Reddit mentions of Principles of Statistics (Dover Books on Mathematics)

You are in a very special position right now where many interesing fields of mathematics are suddenly accessible to you. There are many directions you could head. If your experience is limited to calculus, some of these may look very strange indeed, and perhaps that is enticing. That was certainly the case for me.

Here are a few subject areas in which you may be interested. I'll link you to Dover books on the topics, which are always cheap and generally good.

• The Nature and Power of Mathematics, Donald M. Davis. This book seems to be a survey of some history of mathematics and various modern topics. Check out the table of contents to get an idea. You'll notice a few of the subjects in the list below. It seems like this would be a good buy if you want to taste a few different subjects to see what pleases your palate.
  
  
• Introduction to Graph Theory, Richard J. Trudeau. Check out the Wikipedia entry on graph theory and the one defining graphs to get an idea what the field is about and some history. The reviews on Amazon for this book lead me to believe it would be a perfect match for an interested high school student.
  
  
• Game Theory: A Nontechnical Introduction, Morton D. Davis. Game theory is a very interesting field with broad applications--check out the wiki. This book seems to be written at a level where you would find it very accessible. The actual field uses some heavy math but this seems to give a good introduction.
  
  
• An Introduction to Information Theory, John R. Pierce. This is a light-on-the-maths introduction to a relatively young field of mathematics/computer science which concerns itself with the problems of storing and communicating data. Check out the wiki for some background.
  
  
• Lady Luck: The Theory of Probability, Warren Weaver. This book seems to be a good introduction to probability and covers a lot of important ideas, especially in the later chapters. Seems to be a good match to a high school level.
  
  
• Elementary Number Theory, Underwood Dudley. Number theory is a rich field concerned with properties of numbers. Check out its Wikipedia entry. I own this book and am reading through it like a novel--I love it! The exposition is so clear and thorough you'd think you were sitting in a lecture with a great professor, and the exercises are incredible. The author asks questions in such a way that, after answering them, you can't help but generalize your answers to larger problems. This book really teaches you to think mathematically.
  
  
• A Book of Abstract Algebra, Charles C. Pinter. Abstract algebra formalizes and generalizes the basic rules you know about algebra: commutativity, associativity, inverses of numbers, the distributive law, etc. It turns out that considering these concepts from an abstract standpoint leads to complex structures with very interesting properties. The field is HUGE and seems to bleed into every other field of mathematics in one way or another, revealing its power. I also own this book and it is similarly awesome. The exposition sets you up to expect the definitions before they are given, so the material really does proceed naturally.
  
  
• Introduction to Analysis, Maxwell Rosenlicht. Analysis is essentially the foundations and expansion of calculus. It is an amazing subject which no math student should ignore. Its study generally requires a great deal of time and effort; some students would benefit more from a guided class than from self-study.
  
  
• Principles of Statistics, M. G. Bulmer. In a few words, statistics is the marriage between probability and analysis (calculus). The wiki article explains the context and interpretation of the subject but doesn't seem to give much information on what the math involved is like. This book seems like it would be best read after you are familiar with probability, say from Weaver's book linked above.
  
  
• I have to second sellphone's recommendation of Naive Set Theory by Paul Halmos. It's one of my favorite math books and gives an amazing introduction to the field. It's short and to the point--almost a haiku on the subject.
  
  
• Continued Fractions, A. Ya. Khinchin. Take a look at the wiki for continued fractions. The book is definitely terse at times but it is rewarding; Khinchin is a master of the subject. One review states that, "although the book is rich with insight and information, Khinchin stays one nautical mile ahead of the reader at all times." Another review recommends Carl D. Olds' book on the subject as a better introduction.
  
  Basically, don't limit yourself to the track you see before you. Explore and enjoy.

My intro/grad class used Bulmer's book. Its an enjoyable read, easy to follow and answers to odd exercises in the back... and a hell of alot cheaper than current textbooks.

If you want theoretical / mathematical I would suggest reading a few math, stats or engineering books.

Dover is a great place to find some cheaper reading material. They republish old scientific and math texts that were popular in their time in a smaller sized paperback. They're a nice size to bring around with you and they don't cost much.

Math and stats findings of today build on this knowledge, and much of it is still used in state-of-the-art applications. Or, that math/stats is used as part of some state-of-the-art algorithm. Lots of the newest ML algorithms are blending math from a variety of areas.

Statistical analysis of experimental data

Principals of Statistics

Information Theory

Statistics Manual

Some theory of sampling

Numerical Methods for Scientists and Engineers (Hamming)

Mathematical Handbook for Scientists Engineers

Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables

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There is also the Data-Science Humble Bundle for more technical / practical skill building.

I use [this book]http://www.amazon.com/Statistics-Manual-Edwin-L-Crow/dp/048660599X) as a reference. It's very small and inexpensive (you may have to buy it direct from Dover, though). It won't go through any derivations, but it covers most of the very important, basic, topics. I also have my old textbook on backup as well. Old editions of textbooks are cheaper, and all the information is the same.

Looking through amazon, this one looks pretty good as well, especially for an inexpensive text. A reviewer said it ends with what we just talked about! Any book that lays down the foundations well enough should be fine. See if you can find one that does correlation analysis, since you'll probably use that later.

Getting more advanced than this may depend on your field. If you're in biology, I'd recommend a book on designing and analyzing scientific experiments. I can't recommend a good title, though, because I'm only familiar with computer experiments (which tend to be easier).

I hope this helps!

Principles of Statistics (Bulmer) - this is a very nice introduction to probability and statistics. It takes you through the important distributions (binomial, normal, poisson, etc), laws of probability, central limit theorem, etc. And it's like $10 as an eBook or $15 in paperback.

http://www.amazon.com/Principles-Statistics-Dover-Books-Mathematics/dp/0486637603/ref=sr_1_5?ie=UTF8&qid=1463424228&sr=8-5&keywords=statistics