We present a method for shape matching that approximates group-wise shape registration by reducing the problem to a 1-D registration. First, a novel medial curve method is proposed for computing a 1-D description of the shape. Second, a group-wise registration of the geometric descriptor is performed by directly minimizing the group variance. The resulting registration is used to adjust a global shape feature used to compute the final correspondence. Thus, the problems of description, registration, and statistical analysis are solved in one framework, while reducing the computational problem of group-wise registration of shapes substantially. We validate our method on 620 lateral ventricles extracted from the ADNI MRI dataset and 19 lateral ventricles from patients with HIV/AIDS and matched controls, also scanned with MRI. We show that our group-wise approach leads to improved statistical results, and also compare it to the SPHARM method.