DNA barcoding can identify biological species and provides an important tool in diverse applications, such as conserving species and identifying pathogens, among many others. If combined with statistical tests, DNA barcoding can focus taxonomic scrutiny onto anomalous species identifications based on morphological features. Accordingly, we put nonparametric tests into a taxonomic context to answer questions about our sequence dataset of the formal fungal barcode, the nuclear ribosomal internal transcribed spacer (ITS). For example, does DNA barcoding concur with annotated species identifications significantly better if expert taxonomists produced the annotations? Does species assignment improve significantly if sequences are restricted to lengths greater than 500 bp? Both questions require a figure of merit to measure of the accuracy of species identification, typically provided by the probability of correct identification (PCI). Many articles on DNA barcoding use variants of PCI to measure the accuracy of species identification, but do not provide the variants with names, and the absence of explicit names hinders the recognition that the different variants are not comparable from study to study. We provide four variant PCIs with a name and show that for fixed data they follow systematic inequalities. Despite custom, therefore, their comparison is at a minimum problematic. Some popular PCI variants are particularly vulnerable to errors in species annotation, insensitive to improvements in a barcoding pipeline, and unable to predict identification accuracy as a database grows, making them unsuitable for many purposes. Generally, the Fractional PCI has the best properties as a figure of merit for species identification. The fungal genus Ramaria provides unusual taxonomic difficulties. As a case study, it shows that a good taxonomic background can be combined with the pertinent summary statistics of molecular results to improve the identification of doubtful samples, linking both disciplines synergistically.