Recently many efforts have been made to incorporate persistence diagrams, one of the major tools in topological data analysis (TDA), into machine learning pipelines. To better understand the power and limitation of persistence diagrams, we carry out a range of experiments on both graph data and shape data, aiming to decouple and inspect the effects of different factors involved. To this end, we also propose the so-called \emph{permutation test} for persistence diagrams to delineate critical values and pairings of critical values. For graph classification tasks, we note that while persistence pairing yields consistent improvement over various benchmark datasets, it appears that for various filtration functions tested, most discriminative power comes from critical values. For shape segmentation and classification, however, we note that persistence pairing shows significant power on most of the benchmark datasets, and improves over both summaries based on merely critical values, and those based on permutation tests. Our results help provide insights on when persistence diagram based summaries could be more suitable.