In previous work, we proposed a technique for preserving the privacy of quasi-identifiers in sensitive data when visualized using parallel coordinates. This paper builds on that work by introducing a number of metrics that can be used to assess both the level of privacy and the amount of utility that can be gained from the resulting visualizations. We also generalize our approach beyond parallel coordinates to scatter plots and other visualization techniques. Privacy preservation generally entails a trade-off between privacy and utility: the more the data are protected, the less useful the visualization. Using a visually-oriented approach, we can provide a higher amount of utility than directly applying data anonymization techniques used in data mining. To demonstrate this, we use the visual uncertainty framework for systematically defining metrics based on cluster artifacts and information theoretic principles. In a case study, we demonstrate the effectiveness of our technique as compared to standard data-based clustering in the context of privacy-preserving visualization.