Does one usually measure many particles and obtain a frequency distribution for either position or velocity? Or does one just measure one single particle?
Particle detectors are set up to measure individual particles, tracks (where it passed) and momentum using magnetic fields.
Here is an example:
This is a single pi-proton scattering event in a bubble chamber with a magnetic field perpendicular to the picture. It has four tracks, each is measured. To get an interaction crossection one would have to accumulate four track events and get the distributions and the frequency of appearance.
How does "the probability distribution" phycisists sometimes talk about enter in?
Is it a prediction model for the experimental result?
Quantum mechanical models predict probability distributions. Accumulation of measurements are fitted to these distributions, and the fit checks the accuracy of the models.
Is it related to the frequency distribution for the measurements (if there are many) that is processed further to obtain more realistic results (realistic in an uncertainty sense)?
Accumulating statistics reduces statistical errors. The quantum mechanical probabilities are a different , physics mathematical model.
Maybe this double slit experiment with single photons at a time will help:
single-photon camera recording of photons from a double slit illuminated by very weak laser light. Left to right: single frame, superposition of 200, 1’000, and 500’000 frames.
The predicted probability distribution is for "single photon impinging on double slits" with given boundary conditions. The probability distribution is predicted to show interference, because it is a solution of a quantum mechanical wave equation.
Single photons are measured at a specific (x,y) on the left frame, little dots seemingly random. The accumulation slowly builds up the predicted interference, increase of statistics making it more accurate, and on the right the full statistics gives the probability distribution with good accuracy.