The main objectives of the paper are to decompose Atkinson's measure into within-group and between-group inequality and to compare it with decompositions of inequality measures, like the Gini coefficient, variance of logarithms, Theil's entropy index and the square of the coefficient of variation. Atkinson's inequality index is given an interpretation of mean order ß and then it is decomposed for population subgroups. Various other measures are considered for decompositions and a variety of decomposition schemes for Gini coefficient and other measures in the literature is reviewed and commented upon. Empirical analysis using U.S.A. and U.K. data are performed on the basis of income distribution data classified by family size. For various decomposition schemes, contribution to between size-group and within size-group inequalities are compared in percentage terms. Comparisons are also made over time for the years 1964 and 1974 both for U.S.A. and U.K. and the result indicates that within size-group income inequality has declined while between size group and total income inequality shows an increase. Three important properties of Atkinson's inequality index are proved in the appendix.