Gaussian

The Gaussian1 function (sometimes called the “normal distribution” or “the bell curve,” though both terms are a bit inaccurate in this case) is a standardized curve that frequently comes up in physics; for example, in random processes such as particle decay. The formula for the Gaussian function is:

(3)\[f(x) = e^{- \frac{1}{2}\left( \frac{x - \mu}{\sigma} \right)^{2}}\]

  • \(\mu\) is the mean of the distribution. (For a Gaussian, the mean, mode, and median are the same.)

  • \(\sigma\) = the standard deviation; it’s related to the “full width at half maximum” (FWHM) of the curve by FWHM = \(2\sigma\sqrt{2\ln 2} \approx 2.35\sigma\).

  • \(e\) = Euler’s constant, a transcendental number that occurs often in calculations that relate to growth and increase. It’s formally defined as \(\lim_{n \rightarrow \infty}\left( 1 + \frac{1}{n} \right)^{n}\).

If you want to work with the normal distribution \({\mathcal{N}(x)}\) as a “probability density function” then you’ll need to include a normalization so the integral \(\int_{- \infty}^{\infty}{\mathcal{N}(x)dx =} 1\).

(4)\[ \mathcal{N}(x:\mu,\sigma) = \frac{1}{\sigma\sqrt{2\pi}}e^{- \frac{1}{2}\left( \frac{x - \mu}{\sigma} \right)^{2}} \]

Without the normalization factor, the maximum value (“amplitude”) of \(Ae^{- \frac{1}{2}\left( \frac{x - \mu}{\sigma} \right)^{2}}\) is \(A\), which is easy to read from a graph.2

annotated gaussians

Figure 89: A few Gaussian functions with annotations.

I’ll repeat something I say in The Basics: You see the word “Gaussian” a lot in this tutorial, because it’s an easy distribution to work with as I prepare plots and such. There are many other probability distributions and functional forms that are used in physics. For example, there’s the Landau distribution, which describes energy loss of ionizing particles.

Gaussian distributions come up a lot because when you’re making an observation that depends on the combination of many underlying probability distributions, the combination tends towards a Gaussian.3 But don’t start believing that everything is normal!

xkcd dating_pools

Figure 90: https://xkcd.com/314/ by Randall Munroe

1

You may have noticed I’m sloppy, and use the terms “Gaussian distribution”, “Gaussian function”, or just plain “Gaussian” interchangeably, as if they were all exactly the same thing, or even linguistically correct. All I can do is quote Ralph Waldo Emerson: “A foolish consistency is the hobgoblin of little minds”. Of course, this quote has nothing to do with physics, statistics, or the size of my little mind.

2

When I was a graduate student, I showed a plot very much like this one at one of my first talks to the rest of my experiment’s group. The spokesman thundered “Take that off the screen!” He felt that the information it conveyed was so trivial that I was insulting everyone’s intelligence. Don’t make the same mistake that I did!

On the other hand, if you feel that your intelligence has just been insulted, you may be ready to be an experiment spokesperson. Quick, write a proposal!

3

If you’re curious why, it’s a consequence of the Central Limit Theorem. The mathematician Grant Sanderson has a nice series of animated videos that explain this.