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A friend of mine asked whether I knew of such a book that offers both intuitive and formal explanations of how a wide range of statistical methods were originally derived. She wasn't looking for a cookbook with a host of statistical tests/techniques but a book that dives into the "common core" (for the lack of a better word) or principles behind these methods in a systematic manner.

As a non-statistician, I couldn't think of any but believe there must books or introductory texts like that. I'm more than interested to read them myself! (I also wish to get this book for my friend's birthday!)

I apologize if my question sounds naive on this forum! Any recommendations are greatly appreciated!

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  • $\begingroup$ stats.stackexchange.com/questions/29380/… the link for a potentially helpful thread on statistics books that explain concepts by using more images than equations $\endgroup$ – PsychometStats Nov 7 '19 at 16:34
  • $\begingroup$ @PsychometStats That's super helpful! Thanks for pointing me to that thread!! $\endgroup$ – Yuan Meng Nov 7 '19 at 19:21
  • $\begingroup$ @Yuan_Meng you're welcome $\endgroup$ – PsychometStats Nov 7 '19 at 19:30
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In many parts of the world, undergrads in statistics learn to derive common statistical procedures, so any standard text that has basic mathematical statistics - of which there are dozens of reasonably suitable choices - covers the basics of this. It requires little more than basic calculus and a little linear algebra to cover many of the tests discussed in a basic intro stats course.

A text like Wackerly, Mendenhall and Scheaffer, Mathematical Statistics with Applications has derivations for a number of basic tests (some might be partly covered in exercises). Pretty much any edition will do - you don't need a recent one, and there are any number of good alternatives at a similar level.

I'd probably start with something like that.

If you seek a little more depth (within a still-widely-used text), you might consider Casella and Berger, Statistical Inference. I don't claim either book is the best of its kind, just that they're widely used books.

I'd supplement those with a book on nonparametric statistics. Possibly something like Conover's Practical Nonparametric Statistics, which gives just about enough background to derive a number of popular tests, and to understand something of how permutation tests work.

What you'd need beyond that is a reasonable text on regression. I'll try to come back with a suggestion for that, but from my recollection Fox's book Applied Regression analysis and Generalized Linear Models was reasonable and if I recall correctly has some of the derivations.

If you want to look at the bootstrap, perhaps Davison and Hinkley, Bootstrap Methods and their Application.

What else you might look at (e.g. Bayesian stats, time series, survival analysis etc etc) depends on how broad a coverage you want.

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