In a heavy-tailed model such as the Cauchy distribution, adding observations need not make the sample average converge to a stable population mean in the way it does for independent finite-variance observations.

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But let us not even record those cases. Now, that his recorded misses followed the mild pattern of a bell curve, most would be fairly close to the mark, and very few would be very wide of it. Suppose he shot arrows long enough in successive sets. For each set, he could calculate an average error and standard deviation, even give himself a score for blindfolded archery. But our archer is not in the land of the bell curve. His misses are not mild. All too often, his aim is so bad that the arrow flies almost parallel to the wall and strikes hundreds of yards from the target, or even a mile, if his arm is strong enough. Now, after each shot, let him try to work out his average target score. In the Gaussian environment, even the wildest shots have a negligible contribution to the average. After a certain number of strikes, the running average score will have settled down to one stable value. And there is practically no chance the next shot will change that average perceptibly.

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An instrument governed by such a cauchy, law of probability, would never be put on sale by an ordinary craftsman. One could not even imagine a firm that would manufacture one. Holmes Renaud de l'Académie des Sciences, August 29, 1853 Such has been the argument of most mathematicians and scientists ever since. Gaussian math is easy and fits most forms of reality, or so it seems. But with a sharp hindsight provided by fractal geometry the gaussian case begins to look not so normal after all it was so called only because science tackled it first as ever in science there is a healthy opportunism to begin with the problems easiest to handle. But the difference between the extremes of Gauss and of Cauchy could not be greater. They amount to two different ways of seeing the world. One in which big changes are the result of many small ones, or another in which major events loom disproportionately large. Mild and wild chance, described earlier, are my generalizations from Gauss and Cauchy.