So I've heard two different explanations of the uncertainty principle, both of which make sense on their own, but I'm having a hard time figuring out how they're connected. The first is that the uncertainty principle is really a statistical principle. Basically, we can measure the position of a particle any individual time however precisely our equipment allows. Same with the momentum. However, if we take multiple measurements, there will always be some variation in both data sets and the product of the standard deviations will always be greater than or equal to $h/2$.
The other explanation I've heard explains it in terms of properties of waves. Essentially, a particle is really just the superposition of many different waves in the relevant field. A period wave doesn't HAVE a well-defined position or momentum, so we have to add together many different waves of varying momentums in order to create enough destructive interference that most of the peaks cancel out and we get a wave-packet with a fairly well-defined position, though there will always be SOME variation. A similar process is required for momentum. This is the most intuitive explanation of the uncertainty principle I've heard, but how is it related to the statistical definition? I know it has something to do with the wave-function being a probability function (or, more specifically, its squared magnitude is the probability density function) but I'm not sure exactly what.
I think part of what's got me confused is that I don't understand how it's possible to add up a bunch of waves to (for lack of a better term) "generate" a particle with a better defined position/momentum. The math makes sense, but how does it match up to how we actually perform measurements? I don't need the details of how specific measurement devices work but more the concept of measurement in quantum mechanics and how it corresponds to the idea of taking the superposition of many waves to create a wave-packet and what that has to do with taking multiple measurements and having variation in the data.
I feel like I'm starting to actually get at least a bit of a grip on how quantum mechanics works (I've been fascinated by it for years), but I'm quite confused about this. Any help is greatly appreciated.