The derivation of the momentum/position Heisenberg Uncertainty Principle (HUP) is based on the statistical interpretation which says that if we have a lot of quantum systems in identical states, and measure the momentum in half of them and get a distribution with standard deviation σp, and measure the position of the other half and get a distribution σx, then σp×σx ≥ [itex]\hbar[/itex]/2. Fine. And obviously both σp and σx must be non-zero. But these are statements about a collection of particles. Purely from the point of view of statistics, a non-zero standard deviation in each of two distributions does not prevent one element from each set from being equal to the respective expected value of its distribution at the same time. Nonetheless, a commonly stated corollary is that a single particle cannot have both a determined position and a determined momentum simultaneously. This then transforms the HUP (a) into a statement about a single particle, thereby (b) giving the standard deviation an ontological meaning. The only explanations that I have seen for this corollary are: (1) confounding the HUP with the observer effect: hence not a corollary (2) pointing out that there are macroscopic quantum effects, which may be a reason to look for some form of (a), but it does not justify the collapse from a statement about a collection of particles to the same statement about a single particle, and thereby does not justify (b). (3) hand-waving. Can anyone give me something better? Thanks.