Raising the bar

Chapter 1,  How to Study Statistics?

Units 1.3 and 1.4. Ways to think about things, and questions for repetition

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 $latex a + b = b + a,\ ab = ba$ 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 $latex ax^2 + bx + c = 0$. 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.

Your knowledge of and interest in the subject will directly depend on the percentage of the material you can explain.

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. (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. It may be relieving to learn that

We tend to like what we understand.

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.

To know all 100% of the material, you'll have to study 110%.

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".

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