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.
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