Let's Face It, Human Intelligence Is Boltzmannian, Not Gaussian

In terms of distribution of intelligence across the nation, rather than the assumed normal distribution - a Gaussian bell curve of intelligence (as above) - I think the distribution of people in the UK would more closely resemble the Boltzmann distribution (see below), where on the horizontal axis 0.1 is the lowest level of intelligence and 7 is a genius.

This is quite intriguing, given that as a rule, approximately 68% of measured values (be they height, weight, blood pressure, and so on) fall within one standard deviation of the mean; approximately 95% of the values fall within two standard deviations from the mean; and a whopping 99.7% of all values fall within three standard deviations from the mean (also illustrated in the image above).

On the other hand, as you can see from the image below, the Boltzmann distribution is biased, and decays exponentially. The analogy to intelligence here being that if the y axis on the graph is number of particles, and the x axis is amount of energy, then no particles will have zero energy, some will have very low energy, many will have medium energy, and as we get to the lower part of the curve, we see only a relatively few people have super high energy. If you replace particles with people and energy with applied intelligence you’ll find it’s a pretty good analogy.  

If we take human intelligence, or perhaps more accurately the extent to which humans have ever applied their potential capacity, we find that it seems not to be like a bell curve at all - rather it's heavily skewed towards the low end, like the Boltzmann distribution. 

So, for example, on a scale of 1-100, where 1 is very low on intelligence, and 100 are the most brilliant minds, if you take the entire population of the UK, my experience indicates to me that the vast majority of people are in the category below 50, with about half being under 25. In fact, I'd guess that broken into percentages it would be roughly as follows, where percentage is percentage of the population:

1 - 25 (50%)

26 - 50 (30%)

51 - 75 (15%)

76 - 100 (5%)

(It's possible these figures might need slightly adjusting, but not by much).

Anyway, the result is definitely not a bell curve. Compare that to say, height or weight - these are bell curves because most of the adult population fall between 4ft 9 and 6ft 7, and 5 stone and 21 stone, which means the bell curve peaks at the average of those ranges and slopes downwards either side with a few rare cases outside of that range. Human intelligence appears not to follow the same patterns as weight and height, even though I think the average member of the public, strewn with democratic and egalitarian aspirations, would like to think differently.

But what makes it an even more interesting phenomenon is that in all likelihood random walk is implicated in many of the exponentially decaying statistics one sees - and even though the intelligence curve isn't symmetrical, it is still possible that a random walk model can account for some of the underlying statistical mechanics.

The general form of random walk arises because parameters and measurements are affected by multiple causes. Take nightclubbing as an example. Suppose in Norwich city centre in 2016 there is a sample group that regularly parties down Prince of Wales Road and averages to the value of n.

If we take a month, say July - each member of that group may go out partying in July. Once the causes that affect people's decision to go out are factored in, we can see many instances where n changes. If a person decides to go out partying, that amounts to a kind of step to the right; that is, it affects n by +1.

However. if a person doesn't go partying in July, that counts as a step to the left - meaning n-1. So assuming all the causes are independent, what we have is something that resembles a random walk-like scenario, hence, partying attendance statistics, when taken over the year, will start to follow a "bell" pattern.

However that won't necessarily be symmetrical, because there may also be exogenous influences at work. For example during certain times of the year, after Christmas, and the subsequent two or three months there will be a bias toward the lower end. Conversely, when it's a hot summer weekend, a bank holiday, or there's a major event in the city, the opposite will happen.

To understand why intelligence is Boltzmannian, not Gaussian, you have to understand that "biased" random walks are highly likely to return non-symmetrical curves. The Boltzmann distribution, which measures frequency distributions of particles over various possible states, arises in atmospheric density with altitude results of random walking molecular motions where there is imposed a maximum energy constraint on the atmosphere (this is due to energy conservation).

If the walk is moving into a space where the density of points in the space varies from place to place one again gets non-symmetry.  So given that human mental resources are limited, it could well be that the intelligence curve is a Boltzmann curve on the leeward side because human mental resources are subject to conservation laws, as well as being 'biased' towards the lower end of the intelligence spectrum - something we don't see in height or weight, which returns the bell curve pattern with the average at the peak of the curve.