  A quantum transition can be seen as a result of interference between various
pathways(e.g. Feynman paths) which can be labelled by a variable $f$. An
attempt to determine the value of f without destroying the coherence between
the pathways produces a weak value of $\bar{f}$. We show $\bar{f}$ to be an
average obtained with amplitude distribution which can, in general, take
negative values which, in accordance with the uncertainty principle, need not
contain information about the actual range of the values $f$ which contribute
to the transition. It is also demonstrated that the moments of such alternating
distributions have a number of unusual properties which may lead to
misinterpretation of the weak measurement results.We provide a detailed
analysis of weak measurements with and without post-selection. Examples include
the double slit diffraction experiment,weak von Neumann and von Neumann-like
measurements, traversal time for an elastic collision, the phase time, the
local angular momentum(LAM) and the 'three-box case' of {\it Aharonov et al}
