  The variance of an observable in a pre-selected quantum system, which is
always real and non-negative, appears as an increase in the probe wave packet
width in indirect measurements. Extending this framework to pre- and
post-selected systems, we formulate a complex-valued counterpart of the
variance called "weak variance." In our formulation, the real and imaginary
parts of the weak variance appear as changes in the probe wave packet width in
the vertical-horizontal and diagonal-antidiagonal directions, respectively, on
the quadrature phase plane. Using an optical system, we experimentally
demonstrate these changes in the probe wave packet width caused by the real
negative and purely imaginary weak variances. Furthermore, we show that the
weak variance can be expressed as the variance of the weak-valued probability
distribution in pre- and post-selected systems. These operational and
statistical interpretations support the rationality of formulating the weak
variance as a complex counterpart of the variance in pre- and post-selected
systems.
