Some time ago, I came across a book titled "Statistics Workbook for Dummies". The for-dummies series is well-known and is supposed to be a series of popular math/science/etc books. But this book is, really, for morons, written by morons. On page 102 of the book, the central limit theorem is "explained" or "motivated" thus:

Of course, this is misleading and is not an explanation of the central limit theorem at all.

The central limit theorem is a theorem in mathematics which has some physical consequences. Its proof requires some mathematics and cannot be fully understood without it. Can it be explained, however, to a non-specialist? Sure, but the explanation is not as trivial as the phrase above suggests. For those who want to apply the central limit theorem, understanding what it is about is essential. Like many other "popular" books in mathematics, statistics, phsysics, science, ... this one makes a bad job. Not surprising. It's one of many many others.

But the problem, you might think, is that the book is, indeed, for dummies. After all, it says so in its title. So, you might think, if you go to the university and take a statistics class in a "quantitative department" (by this I mean, mathematics or engineering or physics, or some other department which does not shy away from mathematical symbols....) you will understand the central limit theorem. Wrong. I have seen generations of students graduate from various reputable "quantative departments" who never learn a proof of the central limit theorem nor what the theorem is about. What is the problem? Well, many of the people who teach that stuff do not know themselves what mathematics is about and yet insist in teaching mathematics. Amazing as it may sound, it is not far from the truth.

Summary: "Statistics Workbook for Dummies" is doing a bad job but this bad job is not much worse than the job being done in many self-proclaimed reputable universities.

Of course, this is misleading and is not an explanation of the central limit theorem at all.

The central limit theorem is a theorem in mathematics which has some physical consequences. Its proof requires some mathematics and cannot be fully understood without it. Can it be explained, however, to a non-specialist? Sure, but the explanation is not as trivial as the phrase above suggests. For those who want to apply the central limit theorem, understanding what it is about is essential. Like many other "popular" books in mathematics, statistics, phsysics, science, ... this one makes a bad job. Not surprising. It's one of many many others.

But the problem, you might think, is that the book is, indeed, for dummies. After all, it says so in its title. So, you might think, if you go to the university and take a statistics class in a "quantitative department" (by this I mean, mathematics or engineering or physics, or some other department which does not shy away from mathematical symbols....) you will understand the central limit theorem. Wrong. I have seen generations of students graduate from various reputable "quantative departments" who never learn a proof of the central limit theorem nor what the theorem is about. What is the problem? Well, many of the people who teach that stuff do not know themselves what mathematics is about and yet insist in teaching mathematics. Amazing as it may sound, it is not far from the truth.

Summary: "Statistics Workbook for Dummies" is doing a bad job but this bad job is not much worse than the job being done in many self-proclaimed reputable universities.

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