  In a search for the signature of turbulence in the diffuse interstellar
medium in gas density distributions, we determined the probability distribution
functions (PDFs) of the average volume densities of the diffuse gas. The
densities were derived from dispersion measures and HI column densities towards
pulsars and stars at known distances. The PDFs of the average densities of the
diffuse ionized gas (DIG) and the diffuse atomic gas are close to lognormal,
especially when lines of sight at |b|<5 degrees and |b|>=5 degrees are
considered separately. The PDF of <n_HI> at high |b| is twice as wide as that
at low |b|. The width of the PDF of the DIG is about 30 per cent smaller than
that of the warm HI at the same latitudes. The results reported here provide
strong support for the existence of a lognormal density PDF in the diffuse ISM,
consistent with a turbulent origin of density structure in the diffuse gas.
