Gaussian Distribution

Introduction

The gaussian, or normal, distribution is a useful and common distribution. Imagine looking at a graph of the number of students in a physics class who achieved each grade in a recent test. The top of curve would be the most common grade, the mean, and other marks would be roughly symmetrically distributed either side of this, decreasing in likelihood. You can imagine that it would create a similar “bell curve” as that seen in Fig.1.

Fig.1: A Gaussian distribution

The larger the data set, the closer the convergence of the system towards a normal distribution. This forms the Central Limit Theorem. If you look at the section (Poisson Distribution) below, you may notice that the distribution resembles a bell-curve. Using a Central Limit Theorem argument, the Poisson distribution is well-approximated by the Gaussian for sufficiently large means.

Probability Calculations Using Gaussian Distributions:

We can find out the probability of a measurement lying in a certain range by integrating the Gaussian Distribution function.

Integrating this directly is impossible between finite limits! Luckily, it can be approximated to an extremely high degree of accuracy using numerical techniques. In our case, we’ll use Excel to do this.

Using Excel:

Excel has a function called NORM.DIST(x, mean, standard deviation, 1), which integrates the Gaussian from negative infinity to x. Remembering that a Gaussian is also symmetric about its mean, you can combine these in order to integrate over any desired range.

We will demonstrate how to do this using an example from the required textbook “Measurements and their Uncertainties”:

A box contains 100 Ω resistors which are known to have a standard deviation of 2 Ω.

a) What is the probability of selecting a resistor with a value of 95 Ω or less?
b) What is the probability of finding a resistor in the range 99-101 Ω?

Find a prepared Excel Guide Template here and answers (see Distributions pdf) in case you get stuck!