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 \(e \approx 2.71828\) 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 \(\sigma\) results in a different normal distribution. The value of \(\mu\) determines the center of a normal distribution curve on the horizontal axis, while the value of \(\sigma\) determines 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 practically zero.
