Allometric growth is found in many tagging systems online. That is, the number of new tags (T) is a power law function of the active population (P), or T P^gamma (gamma!=1). According to previous studies, it is the heterogeneity in individual tagging behavior that gives rise to allometric growth. These studies consider the power-law distribution model with an exponent beta, regarding 1/beta as an index for heterogeneity. However, they did not discuss whether power-law is the only distribution that leads to allometric growth, or equivalently, whether the positive correlation between heterogeneity and allometric growth holds in systems of distributions other than power-law. In this paper, the authors systematically examine the growth pattern of systems of six different distributions, and find that both power-law distribution and log-normal distribution lead to allometric growth. Furthermore, by introducing Shannon entropy as an indicator for heterogeneity instead of 1/beta, the authors confirm that the positive relationship between heterogeneity and allometric growth exists in both cases of power-law and log-normal distributions.