Normality

What does it mean to say that the probability of getting a head when you flip an unbiased coin is 1/2? Not that if you flip it a million times it will come down heads for half of them; but that if you flip the coin a million times it is more likely to be near 500,000 heads than it is to be 600,000.  So what does 'more likely' mean in this context?  That if you repeat those million flips ten million times, say, writing down at the end of each million flips how many heads there were, and then plot on a graph how many times each number of heads occurs, the peak number will be very close to 500,000.  In this sense, it is the most likely.

Plotting the number of heads obtained for each of the ten million sets of a million flips will, if the coin is truly unbiased, produce a smooth bell-shaped curve, high in the middle, falling off sharply and going to almost nothing at the edges. This shape - called the Normal distribution - occurs frequently in probability theory and in systems determined by the aggregation of millions of essentially random events.  Its name is partly a reflection of this universality: almost independently of the details of the context, it will pop up time and time again, a demonstration of how larger structures can emerge from the combined action of many small and apparently structureless effects. 

So, for example, the Normal curve is found throughout tests of various kinds of ability mental, physical, scientific, artistic.  Where such skills can be measured, it is generally found that there is this bell-shaped distribution: most people in the middle, clustered around the average - the peak of the bell - with correspondingly fewer people any distance away from that middle area. This obviously corresponds to experience, were we tend to meet many people with more or less average skills, but very few with anything out of the ordinary.

However, there is one facet of the Normal distribution which needs to be emphasised; it is symmetrical, the tiny tail above the average is always balanced by a tiny tail below it.  Thus for any system which approximates to this theoretical ideal, you will always find a match between the extremes.

This has an interesting consequence in the field of human abilities.  If, as the statistics seem to indicate, measurable qualities like intelligence - of whatever kind, not just the very limited type judged by discredited IQ tests - understanding, strength, speed, and some forms of creativity are indeed distributed according to the Normal curve, then the ineluctable logic is that for every genius, every exceptionable performer, every athlete, there will almost certainly be someone who labours under a corresponding lack of intelligence, lack of ability, lack of strength.  In a sense, for every mental or physical advantage we enjoy, someone, somewhere, is paying the price of a conjugate lack of just that mental or physical attribute.  But just because blind chance and unfeeling Nature exact this price does not mean that we should acquiesce in this inequity.  Instead we should be acutely aware that the burden of responsibility for using our inherited ability properly is double: we owe it to ourselves, and, even more, we owe it to our unknown, suffering twin.
 
(25.12.91)

Index