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Recent theoretical proposals concerning non-Gaussian statistics of passive scalars in random velocity fields are tested experimentally, by measuring the probability distributions of fluctuating temperature in an oscillating grid flow across which a steady temperature gradient is maintained. Pronounced exponential tails occur at sufficiently high Reynolds number R, and predominantly Gaussian statistics at low R. When the extended tails are present for the passive scalar, the corresponding velocity power spectrum shows reasonable scaling, and the velocity distribution is not far from Gaussian. The present paper provides a more complete characterization of the flow field than an earlier brief report [Phys. Rev. Lett. 67, 3507 (1991)], and also contains a description of additional features, such as the skewness of the distributions. Finally, the effective or eddy diffusivity of both heat and a molecular impurity are measured and compared. [ABSTRACT FROM AUTHOR]