Analyses of income inequality are often applied to the case of measuring total income inequality in a given population, but often it is more interesting to make research into how income inequality can be related to income differences between population subgroups or how income inequality can be related to the distribution of different income sources. The first category includes the cases, where total inequality can be expressed as the sum of a within-group inequality term and a between-group term, where the within-group contribution is itself a weighted sum of the sub-group inequality values. The second category includes the cases, where the contributions of separate income sources to total income inequality are examined. Furthermore combinations of the two categories can be done. Many empirical studies have decomposed total income inequality by population subgroups or by income source, without being critical of which inequality measures that are able to be used -satisfactorily - in a decomposition. Therefore we begin in section 2 with a number of general principles that we might wish to be satisfied by any decomposition by population subgroups. Section 3 proposes some general principles that we might prefer to be satisfied by any decomposition by income sources. Section 4 gives a short overview of the data used in the following empirical studies. The empirical results of this paper are calculated on the basis of Luxembourg Income Study (LIS). Section 5 and section 6 present the empirical work of this paper. Finally section 7 summarizes the main findings of this study.