Why is the normal distribution so important

Fortunately we have a very good reason for not worrying too much about the absence (in this book) of a rigorous test for evaluating the normahty of a distribution. The statistical techniques that we present are robust with respect to departures from normality. Even if the population of interest is not normal, these techniques stiU can be used, because they continue to be approximately valid. 

This robustness arises from the central limit theorem, one of the fundamental theorems of statistics, which might be stated as follows  

The classic illustration of the central limit theorem comes from throwing dice. The probability that we observe a certain niunber of points throwing a single die is shown in Fig. 2.6(a). The possible outcomes are the integers from 1 to 6, and if the die is not biased all of them have the same chance to occur, leading to a distribution that is far from normal. 

Often the overall error of an experimental result is the aggregate of several essentially independent individual errors, with no individual error predominating. In the titration, for example, we pointed out the 

Perhaps the central limit theorem justifies the rapturous remark of Sir Francis Galton, the inventor of linear regression Hardly will there be something as impressive for the imagination as the admirable form of cosmic order expressed by the Law of the Frequency of Errors that is, the normal distribution). Had the Greeks known of it, they would certainly have personified and deified it . 

---