This paper focuses on the Gini index and presents two additional features which may be characterized as new indices of location and skewness. In section 2, a fast computation algorithm of the Gini index is shown. This algorithm demonstrates that Gini's index belongs to the class of relative mean deviations. In section 3 a ``G-Mean'' is defined as \(y_ G=(y_{GR}+y_{GL})\), where \(y_{GR}\) and \(y_{GL}\) are two income levels which may be computed as partial sums of \(\sum_{i}k_ iy_ i\) for \(i\leq (n/2)\) and \(i>(n/2)\), respectively, and \[ k_ i=| (m- 2i+1)\sum^{n/2}_{i=1}(n-2i+1)|. \] The G-Mean is the income a in the following expression which minimizes \(\sum^{n}_{i=1}c_ i(y_ i-a)^ 2\), with \(c_ i\) specified in Proposition 2. Besides the mean value property, \(y_ G\) takes into account the ranking of the incomes. Thus, the G-Mean is a combination of median and arithmetic mean. In section 4, with \(y_ G\) and Gini's mean difference a relative asymmetry index \(A^ R=2(y_ G-y)/\Delta\) is defined which is translation invariant and homogeneous of degree zero. This index is compared with Pearson's asymmetry index \(A_ p\). \(A^ R\) is included in [-1,1], equals zero when the income distribution is symmetric with respect to the mean income, tends towards \(+1\) when all individuals but one have zero income (Proposition 3). In section 5, the relationship between Gini's inequality index \(I^ R_ G\) and \(A^ R\) is characterized. With symmetric income distributions we have \(I^ R_ G\leq\) (Proposition 4), and \(I^ R_ G>\) \(\Rightarrow\) \(A^ R>0\) (Proposition 5). This implies a necessary condition for the elasticity of social welfare.