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Chapter 1. How to Study Statistics?

Most of the material of the first two chapters of NCT can be safely skipped, not only because it is unimportant but primarily because our brain tries to systematize the incoming information, and there is very little systematic content in the first two chapters. Also, some topics from Chapters 1-2 are better discussed later, when you have the necessary tools at hand. I use this space to discuss some issues deeper than in the book and at the level you will need later. High-school material or the stuff you are supposed to remember from math prerequisites is reviewed. Solve the relevant exercises in NCT only when you find my explanations insufficient.

Unit 1.1. The structure of mathematics

Math consists of definitions, axioms and statements. Definitions are simply names of objects. They don't require proofs. Axioms (also called postulates) are statements that we take for granted; they don't require proofs and are in the very basement of a theory. Statements have to be proved. Depending on the situation and the tastes of the researcher, they may have different other names: theorem, lemma, proposition, criterion etc. Statements are a way to economize on space and thinking efforts. They summarize sometimes very long arguments and serve as building bricks in subsequent constructions.

In math texts definitions take up at most 5 to 10% of space. A common error of students of quantitative subjects is to jump to statements without thinking enough about definitions (when I was young I also committed this error). Definitions not only give names to objects but they also give direction to the theory and reflect the researcher's point of view. Often understanding definitions well allows you to guess or even prove some results.

Unit 1.2. Studying a definition

A definition starts with a preamble which sets a background and allows you to see the existence of objects with distinct properties. Most of the time the preamble is not even mentioned but there are cases when it contains a complex logical argument. After you understand the preamble, understanding the definition proper - that is giving the name - becomes easy. Studying a definition is concluded by considering possible equivalent definitions and deriving immediate consequences.

Example 1

(1) Preamble. Let us consider natural numbers (Kronecker is reported to have said, ''God created the natural numbers; all the rest is the work of man'').  These are numbers 1, 2, 3,... that naturally arise when counting things. We can notice that some of them are divisible by 2 (for example, 4/2 = 2 is an integral number) and others are not (e.g., 3/2 is a fractional number).

(2) Definition proper. The natural numbers that are divisible by 2 (2, 4, 6,...) are called even. The natural numbers that are not divisible by 2 (1, 3, 5, ...) are called odd.

(3) Immediate consequences. (i) Any natural number is either even or odd. (ii) If a natural number is even and divisible by 3 at the same time, then it is divisible by 6.

Some basic objects cannot be defined or their precise definitions are too complex for this course. In such cases it is better to give equivalent (descriptive) names or simply list the objects we name (as it has been done in case of natural numbers).

Example 2. The following names are used interchangeably: set = collection = family = array.

Example 3. Natural numbers are members of the set $latex N={1,\,\,2,\,\,3,\,\,...}$. Integer numbers are members of the set $latex Z={0,\,\pm 1,\,\pm 2,\,...}$. In some applications (see Unit 5.15 and Unit 6.9) we need nonnegative integer numbers listed in $latex Z_+={0,\,1,\,2,\,...}$.

Example 4. The next type of numbers we need, real numbers, cannot be put in a list. As a convenient working definition, we call real numbers those numbers that can be written in a decimal form. For example, the famous number $latex \pi$, is, by definition, the ratio of the length of a circumference to its diameter. Its decimal form, is, up to approximation, $latex \pi = 3.1415926...$ Integer numbers fall into this category: for example, $latex 1 = 1.000...$ The set of all real numbers is denoted $latex R$.

Unit 1.3 Ways to think about things

(1) When you study a definition, don't think about a single representative of a class of objects; try to think about the whole class. Thinking in terms of sets will systematize your knowledge.

(2) In algebra, when we denote numbers by letters, we mean that whatever rules we write with letters are of universal character and apply to all numbers in a given set. For example, the commutativity rules for summation and multiplication $latex a + b = b + a$, $latex ab = ba$ hold for all real numbers and not just for those specific ones you may plug in. Simple algebra rules, like the ones we have just seen, are easy to formulate verbally. Verbal formulation and derivation of complex algebra rules is a nightmare. For example, try to verbalize the rule for finding the roots of the quadratic equation $latex a{x^2} + bx + c = 0$. This is what people did before 1637, when René Descartes introduced modern algebraic notation. This is what people still do in the beginning and intermediate micro- and macroeconomic courses. Excuse me, but this is a stone age! This tradition persists for the only reason that the teaching methodology at the secondary and high schools fails. With the right methodology, every normal person can learn algebra (I studied early education issues, experimented with my children and know what I am talking about).

(3) Believe me, even with zero math background you can quickly learn algebra if you follow a few simple rules:

(a) Pay attention to small details, like the usage of upper- and lower-case letters, the difference between parentheses (,), brackets [,] and braces {,}, punctuation and arithmetic signs ($latex +,\,-,\,\times,\,/$).

(b) Try to find a tangible (or intuitive) interpretation for each element in a mathematical construction. Usually, geometry clarifies many things. For example, a real number can be uniquely identified with a point on a straight line (called a real line in this case); a pair of real numbers can be uniquely identified with a point on a coordinate plane; Venn diagrams help when working with sets.

(c) Similarly, explain in your own words each step in a long algebraic transformation. Don't trust the book or the professor.

(d) Initially it helps to translate formulas to verbal statements and back. Verbalization can also help in case of long, complex definitions and statements (we'll return to this when discussing $latex p$-values, see Unit 9.4). With time you'll be able to do without verbalization, and in complex situations the best thing to do is to shut up and concentrate. Digression: When I brought my son to the USA and enrolled him in the 7th grade, he did not speak English. The math teacher would explain the students the solution for half an hour, they would diligently write down his explanations and my son would sit doing nothing. The teacher would ask why he wasn't writing anything and my son would show a one-line solution to the problem.

(e) In such a theoretical subject as mathematical statistics it is not possible to provide real-life motivating examples for absolutely all parts of the theory. In fact, many theoretical problems are prompted by the internal logic of the theory; even more often solutions to problems are based on mathematical intuition. We are not computers; we need some level of interest in the subject and some level of satisfaction from our efforts to continue our studies.

(f) Try to keep your level of understanding high (say, at 80 to 90%) and the interest in the subject will ensue. If your ambition is to get a high mark in statistics, follow what I call a 110% rule. We inevitably forget a part of what we've studied.

Unit 1.4 Questions for repetition

1. Which of the following is true/false: a sum of natural numbers is natural, a difference of natural numbers is natural, a product of natural numbers is natural, a ratio of natural numbers is natural?

2. Answer the same questions replacing ''natural'' with ''real''.