A simulation study by Monte-Carlo method was used to investigate the e ect of violating the assumption of bivariate normality on the Pearson correlation coe cient. We also examined the impacts of nonlinearity between a pair of variables when they are not independent, even though the population correlation is zero. The results of the study showed that there is a great impact when such violations of normality occur. It also showed the rate of Type I error can either be quite small or large in contrast to the assumed error rate. We made a recommendation that the t statistic be used with great caution or altogether avoided if there are good reasons to believe that a nonlinear relationship exists between the variables.