How should a mathematically-inclined person learn descriptive statistics?

Question

I am interested in learning descriptive statistics. But I am completely baffled, that there seem to be no mathematically rigorous books on this subject, as far as I know at least. The Wikipedia page states, that descriptive statistics is not based on probability theory. If that is the case, then on what is it based on?

It could be that this field is simply not an area of mathematics. However, I believe this idea to be rather absurd, since it is clearly using some level of regular mathematical reasoning. I am not in any way saying that when teaching a subject, mainly appealing to intuition is bad, but rather that developing mathematics "in thin air" is fairly unsatisfactory.

Given all this how do I, a rather mathematically-inclined person, go about learning descriptive statistics?

Some general statements about the nature of said subject with respect to what I talked about may be helpful as well as links to any potentially useful material.

Answer 1

Descriptive statistics is a part of mathematics, but it is generally thought at high school. It is not useful to mathematicians. This also gives the reasons why there is no book:

1. Most mathematicians are not willing to write a book.
2. There is no real need for a book since most mathematicians don't want to study it.
3. People who need it, usually don't like rigorous mathematics.
4. Descriptive statistics is usually in high school books, where it is combined with algebra, geometry and other subjects.

However, you need descriptive statistics to give your information as clearly as possible. For example, when you want to show how many per cent of the tax money is spent at each category, a pie chart is useful. When you want to compare the amount of tax paid per country, a bar chart may be more useful.

The mathematical part of descriptive statistics is learning about how to compute or create all those diagrams or central tendencies. This is not hard: The formulas are actually easy, and creating diagrams is more learning how to use software than learning maths. The non-mathematical part is deciding between all those diagrams and statistics. This is actually the harder part for a mathematician.

Answer 2

I share your concern about studying non-mathematical subjects, but when I have to face a decision like this I ask myself: what for? I guess that you have in mind a specific pattern if you ponder over this choice. Could this subject be useful for what you have in mind? I don't really think that studying something apart from mathematics is bound to be a loss of time: also descriptional/qualitative science can result in a better understanding of phenomena hardly investigable mathematically. And from this enhanced understanding can come an improved mathematical model that can answer the questions not satisfied by your qualitative model.

Of course I would prefer a mathematical subject if an (almost) equivalent one existed and I bet you too.

Answer 3

I would suggest looking into Bayesian Statistics resources, they tend to be more suited for people with a mathematical background.

The reason we usually hear less about Bayesian statistics, is not because of its capabilities, every concept in classical/frequentist statistics has an analogue in Bayesian statistics.

Bayesian statistics requires a certain level of mathematical understanding and formalism, and thus is being taught less outside the science and technology faculties.

If you are looking for a great freebie intro book, my favorite is: Think Bayes

And for the more math-savvy reader I would recommend:

• Bayesian Data Analysis by Gelman, et al

• Sivia and Skilling, Data analysis: a Bayesian tutorial (2ed) 2006 246p 0198568320

• Bolstad, Introduction to Bayesian Statistics.