  We present the results of the $\gamma$-ray flux distribution study on the
brightest blazars which are observed by the \emph{Fermi}-LAT. We selected 50
brightest blazars based on the maximum number of detection reported in the LAT
third AGN catalog. We performed standard unbinned maximum likelihood analysis
on the LAT data during the period between August 2008 and December 2016, in
order to obtain the average monthly flux. After quality cuts, blazars for which
at least 90\% of the total flux was survived were selected for the further
study, and this includes 19 FSRQs and 19 BL Lacs. The Anderson-Darling and
$\chi^2$ tests suggest that the integrated monthly flux follow a log-normal
distribution for all sources, except for three FSRQs for which neither a normal
nor a log-normal distribution was preferred. A double log-normal flux
distribution tendency were observed in these sources, though it has to be
confirmed with improved statistics. We also found that, the standard deviation
of the log-normal flux distribution increases with the mean spectral index of
the blazar, and can be fitted with a line of slope 0.24$\pm$0.04. We repeat our
study on three additional brightest unclassified blazars to identify their flux
distribution properties. Based on the features of their log-normal flux
distribution, we infer these unclassified blazars may be closely associated
with FSRQs. We also highlight that considering the log-normal behavior of the
flux distribution of blazars, averaging their long term flux in linear scale
can largely under estimate the nominal flux and this discrepancy can propagate
down to the estimation of source parameters through spectral modeling.
