Is there any kind of data collected on the percentage of court cases won by court appointed lawyers?

I know of no studies that have been conducted regarding the "success rate" of the public defender system. I doubt that any have been done simply because it would be next to impossible to define "success." For example, if a defendant were charged with 3 crimes and the public defender manages to get 2 dismissed and the other reduced, is that a "success" even though the client was convicted of a crime?

There have been some studies done on how many cases a public defender should have to take in a year. The theory is that by decreasing the number of cases an attorney handles, the quality of representation will increase. I just did a quick search on the Washington Courts website for public defense (here's the link to the search results: http://www.courts.wa.gov/search/index.cfm?fa=search.start_search) and I think there are some documents there that discuss the investigations into public defender caseloads and the appointment process. Hopefully that will be of help to you!

I read a study a while back on the federal system that said it did not make a difference whether you had a public defender or not. But, it all comes down to the private attorneys you use. Some are better, a lot are worse, I have seen the worse a lot.