  This work is concerned with finite range bounds on the variance of individual
eigenvalues of Wigner random matrices, in the bulk and at the edge of the
spectrum, as well as for some intermediate eigenvalues. Relying on the GUE
example, which needs to be investigated first, the main bounds are extended to
families of Hermitian Wigner matrices by means of the Tao and Vu Four Moment
Theorem and recent localization results by Erd\"os, Yau and Yin. The case of
real Wigner matrices is obtained from interlacing formulas. As an application,
bounds on the expected 2-Wasserstein distance between the empirical spectral
measure and the semicircle law are derived. Similar results are available for
random covariance matrices.
