The use of good-quality data to inform decision making is entirely dependent on robust processes to ensure it is fit for purpose. Such processes vary between organisations, and between those tasked with designing and following them. In this paper we report on a survey of 53 data analysts from many industry sectors, 24 of whom also participated in in-depth interviews, about computational and visual methods for characterizing data and investigating data quality. The paper makes contributions in two key areas. The first is to data science fundamentals, because our lists of data profiling tasks and visualization techniques are more comprehensive than those published elsewhere. The second concerns the application question "what does good profiling look like to those who routinely perform it?", which we answer by highlighting the diversity of profiling tasks, unusual practice and exemplars of visualization, and recommendations about formalizing processes and creating rulebooks.