In robust statistics, data depth functions are extremely powerful and can provide measures of central tendency beyond the ordinary means and medians. These functions provide a sense of depth to points in multivariate space, providing by default a center-outward ranking of observations, which is resistant to outliers and which can be applied to complex and high-dimensional data. Various data depth processes are considered to determine the most optimal location measure with real and simulated data. The performance of Mahalanobis Depth (MD), Half-space Depth (HSD), Zonoid Depth (ZD), Projection Depth (PD), and Spatial Depth (SPD) are compared on some health datasets including the Pima Indians Diabetes Dataset and the Wisconsin Breast Cancer (WBCD) Dataset. The results of these procedures are studied based on calculated depth values and error rates in the discriminant analysis. The findings suggest that the highest depth values are always exhibited by Spatial Depth (SPD), with better robustness and stability without losing accuracy, thus making it the best option. Nevertheless, Mahalanobis Depth (MD) also performs well, which is why it is highly applicable to the robust statistical modelling. Moreover, a new Generalized Mahalanobis Depth (GMD) has been proposed, based on robust location and scatter estimators to eliminate the weaknesses of classical MD. The GMD is more robust to contamination and is valid with singular or ill-conditioned covariance structures, and to high-dimensional data of relevance to real-world data, achieving lower misclassification rates.