptokurtic) or near the center (platykurtic). Bell-shaped distributions can therefore still differ from normal in excess kurtosis (normal ~mesokurtic: $\kappa=0$).
Most real-world datasets are not exactly normal. Most data we will encounter in empirical research follow different distributions. However, the log-normal distribution and the logit- resp. logistic-normal distribution are able to provide handy and flexible solutions for a wide range of empirical data.
Citation
The E-Learning project SOGA-Py was developed at the Department of Earth Sciences by Annette Rudolph, Joachim Krois and Kai Hartmann. 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: Rudolph, A., Krois, J., Hartmann, K. (2023): Statistics and Geodata Analysis using Python (SOGA-Py). Department of Earth Sciences, Freie Universitaet Berlin.