  We present an analysis of the distribution of H-alpha emission measures for
the warm ionized medium (WIM) of the Galaxy using data from the Wisconsin
H-Alpha Mapper (WHAM) Northern Sky Survey. Our sample is restricted to Galactic
latitudes |b| > 10. We removed sightlines intersecting nineteen high-latititude
classical H II regions, leaving only sightlines that sample the diffuse WIM.
The distribution of EM sin |b| for the full sample is poorly characterized by a
single normal distribution, but is extraordinarily well fit by a lognormal
distribution, with <log EM sin |b|> = 0.146 +/- 0.001 and standard deviation
0.190 +/- 0.001. <log EM sin |b|> drops from 0.260 +/- 0.002 at Galactic
latitude 10<|b|<30 to 0.038 +/- 0.002 at Galactic latitude 60<|b|<90. The
distribution may widen slightly at low Galactic latitude. We compare the
observed EM distribution function to the predictions of three-dimensional
magnetohydrodynamic simulations of isothermal turbulence within a
non-stratified interstellar medium. We find that the distribution of EM sin |b|
is well described by models of mildy supersonic turbulence with a sonic Mach
number of ~1.4-2.4. The distribution is weakly sensitive to the magnetic field
strength. The model also successfully predicts the distribution of dispersion
measures of pulsars and H-alpha line profiles. In the best fitting model, the
turbulent WIM occupies a vertical path length of 400-500 pc within the 1.0-1.8
kpc scale height of the layer. The WIM gas has a lognormal distribution of
densities with a most probable electron density n_{pk} = 0.03 cm^{-3}. We also
discuss the implications of these results for interpreting the filling factor,
the power requirement, and the magnetic field of the WIM.
