Being requested by name is too ego-boosting to pass up, so let me take a crack at clarifying at least one apparent confusion.
> For example, you can measure momentum, position, energy, spin etc... and each of these observables has a different corresponding mathematical operation that you perform on Ψ to get XΨ, where X is the mean value of the observable...
>The weird thing though is that what you measure isn't always exactly this value. Instead, the mean value of many measurements will be this value. You can also compute the standard deviation of these measurements using Ψ, but that's about it.
A good QM textbook will actually say something much more precise. It says that (a) the set of possible outcomes of the measurement is equal to the spectrum of the observable being measured and (b) the chance of getting a particular outcome is given by the squared inner product of the wavefunction with the appropriate eigenvalue. (The whole business of calculating means and standard deviations is confusing unless you understand that; unfortunately, this is allowed to happen often in into QM courses.) This means that QM doesn't just predict some statistical properties of the outcome distribution, it completely specifies the distribution.
Also, I figure you know this, but I want to mention that when you say
> you can plot and analyze this data, and what you'll notice for two entangled particles separated by thousands of miles or more is that there are statistical correlations between the two sets of data
it's important to emphasize that these are non-local correlations (in the Bell sense). You can generate mere local correlations using everyday classical systems.
> For example, you can measure momentum, position, energy, spin etc... and each of these observables has a different corresponding mathematical operation that you perform on Ψ to get XΨ, where X is the mean value of the observable...
>The weird thing though is that what you measure isn't always exactly this value. Instead, the mean value of many measurements will be this value. You can also compute the standard deviation of these measurements using Ψ, but that's about it.
A good QM textbook will actually say something much more precise. It says that (a) the set of possible outcomes of the measurement is equal to the spectrum of the observable being measured and (b) the chance of getting a particular outcome is given by the squared inner product of the wavefunction with the appropriate eigenvalue. (The whole business of calculating means and standard deviations is confusing unless you understand that; unfortunately, this is allowed to happen often in into QM courses.) This means that QM doesn't just predict some statistical properties of the outcome distribution, it completely specifies the distribution.
Also, I figure you know this, but I want to mention that when you say
> you can plot and analyze this data, and what you'll notice for two entangled particles separated by thousands of miles or more is that there are statistical correlations between the two sets of data
it's important to emphasize that these are non-local correlations (in the Bell sense). You can generate mere local correlations using everyday classical systems.