**normal distribution** is used extensively in probability theory, statistics, and 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.

In [1]:

```
%load_ext lab_black
```

In [2]:

```
# First, let's import all the needed libraries.
import numpy as np
import matplotlib.pyplot as plt
import scipy.stats as stats
```

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 becomes so close to this axis that the area under the curve beyond these points in both directions can be taken as virtually zero.

**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.*