What I'm trying to figure out is, how to I determine whether a source I'm looking at is telling the truth? For an example, let's take this page from Metamed: http://www.metamed.com/vital-facts-and-statistics
At first glance, I see some obvious things I ought to consider. It often gives numbers for how many die in hospitals/year, but for my purposes I ought to interpret it in light of how many hospitals are in the US, as well as how many patients are in each hospital. I also notice that as they are trying to promote their site, they probably selected the data that would best serve that purpose.
So where do I go from here? Evaluating each source they reference seems like a waste of time. I do not think it would be wrong to trust that they are not actively lying to me. But how do I move from here to an accurate picture of general doctor competence?
No, I would be surprised, but that is due to my background knowledge that a family unit implies all sorts of mutual correlations, ranging from growing up (if one's parents are Republicans, one is almost surely a Republican as well) to location (most states are not equally split ideologically), and worries about biases and manipulations and selection effects ("This Iowa district voted for the winning candidate in the last 7 elections!").
On the other hand, if you simply told me that 9 random people split 5-4 for Obama, I would simply shrug and say, "Well, yeah. Obama had the majority, and in a sample of 9 people, a 5-4 split for him is literally the single most likely outcome possible - every other split like 9-0 is further removed from the true underlying probability that ~52% of people voted for him. It's not all that likely, but you could say that about every lottery winner or every single sequence you get when flipping a fair coin n times: each possible winner had just a one in millions chance of winning, or each sequence had a 0.5^n chance of happening. But, something had to happen, someone had to win the lottery, some sequence had to be produced by the final coin flip."