On Outliers: What they represent, and why the Central Limit Theorem is Typically Off.

A Bell Curve

The central limit theorem states that if you have many small, independent, random variables, then their sum is distributed approximately as a bell curve. Strikingly, almost everything is made up of many small parts, and these parts don’t tend to influence each other very much.

So much of what can measure seems to fit a bell curve. This is why the normal distribution works. Because this assumption tends to work well, it is usually taken as a matter of course. Students are taught it, lecturers preach it, researchers apply it, and startlingly few stop to question it.

Suppose the variables are not small, or suppose they’re not independent. Suppose, under certain conditions, the value of one variable would seriously effect another. Suppose we’re talking about the buildup of snow on a mountain slope. Most of the time, snowflakes can gradually build, without significant effect. But once enough builds, you don’t find snowflakes resting calmly upon a drift. What you find is an avalanche.

Violent nonlinearities...

The sum total of snowflake movement isn’t what we might expect. The snowflakes on the top used to be lightly packed by the new, gradually coming down. The snowflakes on the bottom used to just sit there. But they’re not just sitting there. They’re moving fast, and they’re moving down.

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