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So, you’re saying one needs an undergraduate knowledge of probability in order to study statistics?

He is actually right. When you originally posted: "I have a math background", I thought that included probability theory. If not, you have to gain that knowledge before reading James et. al. If reading the book that @ondansetron suggested, focus on chapters 1-10. Chapters 11- 13 are important but can be skipped during the first reading.

The probability book I recommended isn't super rigorous, from a mathematical treatment, but has a great blend of logical thinking/problem solving with mathematical components as needed. It's not an easy course but it isn't the hardest/most dry.

Most of the books I'm looking at usually start a problem with "assume this data is normally distributed" or pick a distribution. I guess I'd just like to learn, at introductory level, how to get from data to distribution. Is that not possible? Do I really need to study at graduate level to do this? I know engineers and scientists use statistics without that rigorous of a background.

Thanks for all responses, but I'm not really up to self-studying Probability for 1+ year.

Most of the books I'm looking at usually start a problem with "assume this data is normally distributed" or pick a distribution. I guess I'd just like to learn, at introductory level, how to get from data to distribution. Is that not possible? Do I really need to study at graduate level to do this? I know engineers and scientists use statistics without that rigorous of a background.

Thanks for all responses, but I'm not really up to self-studying Probability for 1+ year.

I think with your math background you can handle the probability book given. Mathematics is challenging. Statistics and probability are also hard subjects. Despite what you may see done, there is no quick fix for statistics. I know many engineers and scientists who use statistics without much formal background but most of them don't do it well (speaking from personal experience).

Doing statistics poorly can have many bad ramifications (speaking from a biomedical perspective, it can literally be life and death with patient research). Economically, this can costs millions of dollars if decisions are based on poorly done statistics; the goal you're mentioning would be very achievable with the background I suggested.

This may sound over the top but too often statistics is perceived as some set of cookbook calculations to be performed with software, but that couldn't be further from the truth.

I can't offer better recommendations than I already have done.