A marginal homogeneity model tests whether the row and column distributions of a square contingency table have the same sample margins. However, for variables with ordered categories, the marginal homogeneity model does not take into account the ordering information, leading to significant loss of power. Score-based tests have been proposed for ordinal variables. In this paper, we extend the idea of scores and propose a new method based on the standardized scores to test the marginal homogeneity for ordered categories with open ends. Our simulation studies demonstrate that our proposed scores is more powerful than the usual scores.