The mathematics of normal distribution (Video 1 of 3)

Posted by Robin Powell on June 19, 2018

I’m very pleased to announce the launch of my latest documentary. It comes in three parts and explains the mathematics behind one of the most important concepts in investing, namely normal distribution.

I’m going to be writing more about this subject over the next couple of weeks. But for now I’d just like thank my good friend Mark Hebner at Index Fund Advisors in California for commissioning this series. No one has been more inspirational to me in my work, or  more supportive of my efforts to educate investors, than Mark has.

And with that, enjoy Part 1. Parts 2 and 3 will follow shortly.


Video transcript:

You may not have heard of him, but Sir Francis Galton was a Victorian genius.

The renowned polymath was born here in Birmingham, England, in 1822, and was responsible for many notable and varied scientific advancements in the 19th century, including the development of fingerprint classification, psychometric testing and the first weather map. He’s even credited with inventing the questionnaire!

But despite such accomplishments, perhaps Galton’s most important and influential legacy is unknown to most people, hidden inside an obscure contraption named after him: the Galton Board. Invented by Galton in 1876, he used the board to develop some of the most powerful concepts in yet another field he worked in: statistical mathematics.

Dr Snezana Lawrence, a mathematical historian at Anglia Ruskin University says: “He was very interested in biology and medicine and the application of mathematics to it. He was particularly interested in how certain characteristics are inherited. That interest, in passing on genes as it were, was actually how he came to be interested in statistics.”

He was particularly curious about why certain human characteristics such as height rather than randomly varying within a population, appeared to vary in a very orderly, recognisable way.

And specifically according to something called a normal distribution. 

Dr Kit Yates, senior lecturer in mathematics at Bath University, says: “The normal distribution is often known as the bell curve. It’s this curve which has a shape like a bell, so it’s very high in the middle and low towards the edges, and it can be used to describe a wide range of naturally occurring phenomena from height, for example, through to IQ scores.

“Typically, a large number of people will be clustered around the mean behaviour — the average behaviour — for a particular trait, be that the average height or average IQ. But there will be some people who are way out in the tails of this distribution, but not very many of them. And so this bell curve characterises the frequency of occurrence of this particular trait in the population.”

This phenomenon of normal distribution fascinated Galton. And what he wanted to do was to provide a practical demonstration of why it occurs. The Galton Board does just that. 

The design consists of a vertical array of pegs, ordered according to a quincunx pattern. A large number of beads, introduced from a centre point at the top of the board, make their way through the pattern, eventually coming to rest in one of several bins uniformly arranged across the bottom.

Dr Yates says: “What you find is that you get this really beautiful normal distribution mapped out by the shape of the beads. The reason for that is that there are far more ways to get to these central bins than there are to get to bins at the extremes.

“Now each of these beads, every time it gets to a peg, it goes left or right with equal probability — 50-50. So if I’m thinking about getting out to this bin, then every time a bead hits a peg to get out there it has to go the same way every time, and the same if I’m trying to get to this other extreme.

“But in the middle there are lots of different paths, sometimes going left, sometimes going right, and these left and right movements will somehow cancel each other out and they’ll end up tending to head towards the centre of this distribution far more.

“So this is the beauty of this Galton board, it generates this predictable normal distribution from what is effectively a random process.”

Now, to be precise, the beads form what is know as a binomial distribution, not a normal one. But there’s a good reason why we can call it that.   

Dr Yates says: “Binomial distribution occurs when you have to make multiple binary choices, which is exactly what’s happening in the Galton board.

“Now we have a tool in mathematics which is called the Central Limit Theorem, which effectively says that if I have enough of these independent, random variables stacked together – these binary trials when the bead hits a peg, should allow us to say that this binomial distribution is well approximated by a normal distribution.”

The Central Limit Theorem is arguably one of the most important in all of statistics. Essentially it explains why the normal distribution is so commonly seen, particularly in social studies, such as Galton’s into human traits.

Dr Yates says: “We see the normal distribution a lot in the natural world because we have lots of these composite random variables that, adding together, either give us an addition to that trait or a subtraction from that trait.

“It might be that there are a number of different genes which contribute to the final height of an individual, which will be inherited from parents, and there might also be a wide range of different environmental factors which contribute. Each one of these will be a random process, not necessarily determined beforehand.

“If you have very tall parents and you have good nutrition and various other beneficial factors, then you might find yourself being very tall. But actually, it’s likely that a lot of these composite factors will cancel out and you’ll end up much nearer to the mean height, which is where this normal distribution comes in and that’s a consequence of this mathematical Central Limit Theorem.”

The Central Limit Theorem allows us to use the normal distribution in a multitude of scenarios: From quantum physics to biometrics to stock market returns to resource allocation to quality control; customers arriving at a counter, telephone calls coming into a system, even the number of yeast cells used when brewing a Guinness can all be represented by the normal distribution.

Dr Yates says: “These normal distributions occur in everyday life and we can make a link between these abstract mathematical concepts and these real-world concepts.

“I think the real power of this Galton board is to illustrate that we can find patterns even in seemingly random places. It’s a whole series of random events, these balls moving through this array of pegs, that reproduces a reliable normal distribution – a pattern that we can well characterise.

“I think that this is what mathematics is all about, finding patterns in places where you might not expect it. And I hope that this sort of thing will enthuse people to try to understand how ubiquitous mathematics really is in our everyday lives.”

If you would like your own Galton board for use in your school, museum, office or even at home, then go to


The Evidence-Based Investor is produced by Regis Media, a boutique provider of content and social media management to financial advice firms around the world. For more information, visit our website and YouTube channel, or email Sam Willet or Christina Waider.

Robin Powell

Robin is a journalist and campaigner for positive change in global investing. He runs Regis Media, a niche provider of content marketing for financial advice firms with an evidence-based investment philosophy. He also works as a consultant to other disruptive firms in the investing sector.


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