For a hundred years or so, mathematical statisticians have been in love with the fact that the probability distribution of the sum of a very large number of very small random deviations almost always converges to a normal distribution. ... This infatuation tended to focus interest away from the fact that, for real data, the normal distribution is often rather poorly realized, if it is realized at all. We are often taught, rather casually, that, on average, measurements will fall within ±σ of the true value 68% of the time, within ±2σ 95% of the time, and within ±3σ 99.7% of the time. Extending this, one would expect a measurement to be off by ±20σ only one time out of 2 × 10^88. We all know that “glitches” are much more likely than that!
-- W.H. Press et al., Numerical Recipes, Sec. 15.1
I don't think it's fair to blame the mathematical statisticians. Any mathematical statistician worth his / her salt knows that the Central Limit Theorem applies to the sample mean of a collection of independent and identically distributed random variables, not to the random variables themselves. This, and the fact that the t-statistic converges in distribution to the normal distribution as the sample size increases, is the reason we apply any of this normal theory at all.
Press's comment applies more to those who use the statistics blindly, without understa...
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