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I have a good handle on probability theory, at least I understand the book I 1st course in probablity by Ross.

However, I am having a hard time understanding Statistics, I got a prep book to teach me but it seems to be just logical deductions are given as the reason for why $\frac{\sigma}{\sqrt{n}}$ for the st.dev of a sample for a random sample.

I am asking for any good books online that would give some more mathematical reason why statistics work.

My level: I read and mostly understood the book listed above. I am studying for the AP Statistics exam.

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There are many challenges for most learning statistics (which I think you rightly have noticed is not like learning probability or other math.)

Expecting there are mathematical reasons why statistics work is one misperception to get over. There are logical reasons why it works (in hard to fully grasp ways) that can be displayed in mathematics but also in physics. In fact one of the earliest to grasp the logic of Bayesian analysis - Francis Galton - did that by building and running a physical machine a two stage Quincunx.

Stigler, Stephen M. 2010. Darwin, Galton and the statistical enlightenment. Journal of the Royal Statistical Society: Series A 173(3):469-482.

I am interested in working this up, for high school and even grade school students - but not there yet. One thing I am worried about is that an ability to be very abstract is likely a requirement - but you seemed to have that. Have a good look at Fig 5 in the reference above and think about it carefully. I have made an animation of that in a PDF here. You need to play with the forward and rewind buttons to play it.

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