We have obtained the rotation measures (RMs) of 2642 quasars by using cross-identification of the most updated quasar catalogue and the RM catalogue. After discounting the foreground Galactic Faraday rotation of the Milky Way, we obtain the residual rotation measure (RRM) of these quasars. We have carefully discarded the effects from measurement and systematical uncertainties of RRMs as well as large RRMs from RM outliers, and we have obtained marginal evidence for the redshift evolution of the real dispersion of RRMs which steadily increases to 10rad/m^2^ about z~1 and is saturated at higher redshifts. Ionized clouds in the form of galaxy halos, galaxy clusters or cosmological filaments with different RM dispersion widths could produce the observed RRM evolutions. However, current data sets cannot constrain the contributions from galaxy halos and cosmic webs. Future measurements of RMs for a large sample of quasars with high precision are necessary to disentangle these different contributions.