The Sampling Distribution and the
Central Limit Theorem
(In Plain English!)

(For Amazon link, click here.)

Statistics is pretty powerful. For example, it allows me to take a sample and then say things like: I know (with 95% confidence) how close my sample mean is to the true population mean, even though I have no idea what the true mean is. Or even what the population looks like! I know how close I am to the truth without knowing what the truth is! And if that’s not close enough, I know what to do to get a closer estimate. That seems powerful to me.

It is the Central Limit Theorem that allows me to do this. And the Central Limit Theorem deals specifically with the sampling distribution. Brooks has written a booklet explaining these two concepts in everyday English that anyone can understand. The following paragraphs come from the Introduction in that booklet:

I remember when I started taking Statistics, and later when I started teaching it to college students. Approximately the first half of the first course in Statistics is spent talking about basic measures related to Statistics (mean, variance, standard deviation, etc.) and about probabilities and their distributions. Then we talk about the sampling distribution and the Central Limit Theorem. Then we start looking at several different applications of Statistics (estimation, hypothesis testing, t-tests, ANOVA, regression, etc.)

The Central Limit Theorem is sort of a peak. All that first stuff builds up to the peak. Students who get over the peak --- those who really understand the sampling distribution and the Central Limit Theorem --- have very little trouble with anything that follows in Statistics (even over several follow-on courses.) But students who can’t get over the peak --- those who never quite get the Central Limit Theorem and all it represents --- have trouble with everything that follows! The Central Limit Theorem is that fundamental to Statistics. If you get it, the field is easy. If you don’t, well very little of it makes any real sense. 

That is why I wrote this. The first version of this booklet was written for my students back in 1973. I am resurrecting it now in ebook form in hopes that you will find it helpful. 

Most of my students in Basic Statistics were there because they had to be there! It was a required course. And most of them had heard, and believed, it was going to be a very difficult course in mathematics. And that was a terrible misconception. From a mathematics standpoint, statistics is easy. It involves addition, subtraction, multiplication, division, and the occasional square root. All this can be done on today’s spreadsheets, pocket calculators, and even most cell phones! It is not the math that is difficult. 

What is difficult is that it is like (a) a course in a foreign language and (b) a course in logic. There are a large number of new definitions and symbols the student gets exposed to, especially in the first part of the first course. And most of these symbols are (literally) Greek! There are things like mu, rho, sigma (both capital sigma and lower case sigma), and sigma-x and sigma-x-bar, and sometimes the differences between them are subtle. And then there is a whole lot of “if this, then that” and “if not this then not that.” It really can be confusing. Not difficult, but confusing.

Table of Contents:

1.0 Introduction:
2.0 Distributions   
    2.1 Population Distribution:
    2.2 Sample Distribution
    2.3 Sampling Distribution
3.0 Central Limit Theorem:
     3.1 An Example:
         3.1.1 With replacement:
         3.1.2 Without replacement:
     3.2 Back to the Example.
     3.3 The Normal Curve:
4. Sample Size:
     4.1 For Normal Approximation:
     4.2 For Sufficient Precision:
5.  Another Example:
6. Extension to Proportions:
     6.1 Margin of Error:
Appendix 1
About the Author

The booklet is available as an ebook from Amazon, written in the .mobi format. That way it is priced so anyone can afford it. A link to it is here:  The Sampling Distribution and Central Limit Theorem.

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