In the context of black-box numerical codes, it is relevant to use sensitivity analysis in order to assess the influence of each random input X over the output Y. Goal-oriented sensitivity analysis states that one must first focus on a certain probability feature θ(Y) from the distribution of Y (such as its mean, quantile, or a probability of failure etc...), which would be chosen regarding a relevant strategy. The wish is to evaluate the impact of each input over θ(Y). In order to get supplementary information about sensitivity, we set that θ(Y) is the α-level quantile of Y , where α ∈]0, 1[. Throughout some examples, it has been pointed out that in some cases quantile-oriented sensitivity indices can detect some influence that Sobol indices would not. Mainly, the influence over each level of quantile displays how an input distribution entirely propagates through the output. We establish further results for the quantile-oriented indices properties in order to justify their relevancy. The main contribution of this paper comes when a statistical estimator for this index is introduced.