We consider the task of computing shape statistics and classification of 3D anatomical structures (as continuous, parameterized surfaces). This requires a Riemannian metric that allows re-parameterizations of surfaces by isometries, and computations of geodesics. This allows computing Karcher means and covariances of surfaces, which involves optimal re-parameterizations of surfaces and results in a superior alignment of geometric features across surfaces. The resulting means and covariances are better representatives of the original data and lead to parsimonious shape models. These two moments specify a normal probability model on shape classes, which are used for classifying test shapes into control and disease groups. We demonstrate the success of this model through improved random sampling and a higher classification performance. We study brain structures and present classification results for Attention Deficit Hyperactivity Disorder. Using the mean and covariance structure of the data, we are able to attain an 88% classification rate.