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Background: I took multiple statistics classes in both high school and college, but nothing I learned ever stuck. The problem is, things like p-tests, the equations for chi-squared/normal distributions, even the standard deviation are always simply presented as fact, without any proof/justification/motivation for why this equation/method is the correct one.

Often, this is because the books are written for people looking to simply apply statistics rather than truly understand it. Usually, not even a calculus-background is assumed, despite the underlying equations being calculus-heavy.

Non-calculus example: Why is the standard deviation not defined as the average distance from the mean, when that is the more intuitively obvious definition? I still don't quite understand the answer to that one...

I did find some books that do go deeply into proofs in my college's mathematics library, but even those were heavy on symbols and light on justifications/motiviations (as well as real-world examples)

Does anyone know of any statistics books that not only go over the equations/methods, but explain in detail why they are what they are?

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wackerly et. al., Mathematical statistics. Very good! :)

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