Repeating my post from the last open thread, for better visibility:

I want to study probability and statistics in a deeper way than the Probability and Statistics course I had to take in the university. The problem is, my mathematical education isn't very good (on the level of Calculus 101). I'm not afraid of math, but so far all the books I could find are either about pure application, with barely any explanations, or they start with a lot of assumptions about my knowledge and introduce reams of unfamiliar notation.

I want a deeper understanding of the basi... (read more)

I don't think that's really what means are. That intuition might fit the median better. One reason means are nice is that they have really nice properties, e.g. they're linear under addition of random variables. That makes them particularly easy to compute with and/or prove theorems about. Another reason means are nice is related to betting and the interpretation of a mean as an expected value; the theorem justifying this interpretation is the law of large numbers.

Nevertheless in many situations the mean of a random variable is a very bad description of it... (read more)

3pragmatist6yAs a first step, I suggest Dennis Lindley's Understanding Uncertainty. It's
written for the layperson, so there's not much in the way of mathematical
detail, but it is very good for clarifying the basic concepts, and covers some
surprisingly sophisticated topics.
ETA: Ah, I didn't notice that Benito had already recommended this book. Well,
consider this a second opinion then.

3buybuydandavis6yRead Edwin Jaynes.
The problem with most Probability and Statistics courses is the axiomatic
approach. Purely formalism. Here are the rules - you can play by them if you
want to.
Jaynes was such a revelation for me, because he starts with something you want,
not arbitrary rules and conventions. He builds probability theory on basic
desiredata of reason that you that make sense. He had reasons for my "whys?".
Also, standard statistics classes always seemed a bit perverse to me - logically
backward. They always just felt wrong. Jaynes approach replaced that tortured
backward thinking with clear, straight lines going forward. You're always asking
the same basic question "What is the probability of A given that I know B?"
And he also had the best notation. Even if I'm not going to do any math, I'll
often formulate a problem using his notation to clarify my thinking.

Repeating my post from the last open thread, for better visibility:

I want to study probability and statistics in a deeper way than the Probability and Statistics course I had to take in the university. The problem is, my mathematical education isn't very good (on the level of Calculus 101). I'm not afraid of math, but so far all the books I could find are either about pure application, with barely any explanations, or they start with a lot of assumptions about my knowledge and introduce reams of unfamiliar notation.

I want a deeper understanding of the basi... (read more)

I don't think that's really what means are. That intuition might fit the median better. One reason means are nice is that they have really nice properties, e.g. they're linear under addition of random variables. That makes them particularly easy to compute with and/or prove theorems about. Another reason means are nice is related to betting and the interpretation of a mean as an expected value; the theorem justifying this interpretation is the law of large numbers.

Nevertheless in many situations the mean of a random variable is a very bad description of it... (read more)