In this paper we reconsider the Dagum decomposition of the Gini index and we compare this decomposition with the decompositions proposed by Mookherjee and Shorrocks and by Lambert and Aronson. In so doing, a deeper insight into the meaning of the overlapping term is given and an alternative expression for this term is obtained. The results are applied to a sub-sample of the Polish CSO survey: the families taken into consideration are selected and grouped according to the number of their components. The aim is to analyze how the present Polish personal income tax modifies the overall income distribution and to evaluate the changes in the within and between groups income inequalities.