  We show that two chosen ensembles of spin states, which are differently
prepared but are described by the same density matrix in quantum mechanics, do
not fully share the same measurable characteristics. One characteristic on
which they differ is shown to be the variance of the spin along a given
direction. We conclude that the statistical description of an ensemble of
states as given by its density matrix, although sufficient in many cases,
should be considered incomplete, as it does not fully describe the measurable
characteristics of the ensemble. A discussion a posteriori on the problem is
provided.
