And I'd be quite happy to use a test which takes this information into account if I knew a suitable test.

I also wanted to mention (but it got removed in the editing), that I definitely can't make a normality assumption. Some cases look close to log-normal, though I haven't done tests for it, and some are already bimodal.

1. Yes, it would only be applied on new data (and on all new data). 2. They definitely are different. I didn't see the requirement for identical distributions there, but I was thinking that the resulting population could be considered as a sample from a multimodal distribution where we take a random page first and then a measurement from that page. But this doesn't really work, you are correct. Maybe it could be rescued by actually doing "take a random page and take one of its samples at random", but probably not. Thank you!