Ok, so yesterday wasn’t quite as basic as I planned on shooting for in this week or two of working on non-mathematical concepts. But the idea was too cool to resist. This isn’t exactly a mathematically elementary subject either, but the concept can be grasped without needing to see the actual functions involved.
This is a random sample of a large number of women, arranged by height:
There are many people of average height in the world, and a smaller number of very tall and very short people. The more extreme the height, the rarer the people with that height.
On the other hand, we could imagine a species in which there was a certain average height and every added inch dropped the probability by a constant amount. The way the heights were distributed wouldn’t be a smooth curve (which happens to be called a Gaussian distribution, or a bell curve, or a normal distribution) as it is in the picture, but instead sort of a pyramid. But we never see that.
This isn’t confined to biology either. Everything from the frequencies of photons emitted by a laser to the velocity components of a gas molecule do the same thing. That same smooth bell curve happens all throughout the sciences. It’s inescapable. Why?
The answer is a mathematical fact called the central limit theorem. In slightly imprecise nonmathematical language it says the following: any time you have a quantity which is bumped around by a large number of random processes, you end up with a bell curve distribution for that quantity. And it really doesn’t matter what those random processes are. They themselves don’t have to follow the Gaussian distribution. So long as there’s lots of them and they’re small, the overall effect is Gaussian.
This is of dramatic importance in the sciences, and aside from that it happens to be a good thing to watch for – you’ll start seeing it everywhere.