  Weak values are average quantities,therefore investigating their associated
variance is crucial in understanding their place in quantum mechanics. We
develop the concept of a position-postselected weak variance of momentum as
cohesively as possible, building primarily on material from Moyal (Mathematical
Proceedings of the Cambridge Philosophical Society, Cambridge University Press,
Cambridge, 1949) and Sonego (Found Phys 21(10):1135, 1991) . The weak variance
is defined in terms of the Wigner function, using a standard construction from
probability theory. We show this corresponds to a measurable quantity, which is
not itself a weak value. It also leads naturally to a connection between the
imaginary part of the weak value of momentum and the quantum potential. We
study how the negativity of the Wigner function causes negative weak variances,
and the implications this has on a class of `subquantum' theories. We also
discuss the role of weak variances in studying determinism, deriving the
classical limit from a variational principle.
