Type I error and power of the standard independent samples t-test were compared with the trimmed and Winsorized t-test with respect to continuous distributions and various discrete distributions known to occur in applied data. The continuous and discrete distributions were generated with similar levels of skew and kurtosis but the discrete distributions had a variety of structural features not reflected in the continuous distributions. The results showed that the Type I error rates of the t-tests were not seriously affected, but the power rate of the trimmed and Winsorized t-test varied greatly across the considered distributions.