The probability density functions (pdfs) of molecular line centroid velocity fluctuations, and of line centroid velocity fluctuation differences at different spatial lags, are estimated for several nearby molecular clouds with active internal star formation. The data consist of over 75,000 13CO line profiles divided among 12 spatially and/or kinematically distinct regions. These regions range in size from less than 1 to more than 40 pc and are all substantially supersonic, with centroid fluctuation Mach numbers ranging from about 1.5 to 7. The centroid pdfs are constructed using three different types of estimators. Although three regions (all in Mon R2) exhibit nearly Gaussian centroid pdfs, the other regions show strong evidence for non-Gaussian pdfs, often nearly exponential, with possible evidence for power-law contributions in the far tails. Evidence for nearly exponential centroid pdfs in the neutral H I component of the interstellar medium is also presented, based on older published data for optical absorption lines and H I emission and absorption lines. These strongly non-Gaussian pdfs disagree with the nearly Gaussian behavior found for incompressible turbulence (except possibly shear flow turbulence) and simulations of decaying mildly supersonic turbulence. Spatial images of the largest magnitude centroid velocity differences for the star-forming regions appear less filamentary than predicted by decay simulations dominated by vortical interactions. No evidence for the scaling of difference pdf kurtosis with Reynolds number, as found in incompressible turbulence experiments and simulations, is found. We conclude that turbulence in both star-forming molecular clouds and diffuse H I regions involves physical processes that are not adequately captured by incompressible turbulence or by mildly supersonic decay simulations. The variation with lag of the variance and kurtosis of the difference pdfs is presented as a constraint on future simulations, and we evaluate and discuss the implications of the large scale and Taylor scale Reynolds numbers for the regions studied here.
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