I don't know what you can expect. You might be interested in work of Cynthia Dwork on the concept of differential privacy. The idea concerns what information can be revealed by queries to a database in spite of attemts to anonymize or average the data or limit the scope of the queries. It seems some valid statistical analyses can be done even if the answerer throws in a random but intentional error to help protect privacy.
The article at jeremykun.com mentioned in Samuel's comment also mentions Dwork. The other answers should give you enough encouragement to check out the research.