The Normal Distribution – Explained Intuitively

Of all the people that you know how many of them are truly extraordinary in any domain? How many world-class dancers do you know? How of the people in the Forbes list of richest people do you personally know? Unless you are a professional dancer or are extremely wealthy, I already know the answer to these two questions: you most likely don’t know any. How do I know that? Well because of something called the Normal Distribution ((Although clearly wealth is not distributed evenly.)).

In 1795 the German mathematician Carl Friedrich Gauss observed that astronomical errors were always distributed in the same way, therefore the Normal Distribution is often referred to as the Gaussian Distribution named after Gauss. But what is this Normal Distribution and how does it allow me to make pretty well founded assumptions about the type of people you probably do or do not know?

It’s a rule about how often things occur in the world around us. For example, there are many scientists but very few who, like Einstein, came up with theories that impact almost every aspect of the world as we know it. Since I live in the Rocky Mountains let’s use mountains as our example. There are millions of mountains but very few with an elevation close to that of Mount Everest — 29,029 feet tall. The Normal Distribution predicts all of this. It does this by telling us how things, like mountains, are generally distributed. But what is a distribution?

Imagine you are making a peanut butter sandwich. You have your slice of bread and you distribute your peanut butter over the slice. In this case you’d want the peanut butter to be evenly distributed with the same amount spread over the entire surface of the slice. Now imagine the slice of bread is enormous and instead of peanut butter you spread mountains over the slice of bread. And instead of spreading them evenly you distribute the mountains so that mountains of average height are stacked in the middle and the taller and smaller mountains are stacked progressively further away towards the edges of the slice.

What would your slice look like?

Something like this (looks like mountains itself!):
The vast majority of the mountains will be distributed in the middle with very few distributed around the edges ((I have not done the analysis on all mountains in the world so this is an hypothesis of what a histogram of the elevation of all mountains in the world might look like. In a dataset of hills from the UK that I was able to analyze for this post I found that the distribution was somewhat right skewed which means that there were more smaller hills than really tall hills. This makes sense because what constitutes the minimum size of a hill or mountain is arbitrary and since the ground is flat you’d expect there to be more smaller hills or than super tall ones. I also analyzed a dataset of 80 Peaks with Prominence 2,000 ft. and greater in Colorado and I found that was left skewed. But neither of these datasets were representative of all mountains.  The larger UK dataset was more representative of all the hills in the UK and although right skewed seemed to approximate a normal distribution based on my analysis (although it failed the Anderson-Darling normality test but that could have been because it was such a large dataset but the Q-Q plot seemed “approximately” normal). A study of all mountains, however, would be an interesting and fun to conduct–collecting all the data would be the difficult part.)). This pattern works not only for mountains but it also works for most other things that occur in the natural world. If you did this exercise with people’s heights the results would be the same. But the same would be true for extraordinary talent in any domain. Most people fall within the range of average and the more or less a person is of something (taller/smaller, more/less talented etc.) the less there are of them to the degree that the Albert Einsteins or Peyton Mannings of this world are extremely rare ((Intelligence is normally distributed, although I don’t have evidence that the quality of football players are normally distributed.)).

Now isn’t this intuitive? You knew this already didn’t you? Well this simple idea is a fundamental concept in statistics, and it allows statisticians, researchers, pollsters, data scientists etc. to make all kinds of predictions etc. about the world we live in.