In cases where a criminal defendant rejects the service of counsel and defends himself pro se, how often is he acquitted?
This paper studies the question, in a restricted context of federal cases (the main problem is getting data, but turns out that some federal court clerks have been obliging). From that database, between 1998-2011 1,156,460 felony cases were resolved. 0.2% of those cases were pro se representations with virtually all cases having professional representation ("data missing" is 10 times more frequent than pro se, and the only thing rarer was pro bono attorney at 0.03%).
As for outcomes, it is possible that pro se defendants had the charges dismissed at a higher rate (16.9% pro se vs. 6.1% represented -- "possible" is a way of simplifying the statistical problems in interpreting these numbers); with acquittals it's a wash, and with convictions pro se defendants may have done worse (14.8% pro se, 4.9% represented). There also seems to be a trend that juries convict pro se defendants more often than they do represented defendants (90% vs. 77.9%).
Ah, and this paper follows up and contradicts the findings of the Hashimoto study, linked in the comments.