Reddit mentions of Statistics

Check out Statistics by William Hays. It is not an intuitive or pleasant read, but it routinely works through proofs of the underlying models. The top amazon review sums it up well:
> This text is for the mathematically inclined-it is not a "how-to" book, rather a "why" book that thoroughly explains the theory behind a variety of popular inferential techniques

Regarding the n-1 correction (assuming you are talking about variance), you can see a full proof on wikipedia's Bessel correction entry. I personally have struggled to reach an intuitive understanding of this proof, which I think is what you are getting at. Best of luck with it.

I don't have an answer to your question on proportion samples.

If you are looking for something very calculus-based, this is the book I am familiar with that is most grounded in that. Though, you will need some serious probability knowledge, as well.

If you are looking for something somewhat less theoretical but still mathematical, I have to suggest my favorite. Statistics by William L. Hays is great. Look at the top couple of reviews on Amazon; they characterize it well. (And yes, the price is heavy for both books.... I think that is the cost of admission for such things. However, considering the comparable cost of much more vapid texts, it might be worth springing for it.)