Face image quality assessment (FIQA) attempts to improve face recognition (FR) performance by providing additional information about sample quality. Because FIQA methods attempt to estimate the utility of a sample for face recognition, it is reasonable to assume that these methods are heavily influenced by the underlying face recognition system. Although modern face recognition systems are known to perform well, several studies have found that such systems often exhibit problems with demographic bias. It is therefore likely that such problems are also present with FIQA techniques. To investigate the demographic biases associated with FIQA approaches, this paper presents a comprehensive study involving a variety of quality assessment methods (general-purpose image quality assessment, supervised face quality assessment, and unsupervised face quality assessment methods) and three diverse state-of-theart FR models. Our analysis on the Balanced Faces in the Wild (BFW) dataset shows that all techniques considered are affected more by variations in race than sex. While the general-purpose image quality assessment methods appear to be less biased with respect to the two demographic factors considered, the supervised and unsupervised face image quality assessment methods both show strong bias with a tendency to favor white individuals (of either sex). In addition, we found that methods that are less racially biased perform worse overall. This suggests that the observed bias in FIQA methods is to a significant extent related to the underlying face recognition system.