Berkeley STAT 157 - Fragments (4 pages)

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Fragments

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Lecture Notes


Pages:
4
School:
University of California, Berkeley
Course:
Stat 157 - Seminar on Topics in Probability and Statistics

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Chapter 6 Fragments 6 0 1 The Normal distribution for data do textbooks give a misleading impression of its ubiquity Almost all textbooks on introductory statistics have a chapter on the Normal distribution mainly motivated as a prelude to the mathematical theory around probability distributions for sample averages The latter in proper context is uncontroversial Let s consider the setting where we have observational data of the type1 that can be represented in a histogram Question For what kinds of data is it empirically true or reasonable to suppose that histograms are approximately Normal Giving a serious answer to this intrinsically rather vague question is surprisingly hard What do textbooks say I read the relevant chapter in three textbooks 48 Chapter 6 45 Chapter 5 16 Chapter 9 What they explicitly say in the chapter is brief and unobjectionable In fact 16 says nothing 48 says The normal distribution is used as a model for a variety of physical measurements since it has been discovered that many such measurements have distributions that are normally distributed or at least approximately so and 45 adroitly sidesteps the issue by quoting a folklore phrase The fuzzy central limit theorem says that data which are influenced by many small and unrelated random effects are approximately normally distributed Having said this and no more about the general question each proceeds to in text examples and student exercises many of which involve data having a Normal distribution The subjects of the examples and exercises in those 1 Briefly the same attribute of different individuals interpreted broadly 77 three chapters are listed essentially completely in Table xxx My supposition is that to a student the memory of the one explicit sentence is crowded out by many examples and exercises so the student comes away consciously or unconsciously with the idea that data on these kind of subjects follows the Normal distribution Notes on Table xxx I have categorized these textbook



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