Concordance indices are used to assess the degree of agreement between different methods that measure the same characteristic. In this regard, the total deviation index (TDI) is a widely used measure of agreement whose main advantage is that it results in the same measurement scale as that of the variable of interest, allowing a straightforward interpretation. First of all, the present work introduces the three already established methods for estimation and inference about the TDI. These methodologies assume that data are normally distributed and also linearity between the response and the effects. Then, as an alternative for assessing agreement in those situations in which these assumptions are not fulfilled, we propose here a new non-parametric approach that can deal with any kind of data. It must be pointed out that such approach has not been considered in any previous work. In the present study, the main goals are evaluating the performance of the established methodologies in contexts of normal and non-normal data and also comparing them with our non-parametric proposal. We apply the four methods to three data sets that represent contexts of very different natures: normal data, continuous extreme data and count data. The results obtained when assessing agreement in these data sets are discussed. Moreover, a simulation study is carried out in scenarios that emulate the aforesaid contexts. We conclude that when the assumptions of the parametric techniques hold, the non-parametric approach lacks efficiency compared to the established methods. However, when evaluating agreement in non-normal data, our non-parametric proposal is shown to have better properties and to give more reliable estimates than the already established methodologies.